Probabilistic Behavior of Harmonic Functions

00.

Proof: It follows immediately from Jensen's formula that if B(a,r) ~ log If(a)1

~

IB( 1

a,r

)1 (

JB(a,r)

Then

IflP

n then

log Ifldxdy.

Multiply this equation by p, exponentiate each side, and apply Jensen's inequality to the integral in the resulting equation to obtain:

If(a)IP

~

IB( 1

a, r

)1 }{B(a,r) IflPdxdy.

Proof of Theorem 1.5.3: For p > 1, this is a consequence of what we've shown about harmonic functions: If f E HP, then Theorem 1.5.1 (or rather, its analogue for the disk) implies that f is the Poisson integral of a function 9 E V(aD) and that furthermore, f converges nontangentially to 9 a.e. and fr((}) = f(re iO ) converges in LP(aD) to g. Then Naf(e iO ) ~ AaMg(eiO) (the analogue of Theorem 1.4.7 for the disk) and thus, IINafilp ~ AallMgllp ~ Ca,pllgllp = Ca,pllfIIHv. Here for the last equality we have used the fact that fr ---7 9 in V(aD). We now assume that 0 < p < 00. Let 0 < r < 1 and set fr(z) = f(rz). Then fr is analytic on a neighborhood of D and so Ifr(z)l~ is subharmonic on this neighborhood. Let h(z) be the harmonic extension of Ifr(z)l~ to D. Then Ifr(z)l~ - h(z) is subharmonic on a neighborhood of D, identically zero on aD, so that by the maximum principle (Remarks 1.1.8), Ifr(z)l~ ~ h(z) for all zED. Then

IINafrll~ = IlNa(Jr~)II~ ~ IINahll~ ~ CalIMhll~ ~ Callhll~ = Callfrll~ ~ Callflli{p" As r

iI, Nafr(e iO ) i Naf(e ifJ ), so by Fatou's lemma, IINafilp ~ CallfllHv.

Corollary 1.5.5 Suppose f E HP, 1 integral of its nontangential limit.

~

p

~ 00,

on the disk. Then f is the Poisson

Proof: If 1 < p ~ 00 this follows immediately by applying the analogue for the disk of Theorem 1.5.1(a) to the real and imaginary parts of f. For the p = 1 case, the analogue for the disk of Theorem 1.5.1(b) implies f has a nontangentiallimit a.e., call it f(e ifJ ). Fix 0 < r < 1 and set fr(z) = f(rz). This is analytic on D, and thus,

24

1. Basic Ideas and Tools

That is,

By continuity, limrll f(rse i9 ) = f(se it ). Also, since f(re i9 ) ::; C aN a f(e i9 ) and the latter function is in £1, we may apply Lebesgue's dominated convergence theorem to conclude that

as desired. As the proof showed, the case p = I distinguishes analytic functions in HP from harmonic functions which satisfy a condition like that in the definition of HP. And clearly, in this p = I case, the key was Theorem 1.5.3. A slightly alternative approach to the proof of Theorem 1.5.3 is as follows: Let f E HP and let B denote the Blaschke product of the zeros of f· Then 9 = has no zeros in D and since IB(z)1 = I for z E aD then IlgllHP = IlfIIHP. In the interior of D, IBI ::; I so If I ::; Igl. Suppose we can show IlNagllp ::; CallgllHp. Then IINafilp ::; IINagilp ::; CallgllHP. In other words, we have reduced matters to showing the theorem for functions 9 E HP which have no zeros in D. But for such 9 we may consider g~; this is in H2, so we may apply the first part of the proof of Theorem 1.5.3, that is, the p > I case. This alternative approach requires a detailed knowledge of the zeros of an analytic function in HP on the unit disk. But the proof we gave essentially required the same: we used the subharmonicity of Ifl P which itself is a consequence of Jensen's formula. Jensen's formula is simply a statement about the distribution of zeros of an analytic function, and in fact, in most presentations of the theory of one complex variable, Jensen's formula is used to prove that the zeros of an HP function can be used to form a convergent Blaschke product. In any case, what is of decisive importance here is that the zeros of an analytic function are isolated. This is not true if we only assume that the functions we are considering are harmonic. In fact, if u is a real valued harmonic function defined on a domain 0, and if u(a) = 0, then the mean value property and continuity of u imply that u must vanish at least once on aB(a,r) whenever B(a,r) S;;; O. Consequently, the zeros of a real valued harmonic function can never be isolated. It is also not true that analytic functions of several complex variables have isolated zeros. For a number of years, these difficulties thwarted efforts to extend these ideas to higher dimensions. Finally, Stein and Weiss [SWI], [St2], found the correct viewpoint. In one dimension, the Cauchy-Riemann equations imply that the pair (v, u), consisting ofthe imaginary part and the real part of an analytic function F = u + iv, is locally the gradient of a harmonic function. In several variables, on lR~+1, the gradient of a harmonic function is an n + I-tuple of functions. Stein and Weiss found that certain powers of the norm of such an n+ I-tuple are subharmonic; this is exactly what is required

i

25

1.5 HP spaces on the upper half space

to take the proof of Theorem 1.5.3 and extend it to higher dimensions. We will now begin to discuss these ideas in detail. Definition 1.5.6 Let F(x,y) = (uo(x,y), ... ,un(x,y)), x E ~n,y > 0, be an n+ 1 tuple of C 1 functions defined on ~~+1. Write Xo = y. We say that F(x, y) satisfies the Cauchy-Riemann equations (or generalized or Stein-Weiss Cauchy-Riemann equations) if EJ~-o ~au: = 0 and ~aau. = ~au. for every j and k. Xj Xk XJ If we consider the matrix M = [~], j, k = 0, ... ,n, the definition simply says that M has zero trace and is symmetric. If n = 2, these are just the usual Cauchy-Riemann equations. Note that the conditions in the definition can be rewritten as div F = 0 and curl F = O. That curl F = 0 implies F is locally the gradient of a function, and div F = 0 implies this function is harmonic. Hence F is locally the gradient of a harmonic function. But all derivatives of a harmonic function are harmonic, in particular, the Uj are harmonic. 1

Definition 1.5.7 For F

=

(uo, . ..

,Un)

we set

IFI = (Ej=o IUj 12) "2.

For

0:

> 0 we

define NaF = NalFl. The key observation is: Theorem 1.5.8 If F = (uo, ... ,un) satisfies the Cauchy-Riemann equations and if q 2: n~l, then IFlq is subharmonic. We will not prove this theorem. Its proof is lengthy, mostly computational, but also clever. It would lead us too far off track to present the proof here. See [SWl] or [St4] for a complete presentation of the proof. Definition 1.5.9 We say that F = (uo, ... ,un) is in HP(~~+l) if it satisfies the Cauchy-Riemann equations on ~~+1 and 1

IIFIIHP: = sup ( y>O

(

J~n

IF(x, y)IPdX) Ii <

00.

Corollary 1.5.5 showed that in the case of the disk, HP functions, 1 :::; P :::; 00, can be represented as the Poisson integrals of their boundary values. We want to do the same here. In the setting of the disk, this result in the case p > 1 was a consequence of results on harmonic functions, but the p = 1 case required the estimate of the nontangential maximal function, Theorem 1.5.3. (Which in turn was a consequence of the subharmonicity of powers of the modulus of an analytic function.) Now in higher dimensions, the same will be true: the case p > 1 will be a consequence of results on harmonic functions but the case p = 1 will need an estimate of the nontangential maximal function. (Which again, will be a consequence of the subharmonicity of powers of a function satisfying the Cauchy-Riemann equations.) We do this estimate now.

26

1. Basic Ideas and Tools

Theorem 1.5.10 Suppose F E HP(IR++ 1 ) , n~l < p ::; 00 and a > O. Then NaF E LP(lRn). In fact, there exists a constant Ca,n, depending only on a and n, such that IINaFllp ::; Ca,nIIFIIHP. The proof is similar to that of Theorem 1.5.3 although we need to take a little more care here because of the noncompactness of 1R++1 . Before presenting the proof of the theorem, we prove a lemma that will help us dispense with the difficulties that arise in this noncompact case. Lemma 1.5.11 Let R > 0 and set ER = {(x, y) E IR n+1 : Ixl 2 + y2 = 2R2} n 1R++l. Let v be a Borel measure on IRn with Ivl(lRn) < 00. Let c > 0 be fixed and set

v(x, y) Then as R

i

00,

v(X,Y)IER

---->

=

(

llRn

Py+" (x - t)dv(t).

o.

Proof: Let 17 > O. Choose R o so that JlRn\B(o,R o) dlvl let (x, y) E E R . There are two possibilities: Case 1. Ixl > y. Then Ixl > R and thus, Iv(x, y)l::; ( Py+" (x - t)dlvl(t) 1B(O,Ro)

+ (

llRn\B(O,Ro)

< 17· Suppose R > 2Ro, and

Py+,,(x - t)dlvl

= I + II.

For I, note IPy+,,(x - t)1 ::; l~nJrl!!~' which implies I::; Ivl(B(O, Ro)) (1~~~~)"2+1

::;

Ivl(lRn)(Tjl~'~:l (R-ko)n < 17 if R is large. To estimate II, we estimate IPy+,,(x-t)1 ::; (y~'E)n so that II::; ~lvl(lRn\B(o,Ro)) < C(n,c)17. Thus, in this case, Iv(x,y)1 ::; 17 + C(n, c)17 on ER if R is large. Case 2. Ixl ::; y. Then y ~ Rand

Iv(x, y)l::; { Py+" (x - t)dlvl(t)::; { ( en )n dlvl(t) llRn llRn y + c ::; (y :nc)n Ivl(lRn) ::; 17 if R is large.

Proof of Theorem 1.5.10: Let F = (uo, ... ,un); fix p, n~l < p ::; 00. First note that if p > 1, FE HP(IR++ 1 ) implies that SUPy>o Iluj(·,y)llp < 00 for each j and hence F is the Poisson integral of its nontangentiallimit by Theorem 1.5.1(a). The result then follows from Corollary 1.4.9. So we now assume n~l < p ::; 1. Set r = n;;:l. Let c > 0 and set F,,(x, y) = F(x, y + c) and Uj" = Uj(x, y + c). Since IF(x, yW is subharmonic on 1R++1, then fory~~,

1.5 HP spaces on the upper half space

27

(1.5.1)

We now define ge(x, y) = JlRn Py(x - t)lFe(t, OW dt. We would like to show that lFe(x, yW ~ ge(x, y) for all (x, y) E ffi.~+1. Note that lFe(x, OW = g€(x, 0) and that lFe(x, y)IT is subharmonic and ge(x, y) is harmonic. This is not enough, however, to conclude lFe(x,yW ~ ge(x,y) on ffi.~+1. (For example, consider the functions and y on ffi.~+1.)

°

The above computation yields 1F~(x,y)1 :S C(E)IIFIIHP. Writing 1F~(x,y)1 = IF~ (x, y)IPIF~ (x, y)11-P leads to the conclusion that F~ E Hl(ffi.~+1) and therefore for each j E {O, ... ,n}, SUPy>o Iluj.<: (., y)lll < 00. So by Theorem 1.5.1, each Uj~

'kn

may be expressed as Uj~(x,y) = Py(x - t)dvj(t) for some Borel measure Vj on ffi. n . Consequently, Uje(x,y) = JlRn Py+~(x - t)dvj for each j. For R > set DR = {(x, y) E ffi.~+1: Ixl 2 + y2 = 2R2} n ffi.~+l. Then aD R consists of two pieces: ER (notation as in Lemma 1.5.11) and the ball B(O, V2R) in ffi. n .

°

By Lemma 1.5.11, given ry > 0, we can choose R so large so that IUjel < ry/Vn+1 on E R · Then lFe(x, yW < ryT on E R , and lFe(x, OW = ge(x,O) on B(O, V2R), so that lFe(x, yW ~ ge(x, y) + ryT throughout DR by the maximum principle (see Remark 1.1.8). Letting R i 00 and ry 1 gives lFe(x, yW ~ ge(x, y) on ffi.~+1. Finally, we have

°

r INaFe(x)lPdx JRnr (NalFen~dx ~ J~nr INage(x)l~ dx ~ C r Ige(x,O)I~ dx C r lFe(x,O)iP dx ~ CIIFIIHP. ~n ~n =

Jilin

=

Note that lime->o NaFe(x) = NaF(x) monotonically so the monotone convergence theorem completes the proof.

Theorem 1.5.12 Suppose 1 ~ p ~ 00 and F E HP (ffi.~+1 ). Then for almost every x E ffi. n , the nontangential limit, call it F(x), exists and has IFI E p(ffi.n). Furthermore,

r Py(x - t)F(t)dt. we also have IlFy - Flip °as y 1 0. F(x, y) =

If 1 ~ p <

00,

JlRn

-+

28

1. Basic Ideas and Tools

Proof: Write F = (uo, ... ,un)' First suppose p > 1. Then by Theorem 1.5.1(a), each Uj(x, y) has a nontangential limit, call it Uj(x) E LP(lRn ), and Uj(x, y) = flR n Py(x - t)uj(x)dt. Hence, setting F(x) = (uo(x), ... ,Un (x)), F(x) is the nontangential limit of F(x,y), IFI E LP(lRn) and F(x, y) =

r Py(x - t)F(t)dt.

JlRn

Now suppose p = 1. Then by Theorem 1.5.1(b), each Uj has a nontangential limit, call it Uj(x) E Ll(lRn), hence F(x, y) has a nontangential limit a.e., call it F(x). Let s > O. Then, estimating as in (1.5.1) (now with p = r = 1), we obtain IFe(x,y)1 :::; C(s)IIFIIHl and hence conclude lFel E H2(lR:;:+l). Then F(x, y + s) = flR n Py(x - t)F(t, s)dt. Fix a > O. By continuity, limc--+o F(x, y + s) = F(x,y). Furthermore, since for every s > 0, IF(t,s)1 :::; CaNaF(t) , and IINaFlll :::; Ca,nllFllHl by Theorem 1.5.10, we may let s ----+ 0 and apply Lebesgue's dominated convergence theorem to conclude

F(x, y) =

r Py(x - t)F(t)dt.

JlRn

That IlFy - Flip ----+ 0 as y ! 0 when 1 :::; p < 00 is a consequence of the same result for the Uj, Theorem 1.4.11. Crucial to the last proof was the estimate IINaFllp :::; CallFllHP provided by Theorem 1.5.10. Note that when n = 2, Theorem 1.5.10 states IINaFllp :::; CallFllHP for 0 < p < 00 and this is what we expect in view of Theorem 1.5.3. It would be natural to ask if Theorem 1.5.10 remains valid for a larger range of pi s. Unfortunately, this is not true, although there is a way to repeatedly expand the definition of HP(lR~:+l) so as to obtain a version of Theorem 1.5.10 valid for larger and larger range of pi s. Each successive expansion encompasses a larger range of pi s in such a way that this new definition of HP (lR~+l) agrees with the previous definitions on the previous range of piS. See Fefferman and Stein [FSj. However, it is only when 1 :::; p :::; 00 that we will be able to recover HP(lR~+l) functions as the Poisson integrals of their nontangentiallimits.

1.6

Some basics on singular integrals

In this section, we will discuss the most rudimentary facts concerning singular integral operators. These will be used in the next section when we begin our study of the Littlewood-Paley g-function and the Lusin area function. Here and throughout we will define the Fourier transform of a function f E Ll(lRn) by j(y)

=

r

JlRn

e2TCix.y f(x)dx.

1.6 Some basics on singular integrals

29

Theorem 1.6.1 Let K(x) E L2(]Rn). Suppose there exists a constant B such that

(a)

IK(x)l::::; B

(b)

K is CIon ]Rn\{o} and IV'K(x)1 ::::; Ixl~+l when x -=I- o.

Suppose 1 < p <

00.

For f E p(]Rn) n L1(]Rn) set Tf(x)

=

J IRn

K(x - y)f(y)dy.

Then there exists a constant Ap such that

Here Ap depends only on p, B, n but not on IIKII£>. We will not prove this here. The theorem is due to Calderon and Zygmund [CZ1]. (See either their original paper or Stein [St4] for a proof.) The first step of the proof is to show IITfl12 : : ; Bllfl12; this is an immediate consequence of hypothesis (a) and Plancherel's theorem. This is the only use of hypothesis (a) in the entire proof of the theorem. Thus, instead of (a) we could have merely assumed that T satisfied such an L2 estimate and the conclusion of the theorem would remain valid. We stated the theorem this way simply because it is the most common way to do so. We shall ultimately need a version of Theorem 1.6.1 for Hilbert space valued functions. We recall the necessary concepts and definitions. Let 1i be a separable Hilbert space, that is, 1i contains a countable dense subset, and consequently, a complete orthonormal system {h n } which is countable. A function f: ]Rn ---+ 1i is said to be measurable if whenever h E 1i, (f(x), h) is measurable (as a function of x) on ]Rn. Letting I· I denote the norm of 1i, we see that if f is measurable, then 1

If I = (2:~=1 I(f, hn ) 12» is measurable as a function from ]Rn to R As in the case of complex valued functions we may define a theory of integration of Hilbert space valued functions by first considering simple functions and then taking limits. (Here a simple function has the form 2:~1 hiXEi (x) where Ei ~ ]Rn and hi E 1i.) With this definition IIRn f(x)dx = Y E 1i implies IIRn (f(x), h)dx = (y, h) for every h E 1i since this latter equation holds if f is a function of the form O:XE(X) (0: E 1i, E ~ ]Rn) and both sides are linear. Or, we may simply consider f :]Rn ---+ 1i such that If I E L1(]Rn) and use this relation to define IIRn f(x)dx: We say IIRn f(x)dx = Y E 1i if IIRn (f(x), h)dx = (y, h) for every h E 1i. (If If I E L1(]Rn) it is easy to prove such a y exists.) These definitions are equivalent and in any case, we obtain an integral for 1i valued functions which has all the usual properties of the Lebesgue integral for complex valued functions. Furthermore, for 1 ::::; P ::::; 00 we may define LP (]Rn , 1i) analogously to LP (]Rn): We consider all f: ]Rn ---+ 1i which are measurable and say that such an f E p(]Rn,1i) if IlfIILP(IRn,'H): = IllfIIILP(IRn) < 00. As in the case of complex valued

1. Basic Ideas and Tools

30

functions, we identify two functions if they agree a.e., so that LP (JRn, H) is a set of equivalence classes under this identification. Suppose HI and H2 are separable Hilbert spaces and let B(Hl' H2) denote the Banach space of bounded operators from HI to H 2. If K: JRn ---+ B(Hl' H 2), we say K is measurable if K(x)cp is a measurable H2 valued function for every cp E HI. Let {k n } denote a countable dense subset of HI. Then IK(x)1 = sup{IK(x)knl: Iknl = I}. Thus, IKI is the supremum of a countable set of measurable functions and is therefore measurable if K is. This allows us to define LP(JRn,B(H 1 ,H2)) as the space of all B(H 1 ,H2) valued measurable functions K on JRn such that

Also, using I I, we may define continuity of functions that are B(Hl' H2) valued, and we may define derivatives of such functions in the usual way using difference quotients. Let K E Lq(JRn, B(Hl' H2)), f E LP(JRn, Hd, and suppose 1 :S r :S 00 and ~ = ~ + -1. Then flRn IK(x - y)f(y)ldy :S flR n IK(x - y)llf(y)ldy which is finite for a.e. value of x so that for such X, flR n K(x - y)f(y)dy defines an element of H2 which we denote by K * f(x). Then

*

11K * fllu(lR

n

,H2)

= IIIK * flllu(lRn) :S IIIKIIILq(lRn)lllfIIILP(lRn) =

IIKIILQ(lR n ,B(Hl,H2)) IlfIILP(lRn,H,)

that is, the usual inequalities for convolution are valid in this setting, and are a consequence of similar inequalities for complex valued functions. With these definitions and facts collected, we can now state a vector valued version of Theorem 1.6.1. Theorem 1.6.2 Let HI and H2 be separable Hilbert spaces and suppose that K (x) E

L2(JRn, B(Hl' H 2)) and 1 < p < 00. For f E LP(JRn, HI) n£l(JRn, HI) set T f(x) flR n K(x - y)f(y)dy. Suppose also that there exists a constant B such that (a)

IITfllu(lRn,H2):S Bllfllu(lRn,H')

(b)

K is 0 1 on JRn\{o} and IV'K(x)1

for every f E L2(JRn, Hd

:S

Ixl~+l when x

=

n Ll(JRn, Hd

-I O.

Then there exists a constant Ap such that liT fIILP(IR n ,H2) :S ApllfIILP(lRn,Htl. Here Ap depends only onp,B,n but not on the L 2(JRn,B(Hl,H 2)) norm of K. We will not prove this theorem. Although in Theorem 1.6.2 the functions involved take values in Hilbert spaces, its proof is the same as that of Theorem 1.6.1 and is, in fact, essentially typographically indistinguishable from that proof. See [Hor], [Jou] or [St4]. Note that our statement of Theorem 1.6.2 is not quite parallel with the statement of Theorem 1.6.1. This can be readily reconciled in view of the comments made after that theorem.

1.7 The g-function and area function

31

1.7

The g-function and area function Consider a G I function h: lRn ----> lRm , not identically zero, which satisfies: (i)

There exists G > 0 such that for all x

(ii)

Ihl is radial. That is, componentwise.

l\7h(x)1 ::; G(l + Ixl)-n-2.

E

lRn,lh(x)1 ::;

G(l

+ Ixl)-n-l,

Ih(x)1 = Ih(y)1 whenever Ixl = Iyl where h is defined

Jo

Note that (ii) implies oo Ih(t~W~, ~ =J 0 is constant and we further require the finiteness of this constant: (iii)

For ~ =J 0,

As usual, set ht(x)

=

t;, h

(f)·

Theorem 1.7.1 Suppose 1 < p <

00

and for f E LP(lR n ) set

(Here the convolution is taken componentwise: writing h = (hI"'" h m ), h t * f = (hIt * f, ... ,hmt * f).) Then there exists a constant A p , depending only on p, n and h such that IITfllp ::; Apllfllp. Furthermore, T is, up to a constant factor, an isometry on L2 (lRn). Proof: Fix c > O. Let lRt = {y E lR: y > c}. Set HI =!C, H2 = EB?=I L2 (lRt, ~), the direct sum of m copies of L2 (lRt ) ~). HI and H2 are separable Hilbert spaces. Consider the kernel K (x) = t~ h (f). We will show that for each x, K (x) is an element of H 2 , and hence, since HI =!C, K(x) may be identified as an element of B(HI' H2)' (K(x) acting on a E !C is given by the multiplication aK(x).) We will then show that K E L2(lRn, B(HI) H 2)) and K satisfies the hypotheses of Theorem 1.6.2. This will then show the boundedness of the operator Td(x)

=

r

JlRn

K(x - y)f(y)dy.

This will then be used to show the desired estimate for T f. We remark that we will not try to show that T itself is represented by a kernel which satisfies the hypotheses of Theorem 1.6.2.

32

1. Basic Ideas and Tools

r

Routine computation using the first estimate in (i) shows that

IIK(x)II", :;

I~n (l (u + ~)'n+,dU

1

If Ixl :::; 1 it is easy to estimate IIK(x)II1-i2 :::; CE;' Thus, for all x E JRn, K(x) E H2. Furthermore, our estimates show that IK(x)1 = IIK(x)II1-i2 E L2(JRn ), which by definition, means K E L2(JRn, B(Hl' H2))' Using the same reasoning, and the second estimate in (ii) we have

IVK(x)l:;

IXI~H

(1

(u+

~)'n+,dUr

Suppose f E L2(JRn) n Ll(JRn). Then

This and condition (iii) shows IITdll£2(IRn,1-i2) :::; cllfIIL2(IRn). Consequently, by Theorem 1.6.2, IITd IILP (IRn,1-i2) :::; ApllfllLP(IRn) whenever f E LP(JRn) n Ll(JRn). Explicitly, (1.7.2) whenever f E V (JRn) nLl(JRn). Suppose f E LP(JRn ). By choosing fn E V(JR n ) n Ll(JRn ) such that fn - t f in V(JRn) and a.e., Fatou's lemma shows that (1.7.2) continues to be valid for all f E LP(JRn). Let E 1 0 and again use Fatou's lemma to obtain the desired estimate IITfllp :::; Apllfllp. We note that Ap in 1.7.2 depends only on p, n and some constants associated with the function h. Finally, let E 1 0 in (1.7.1) to obtain

IITfl12 = cllfl12 whenever f E L2(JRn) n Ll(JRn). That this is true for all f E L2(JRn) can be seen by choosing fn E L2(JRn )nLl(JRn) with fn ----) fin L2(JRn) and noting that the sublinearity of T then implies Tfn ----) Tf in L2(JRn).

Hence

33

1.7 The g-function and area function

We note that c in the end of the proof is just the constant c occurring in (iii) so that by renormalizing, we could have simply required that c = 1, in which case T would be an isometry. The integrability condition (iii) forces h(O) = 0, which is the same as requiring that the integral of h be zero. It is possible to show that if h satisfies (i) above and has integral zero on JRn then 1000 Ih(tl;)I2~ < c for every l; E JRn for some finite constant c. Using (1.7.1), this shows T is bounded on L2 and as in the above proof then yields the LP estimate. Then if we further assume Ihl is radial, we are back in the previous case and T is, up to a constant, an isometry on L2. See Journe [Jou]. Let us again consider a function h satisfying (i)-(iii) at the beginning of this section. Fix a> 0 and consider the family offunctions h(y) (x) = h(x-y), Iyl < a.

< p < 00

Theorem 1.7.2 Suppose 1

and for f E U(JRn ) set

Then there exists a constant Ap depending only on p, nand h such that liT flip Apllfllp. Furthermore, T is, up to a constant factor, an isometry on L2(JR n ).

:s;

Proof: The proof is essentially the same as the proof of Theorem 1.7.1 so we quickly sketch it, noting only the changes. Fix E > 0, and define Hilbert spaces 111 = C,

112 =

EB7=1 L2 (

(JRt,

~)

x (B(O, a), dY )). Consider the kernel K(x)

=

t~h(y) (~).

Iyl < a, the estimates in (i) above. Thus, as in the previous proof, we conclude that K E L2(JRn , B(111' 112)) and IVK(x)l:s; Ixl<;;'+l.

It is readily checked that the h(y) (x) satisfy, uniformly in

Note that for y fixed,

Again, since

Ihl

IhMl = Ihl

so reasoning as in (1.7.1) yields:

is radial and satisfies (iii),

(7J Ih(tl;Wdy~t) o

1

2"

=

cIB(O,a)l~ <

00.

B(O,a)

This shows the boundedness of To : L2(JR n ,11d ---; L2(JRn, 112) and again this leads, via Theorem 1.6.2 to the conclusion IITofIILP(lRn,Ji2) :s; Apllflb(lRn) whenever f E

1. Basic Ideas and Tools

34

V(Rn)nLl(Rn). The passage to the limit to obtain IITfllp ~ Apllfllp is as before. That T is an isometry on L2(Rn) is also proved as previously. We would like to reverse the LP inequalities in Theorems 1.7.1 and 1.7.2. This will be done with the following lemma which we state in sufficient generality to cover the situation of each theorem. Let (A, v) be a measure space and 1{ a separable Hilbert space. Suppose 1 < p < 00 and that for each u E A there is a linear 1 operator Tu: LP(Rn ) ~ 1{. For f E VeRn) set Tf(x) = (fA ITuf(x)l2dv(u)) 2.

= cllfl12 for every f E L2(Rn). Suppose also that T is bounded on LP(Rn) for all p, 1 < p < 00: IITfllp ~ Apllfllp" Then Ilfllq ~ ~ApllTfllq for all f E U(Rn), where Lemma 1.7.3 Suppose that there exists a constant c such that IITfl12

1.+1.=1. p q

Proof: Let f, 9 E L2(Rn). Then IIT(f +

g)ll~ = IITfll~ + IITgll~ + 2Re

ilR[ niA[(Tuf(x), Tug(x))dv(u)dx

= c211fll~ + c211gll~ + 2Re [

[(Tuf(x), Tug(x))dv(u)dx.

ilRniA

But also,

Consequently, Re [

f(x)g(x)dx = 12 Re [

ilRn

C

[(Tuf(x), Tug(x))dv(u)dx.

ilRni A

Using Cauchy-Schwarz on the inner integral on the right we obtain: [ f(x)g(x)dxl ~ IRe ilRn

12 [ Tf(x)Tg(x)dx. C

ilRn

Taking the supremum over all 9 E VeRn) n L2(Rn) with Ilgllp 1 IIfllq ~ c2ApilTfllq

= 1 yields: (1.7.3)

1.7 The g-function and area function

35

Consider now f E Lq(l~n) and take fn E Lq(JRn) n L2(JRn) with fn ----; f in Lq. Plainly T is sublinear and thus, ITfn - Tfl :S TUn - f). Since T is bounded on Lq, this implies Tfn ----; Tf in Lq. Thus, since (1.7.3) is valid for each fn, we may pass to the limit to obtain (1.7.3) for general f E Lq(JRn). Theorems 1.7.1 and 1.7.2 now provide a bountiful harvest of examples. Some of these playa prominent role in this monograph. We consider the Poisson kernel P(x) = en ~ and set hl(X) = V'xP(x). A computation shows that (x) = (1+lxI 2 )

2

_

gP

XJ

-3

-(n + l)c n Xj(l + IxI so that hl(X) satisfies the first estimate of condition (i). Similar computation shows that hl(X) also satisfies the second estimate in (i). Although we have not done so, it can be computed that P(~) = e-27r1~1 (see (~) = -27ri~jP(~) so that Ihl(~)1 = Stein [St4] for this computation). Then 2 )--T-

g;: J

27r1~IIP(~)1 which is radial. In addition, for ~ =I- 0,

Thus, hI satisfies conditions (i)-(iii). Let h2(X) = C n IxI2_~ , where (1+IxI2)

2

Cn

is the constant appearing in the for-

mula for the Poisson kernel. With this definition, (h2)y(X) = Y !Jy P y(x) for x E JRn, y > 0. Note that h2(X) = -nP(x) - 2:.7=1 Xj (Compare to formula 2.2.1 below which is similar yet slightly different; h2(X) has a different purpose than the kernel there.) Plainly, h2(X) satisfies the first estimate in condition (i) and routine computation shows that it also satisfies the second condition. Also,

g:,.

10

which is radial. Note that Ih2(~)1 = Ihl(~)1 so for ~ =I- 0, 00 Ihdt~W~ h2 also satisfies conditions (i)-(iii). For f E LP(JR n ), 1 < p < 00, set u(x, y) = Py * f(x). Then

I(hdy

* f(xW = y21V' xu(x, yW

I(h2)y

* f (x W= y21 ~~ (x, y) 12

Definition 1.7.4 Suppose u is harmonic on JR~+I. For x E JRn we define

=

i. Thus,

(1. 7.4)

1. Basic Ideas and Tools

36

and

When u is the Poisson extension of a function f, we will write g(1), gl (1) and g2 (1) for g( u), gl (u), g2 (u) respectively. The function g(1) is called the LittlewoodPaley 9 function of f. Theorem 1.7.5

(b)

Suppose 1 < p < 00. Then there exists constants ap and A p , depending only on p and n such that if f E LP (JRn),

apllfllp:S: Ilg(1)llp :s: Apllfllp apllfllp:S: Ilgl(1)llp :s: Apllfllp apllfllp:S: Ilg2(1)llp :s: Apllfllp" Proof: We have noted that hI and h2 satisfy (i)-(iii) at the beginning of this section and we explicitly computed that in each case the constant c appearing in (iii) is t. Noting (1.7.4), we apply Theorem 1.7.1 with T = gl and T = g2 to conclude that both gl and g2 are bounded on LP(JRn) and, in view of the comment immediately after the proof of that theorem, satisfy IlgI(f)112 = tllfl12 and Ilg2(1)112 = tllfl12' The other LP inequalities: apllfllp :s: IlgI(f)llp and apllfllp :s: Ilg2(1)llp follow from and 1i = en for Lemma 1.7.3. (Using, in the notation there, A = [0,00), v = gl and 1i = e for g2.) Thus, we've shown (a) and (b) for gl(1) and g2(1) and to finish the proof we need to do the same for g(1). Consider h = (hI, h 2) : JRn ----t JRn+l. Since both hI and h2 satisfy the es-

!if

timates in (i) then so does h. Also,

v2lhll.

Then

Ihl

h = (hl,h2)

and so

Ihl = Vlh112+ Ih212 =

is radial and for ~ =I- 0

Then by Definition 1.7.4, (1.7.4) and Theorem 1.7.1, Ilg(1)112 = 2-~ IIfl12 and 9 is bounded on LP(JR n ). The other LP inequality follows from Lemma 1.7.3.

1.7 The g-function and area function

37

Definition 1.7.6 Let a > O. Suppose u is harmonic on JR.~+1. For x E JR.n we define 1

A"u(x) = ( (

1ra (x)

lV'u(s, tWt1-ndsdt)

"2 1

(At)au(x) = ( (

1ra (x)

(A2)aU(X) =

lV'su(s, tWtl-ndsdt)

(lra(X) I: (s, t)1

"2 1

2

t1-ndsdt) "2

When u is the Poisson extension of a function f, we will write Aaf, (Adaf and (A 2 )af for Aau, (Al)aU, (A2)aU respectively. The function Aaf is called the Lusin area function of f, or the square function of f. It is a consequence of the mean value theorem for harmonic functions that g(u)(x) :S CaAau(x) where Ca depends only on a and n. See Zygmund [Zy2, vol. II, p. 210] or Stein [St4, p. 90] for a proof. We will apply Theorem 1.7.2 to these to prove estimates similar to those of Theorem 1.7.5. First, a calculation: Let B = B(O, a) ~ JR.n. Then for f E V(JR.n),

11 1 liln 00

o

=

=

B

l(h1(y))t

00

* f(xWdy

-dt t

2 (h1(y)Mx - W)f(W)dWI dy

~t

{ 1 (x - w) 12 dt 10roo 1B{ 11R.n t n hl - t - - Y f(w)dw dy t

(1. 7.5) Here for the next to last equality we have made the substitution s in the step previous we made use of (1.7.4). Likewise,

1 11 00

o

B

(h(y))t

* f(x)1 2 dy

-dt = (Aaf(x)) 2 .

t

=

x - ty, and

1. Basic Ideas and Tools

38 Theorem 1.7.7

(a)

Suppose f E L2(1R.n). Then there exists a constant co, depending only on a, n such that

(b)

Suppose 1 < p < 00. Then there exists constants bp and B p , depending only on p, n and a such that if f E LP (JR.n)

bpllfllp :::; IIAofilp :::; Bpllfllp bpllfllp:::; II(Al)ofllp :::; Bpllfllp bpllfllp:::; II(A2 )ofllp :::; Bpllfllp· We will not prove this theorem. Its proof follows the proof of Theorem 1.7.5, using Theorem 1.7.2 instead of Theorem 1.7.1 used in the proof of Theorem 1.7.5. The proof of Theorem 1.7.2 and (1.7.5) reveal that Co = IB(O, a)I-!. The approach we have followed, via singular integrals taking values in a Hilbert space, is due to Harmander [HarJ. As we have seen, this approach yields LP inequalities such as Theorems 1.7.1 and 1.7.2 that hold in greater generality than the setting of harmonic functions. However, in most of this text, we will use techniques and ideas from works (many mentioned below) that have considered only the harmonic case. We followed the Harmander approach here only because it was the most expedient way to obtain the LP inequalities of Theorems 1.7.5 and 1.7.7. The disadvantage to this approach, is that, because it uses singular integrals, it seems to constrain us to the range 1 < p < 00 and it does not yield the local estimates we will see throughout the text. See also Question 4.2.16 below. In Theorems 1.7.5(b) and 1.7.7(b) we have made the assumption that f E LP(JR.n). This does not diminish the thrust of the inequalities, Ilg(f)llp :::; Apllfllp, IIAoflip :::; Bpllfllp and the others similar to this, but is an unnecessary apriori assumption for the reverse inequalities. We will now remedy this situation. We do this only for g and Ao since these are of greatest interest to us. Theorem 1.7.8 Let 1 < p as y -+ 00. Then

< 00.

Suppose u is harmonic on JR.+.+1 and u(x, y)

-+

(a)

If g(u) E LP(JR.n) then u is the Poisson integral of a function f E LP(JR.n). Furthermore,

(b)

If Aou E LP(JR.n) for some a > 0, then u is the Poisson integral of a function f E LP(JR.n). Furthermore,

°

1.7 The g-function and area function

39

Proof: We will show (a); (b) is similar. Fix 0 < c < L < 00 and set v(x, y) = u(x, y +c) - u(x, y+ L). The function v(x, y) is harmonic on JR~+l and continuous on JR~+l. Furthermore,

Iv(x, y)1

(y+L 8u = lu(x, y + c) - u(x, y + L)I ::; Jy+e: 8t (x, t) dt 1

1

::;

r+L 18 12)! ( Jy+e: r+L 1 ) ! ( Jy+e: t 8~ (x, t) dt t dt ::; Ce:,L g(u)(x).

Then by Theorem 1.5.1, v(x, y) is the Poisson extension of its nontangentiallimit, that is, v(x, y) = Py * h(x) where h(x) = u(x, c) - u(x, L) E U(JRn). Also,

(g2(h)(x))2

=

1 tl ~~ 00

: ; 21

00

(x, t) 12 dt

~~ (x, c + t) 12 dt +

t1

21

00

~~ (x, L + t) 12 dt

t1

so that (g2(h)(x))2 ::; 4(g(u)(x))2. This, plus Theorem 1.7.5 gives:

Ilhllp ::; Apllg2(h)llp ::; 2Apllg(u)llp, or

(

JIltn Letting L .......

00

lu(x, c)

-

u(x, L)lPdx ::; 2PA~llg(u)II~.

we obtain

This is true for every c > O. Hence, by Theorem 1.5.1, there exists an f E LP(lR n ) with u(x, y) = Py * f(x). Since u(·, c) ....... fO in LP(lR n ) (Theorem 1.4.11) the result follows. The above argument is from Stein [Stl]. We remark that if u is harmonic on JR~+l, adding a constant to u leaves g(u) and Aau unaltered. This the reason for the hypothesis on the behavior of u at infinity. The function g(f) first arose in Littlewood and Paley's study [LP], of the decomposition of Fourier series into lacunary blocks. (See Question 4.2.15 below.) They considered a version of 9 defined on the disk and showed that if F E HP on the disk, 0 < p < 00, then Ilg(F) lip::; ApllFllp. As is the case with many of the results we've discussed concerning HP, this inequality, in the range 0 < p ::; 1, depends on the analyticity of F. They also showed the reverse inequality if F(O) = 0 and 1 < p < 00.

1. Basic Ideas and Tools

40

Suppose f E LP(aD), 1 < p < 00, and f has mean value 0 on aD. Set P[J] and let v be a harmonic conjugate of U on D with v(O) = O. Write F = u+iv. By the M. Riesz theorem (see discussion below) Ilulip ~ Ilvllp. Observe that the Cauchy-Riemann equations imply g(u) = g(F). Thus, Littlewood and Paley's results show that on the disk, Ilg(f)llp ~ Ilfllp, 1 < p < 00. Later, T .M. Flett [FI] gave a partial extension of one of Littlewood and Paley's inequalities. He showed: If FE HP, 0 < p::; 1, and if F has no zeros on the disk, then 1!Fllp ::; Ap(llg(F)llp+IF(O)l). At about the same time, Waterman [Wat] gave analogues for Littlewood and Paley's results in the setting of the upper half plane JR~. Finally, a few years later, Stein [Stl] showed the results on g we've stated here in Theorems 1.7.5 and 1.7.8. Gasper [Gas] later extended one of Stein's inequalities by showing: if FE HP on the ball in JRn, n~l < p < 00, then Ilg(F)llp ::; Apl!Fllp. It was Lusin [Lu] who initiated the study of the area integral. He considered a version of A, defined on the disk and showed that if F is analytic on the disk with F(O) = 0, then IIAa(F)112 = Cal!F112. Marcinkiewicz and Zygmund [MZ2] showed that if F E HP on the disk, 0 < p < 00, then IIAaFllp ::; Cal!Fllp" They showed the pointwise majorization g(F)(x) ::; CaAaF(x) and thus concluded the reverse inequality, 1!Fllp ::; CailAaFIlp if F(O) = 0, 1 < p < 00, as a consequence of Littlewood and Paley's result mentioned above. Stein [Stl] showed the inequalities for the area integral that we've shown here. Calderon [Ca3] completed a last piece that Marcinkiewicz and Zygmund hadn't shown by showing that if F E HP(JR~), 0 < p ::; 1, then 1!Fllp ::; CaiIAaFllp. Shortly thereafter, Segovia [Se] extended this to higher dimensions: if F E HP on JRf+l, n~l < p < 00, then I!FIIHP ~ IIAaFIIp" Gasper [Gas] showed a version of this for functions in HP of the unit ball in JR n . Let f E V(JR), 1 < p < 00, Set u(x, y) = Py * f(x) and let v(x, y) be a harmonic conjugate to u on JR~ with v(x, y) - t 0 as y - t 00. (So U + iv is analytic on JR~). The Cauchy-Riemann equations imply Aau = Aav pointwise, so that Theorems 1.7.7(b) and 1.7.8(b) imply that v is the Poisson integral of a function in LP (JR); this is usually denoted as H f and is called the Hilbert transform of f. Thus, these theorems show the theorem of M. Riesz: if f E LP(JR) , 1 < p < 00, then IIHfllp ::; Cpllfllp" Similar considerations show an analogous result on the disk. This is not at all surprising; H f can be represented as a singular integral: Hf(x) = lime->o ~ ~x-YI>e ~~; dy (see Garnett [Gal, Katznelson [Kat] or Stein [St4] for this) and the theory of singular integrals, used in our proofs of the theorems on the area integral, is a generalization (albeit a deep one) of the study of the Hilbert transform. Similar ideas hold for the notion of conjugacy in higher dimensions. For f E V(JRn), 1 < p < 00, form uo(x, y) = Py * f(x). Given this Uo, it is possible to create Ul, ... ,Un, harmonic on JR+.+ 1 and vanishing at infinity, so that Uo, Ul, ... ,Un satisfy the generalized Cauchy-Riemann equations. (See Fefferman and Stein [FS, p. 169].) However, here the Cauchy-Riemann equations do not allow us to conclude equality of the area functions of the Uj as in the case when n = 1, but it can be U

=

1.7 The g-function and area function

41

shown that A",uj(x) ::=; GA,auo(x) whenever 0' < /3, j E {I, ... ,n}. (Stein [St4, p. 213].) Here G depends only on 0', /3 and n. Thus, fixing 0' < /3 we conclude IIA",ujllp ::=; GIIA,auollp ::=; Gllfllp and hence that Uj is the Poisson integral of a function in LP(lRn). This function is denoted by Rjf and is called the jth Riesz transform of f. Thus, IIRjflip ::=; Gllfllp, Each R j can be represented as a singular integral operator. In fact, the singular integral theory of the previous section can be used to show directly that IIRjflip ::=; Gpllfllp. Then the Uj may be created as Uj(x, y) = Py * Rjf(x). See Stein [St4] for this circle of ideas. In Theorem 1.5.10 we showed that if F E HP(lR~+l), n;;:l < p ::=; 00, then Writing F = (UO,Ul,'" ,un) this immediately implies When n = 1, Burkholder, Gundy and Silverstein [BGS] proved a converse: If U is harmonic on lR~, N",u E LP(lR), 0 < p < 00, then there 1 exists a harmonic conjugate v for U on lR~ such that SUPy2:o (flR Iv(x + iy)IPdx) P ::=; G""pIIN",ullp' (The choice of 0' is irrelevant; see Lemma 4.1.2 below.) Thus, F = U + iv is in HP(lR~) and IIFIIHP ::=; G""pIIN",ullp' This establishes a one-to-one correspondence between functions in HP(lR~) and harmonic functions on lR~ which have nontangential maximal function in LP(lR) - with an equivalence of norms.

IIN",Fllp ::=; G""pIIFIIHP' IIN",uollp ::=; G""pllFllm.

00, U is harmonic on lR~ and has N",u E LP(lR). Then so that by Theorem 1.5.1, u(x,y) = Py * f(x) for some f E LP(lR). By the M. Riesz theorem, Hf E LP(lR) with IIHfllp ::=; Gpllfllp. On lR~ we define v(x,y) = Py * Hf(x). Then SUPy>o Ilvyllp ::=; IIHfllp so that F = U + iv E HP(lR~) and 1IFIlm ::=; II flip + IIHfllp ::=; Gpllfllp ::=; GpIIN",ullp. But, since 1 < p < 00, IINauilp ::=; Gp,,,,llfllp ::=; IIFIIHP. The conclusion here is that in the range 1 < p < 00 this one-to-one correspondence is a consequence of the M. Riesz theorem and estimates for the nontangential maximal function which followed immediately from the theorem of Hardy and Littlewood on the maximal function. In the range 0 < p ::=; 1 these tools are no longer available and this is the subtlety of the theorem of Burkholder, Gundy and Silverstein. The original proof of these authors used Brownian motion; later Koosis [Koo, p. 249] gave an analytic proof based on ideas from Fefferman and Stein [FS].

Suppose 1

SUPy>o Iluyllp <

00

/3 > 0 and

U is a harmonic function on lR~ which has N",u E Then by the theorem of Burkholder, Gundy and Silverstein, there exists a harmonic conjugate v with F = u + iv E HP(lR~) and with IIFIIHP ::=; IIN",ullp' But, by a previously mentioned result of Marcinkiewicz and Zygmund (or rather the version of their theorem for lR~, due to Waterman [Wat]) and the fact that A,au = A,av, this then yields: IIA,aullp ::=; IIA,aFllp ::=; GpllFllm ::=; GpIIN",ullp. Conversely, if u is harmonic on lR~ with A,au E LP(lR) and with u(x, y) ---+ 0 as y ---+ 00, we let v be a harmonic conjugate of u which also vanishes at infinity and set F = u + iv. Then IIA,aFlip = 2l1A,aullp and by Marcinkiewicz and Zygmund, and Calderon, IIFIIHP ::=; GpIlA",Fllp = 2GpIlA",ullp' (The apriori assumption in these theorems that F E HP may be removed with the assumption that F vanishes at infinity and a limiting argument which was essentially given in the proof of Theorem 1.7.8. See Fefferman and Stein [FS, p. 165].) Then by the

Suppose

0',

LP(lR), where 0 < p <

00.

42

1. Basic Ideas and Tools

Theorem 1.5.10, IINauilp :S IINaFllp :S IIFIIHP; thus, IlNaullp :S CpIIA,eullp" Succinctly: if 0 < p < 00, a, j3 > 0 and u is harmonic on ~~, then IIA,eullp :S CpllNaullp" If, in addition, u vanishes at infinity, then IINauilp :S CpIIA,eullp. Fefferman and Stein [FS] showed the higher dimension analogue of the theorem of Burkholder, Gundy and Silverstein. They did this by first proving a higher dimensional analogue of this last statement and essentially reversing the reasoning of the last paragraph. That is, they first showed: if 0 < p < 00, a, j3 > 0 and u is harmonic on ~~+1, then IIA,eullp :S CpllNaull p and, if in addition, u vanishes at infinity, then IINauilp :S CpIIA,eull p. Now consider a harmonic function Uo on ~~+1 with Nauo E LP(~n), n~l < p < 00. Then IIA,euollp :S CpllNauollp. It is possible to form Ul, ... ,Un harmonic on ~~+ 1 , vanishing at infinity, and so that uo, Ul, .•. ,un satisfy the generalized Cauchy-Riemann equations. As mentioned previously, it is not the case that A,euo and A,euj, j E {I, ... ,n} are equal pointwise, but it can be shown that Ayuj(x) :S CA,euo(x) whenever'Y < j3. (Here C = C(,,!,j3,n).) Thus, for each j, IINaujllp :S CpIIA1'uj(x)llp :S CpIIA,euo(x)llp :S CpllNauollp· Consequently, F = (uo, Ub ... ,un) E HP(~n) and IIFIIHP :S CpllNauollp. Note that Theorem 1.5.10 provides the reverse inequality: IIN",uollp :S CpIIFIIHP. Thus, as in the case of ~~, there is a one-to-one correspondence between functions in HP(~~+1), n~l < p < 00 and harmonic functions on ~~+l which have nontangential maximal function in LP(~n) - with an equivalence of norms. We remark that Segovia's [Se] result IIFIIHP ~ IIA",Fllp, n~l < p < 00, follows immediately from these estimates. We further note that as we mentioned after the proof of Theorem 1.5.10, it is possible to successively expand the definition of HP(~~+l) so as to obtain a version of that theorem valid for larger and larger ranges of p. With this definition of HP (~~+1 ), a proof similar to that given above shows that this one-to-one correspondence is valid on the entire range 0 < p < 00. See Fefferman and Stein [FS] for details. Burkholder and Gundy [BG2] generalized Fefferman and Stein's result relating the V norms of the area function and nontangential maximal function. Consider a nondecreasing continuous function IP defined on [0,00] with IP(O) = 0, IP not identically zero and which satisfies the growth condition: 1P(2A) :S CIP(A) for every A > 0, where c is a fixed constant. They showed: Theorem 1.7.9 Suppose a, j3 > 0 and IP is as above. If u is harmonic on ~~+1 then

{ IP(Aau(x))dx:S C { IP(N,eu(x))dx. A~n JlRn If, in addition, limy_+<Xl u (x, y)

= 0 for all x

E ~ n, then

Here C depends only on a ,j3, n, and the growth constant c of IP.

43

1.8 Classical results on boundary behavior

A special case of this is (A) = AP, 0 < p < 00. Therefore, this result includes all previously mentioned estimates regarding the area function. Burkholder and Gundy proved this by means of inequalities relating the distribution functions of the nontangential maximal function and area function. These authors had previously developed and used similar inequalities in treating martingale analogues of these functions [BGl]. These inequalities, which came to be known as good-A inequalities, were later sharpened by R. Fefferman, Gundy, Silverstein and Stein [FGSS] and by Murai and Uchiyama [MU]. It is a major goal ofthis book to state and prove sharper and more general versions of these. This is done in Chapter 4. We also include there a discussion of the probabilistic roots of these inequalities. We will not give the proof of Burkholder and Gundy's theorem here, but will present it in Chapter 4 as a consequence of the good-A inequalities we show there. Chapter 4 also contains applications of good-A inequalities to several other estimates.

1.8

Classical results on boundary behavior

Let u be harmonic on lR~+l and bounded there. Then by Theorem 1.5.1, u is the Poisson integral of a function f E £CXl (lRn) and the nontangentiallimit of u is f a.e. In other words, if u is harmonic and bounded on lR~+l, then it has a nontangential limit a.e. In this section we discuss local versions of this result. We recall (Definition 1.4.10) that if u is harmonic on lR~+1 and x E lRn , we say that the nontangentiallimit of u at x is L if for every a > 0, lim u(s, t) = L. (s,t)-+x (s,t)Era(x)

Definition 1.8.1 For a > 0, h > 0 we define the truncated cone r~(x) = {(s, t) E lR~+l: Ix-sl < at, 0 < t < h}. We say u is nontangentially bounded at x E lR n if u is bounded on some r~(x). Equivalently, if we set N~u(x) = sup{lu(s, t)l: (s, t) E r~(x)}, then u is nontangentially bounded at x E lR n if N~u(x) < 00 for some a, h. In the case of the unit disk we can make analogous definitions by using the cones r a(O). Privalov [Prj (or see Zygmund [Zy2, vol. II, p. 199]) showed that in the case of the disk, a harmonic function u has a nontangentiallimit at a.e. point at which u is nontangentially bounded. Calderon [Cal] extended this to harmonic functions on lR~+1. Succinctly: if u is harmonic on IR~+l then, {x E IR n : u has a nontangential limit at x} = {x E IRn: u is nontangentially bounded at x} a.e. As we noted earlier, Lusin [Lu] showed that IIAaFI12 = calIFI12 whenever F is analytic on the unit disk and has F(O) = O. Marcinkiewicz and Zygmund [MZ2] showed a local version: Suppose that F is analytic on D and E c:;;; aD is the set of points at which F is nontangentially bounded. Then for a.e. 0 E E, AaF(O) < 00 for every a, 0 < a < 1. Later, Spencer [Sp] proved a converse: Suppose that F is analytic on D and E c:;;; aD is a set with the property that for each 0 E E there is an a = 0'(0) such that AaF(O) < 00. Then F is nontangentially bounded at a.e. 0 E E.

44

1. Basic Ideas and Tools

On lR~+l the cones r~(x) allow the definition of a "truncated" version of A,u(x):

Using this, Calderon [Ca2] showed a generalization of the Marcinkiewicz and Zygmund theorem: If u is harmonic on lR~+1 and if E ~ lR n is the set of points at which u is nontangentially bounded, then for a.e. x E E, A~u(x) < 00 for every a, h > O. Stein [St2] showed the converse: Suppose u is harmonic on lR~+1 and E ~ lR is a set with the property that for each x E E there are a = a(x) and h = h(x) such that A~u(x) < 00. Then for a.e. x E E, u is nontangentially bounded at x. We will not prove any of these theorems. For the proofs of the theorems on the disk see Zygmund [Zy2, Chapter XIV]. For the proofs of the versions in lR~+1 see Stein [St4, Chapter VII]. The techniques used in these proofs are not dissimilar to those used in the LP inequalities relating these functions and likewise have much in common with the ideas and techniques used throughout this book. However, it would lead us too far astray to present these proofs here. Nevertheless, these results are of great relevance to this monograph. We will obtain results that quantify, in a very precise way, the behavior of harmonic functions on the complement of the set where A~u(x) < 00. By the above results, at a.e. point where A~u(x) = 00, u fails to be nontangentially bounded; the laws of the iterated logarithm in Chapter 3 will measure the relative growth of these quantities near the boundary. As a corollary of the proof, we will obtain Stein's result mentioned above, but our true focus is to quantify the behavior of a harmonic function on the complement of the sets mentioned in the theorems in this section. See Chapter 3 for details.

Chapter 2 Decomposition into Martingales: An Invariance Principle Our goal in this chapter is to show that arbitrary harmonic functions in the upper half space can be approximated by dyadic martingales so that both the error terms in this approximation and the area function of the martingale are controlled by the area function of the harmonic function. This technique is very similar to the invariance principles discussed in Chapter 6, hence the title for this chapter. Much of what we do here was done in Bafiuelos, Klemes, and Moore [BKM1], and in Banuelos and Moore [BM2]. However, it is to be emphasized that the techniques of these two references are essentially a refinement of those found in Chang, Wilson, and Wolff [CWW]. In order to understand the motivation for the results and techniques here, as well as the historical precedents, we first discuss the Chang-Wilson-Wolff result. We will need some terminology. Throughout this monograph, a cube Q <;;;; jRn will be called dyadic if it has the form Q = [kI2£, (k l +1)2£) x ... x [k n 2£, (kn+1)2£) for some l, kl , ... kn E Z; for such a cube Q we say that Q has sidelength 2£ and denote this as £(Q) = 2£. For m E Z we let Fm denote the set of all dyadic cubes in jRn of sidelength 2- m and we will let F denote the set of all dyadic cubes in jRn of sidelength ~ 1. By a slight abuse of notation, we will also use F m to denote the a-field generated by the set of all dyadic cubes in jRn of sidelength 2- m . Suppose Q is the unit cube Q = [0, 1] x ... x [0, 1]. A dyadic martingale on Q is a sequence of integrable functions {fm};;';=o on Q such that each 1m is Fm measurable and 1m = E(fm+1!Fm) for every m. We define the maximal function of the martingale {fm} by (2.0.1) For k = 1,2, ... we set dk = A - A-I; this is called the martingale difference sequence. For x E Q, let us denote by Qk(X) the unique dyadic cube of sidelength 2- k +1 containing x. With this notation we set (2.0.2)

45 R. Bañelos et al., Probabilistic Behavior of Harmonic Functions © Birkhäuser Verlag 1999

46

2. Decomposition into Martingales

This is called the square function of the martingale Urn}. The conditioned square function is defined by (2.0.3) Since

TnlldkXQk II~

:::; I~kl

hk

d~dx :::; IldkXQk II~,

S(Jrn) and dfm) are pointwise comparable. In one dimension both square functions above are pointwise comparable to the more traditional square function (2.0.4) Our results here work better if we take the square function defined by (2.0.2). If m = 00, we simply write f* and S(J) for f:x, and S(Joo). We will sometimes write S2(Jn) for (S(Jn))2 and similarly write S2(J) for (S(J))2. In general, if Q ~ ]Rn is any cube, we may define a martingale on Q by first scaling and translating it to the unit cube. The martingale maximal function and square functions defined above can then be translated and scaled back to define a martingale maximal function and square functions on Q. In [Fe], C. Fefferman proved a sufficient condition for the nonnegativity of a Schr6dinger operator. In the course of this proof is the following lemma: Suppose f is supported on a dyadic cube Q and IQ f(x)dx = O. Then for all v ;::: 0, p > 1,

h

(J*)2v dx :::; Cp,n

h

(S(J))2 Mpv dx.

(2.0.5)

Here Mpv is a variant of the dyadic maximal function:

Mpv

=

sup Q' dyadic xEQ'

( -1,

1

IQ 1 QI

v P dx )f>

Note that if p = 1, this is just the usual martingale maximal function as in (2.0.1). Fefferman noted that (2.0.5) follows readily from well-known results, in particular, the fact that for p > 1, Mpv E A oo , [To, p. 229]. Fefferman then asks if (2.0.5) remains valid for p = 1. In [CWW], Chang, Wilson and Wolff answered this in the negative. However, let us suppose, just for a moment, that (2.0.5) is valid for p = 1. A simple duality argument then shows that for r > 1,

47

2. Decomposition into Martingales

where r' is the index conjugate to rand Cr' is the constant appearing in the Hardy-Littlewood inequality: IIMvllu' : : : Cr,llvIILr,. It is known that as r' ! 1, Cr'

= 0 (r'~l) = O(r). Consequently, 1 {( If I ) 2r < r r TQT JQ 118(1)1100 dx - 'Y r

for some constant 'Y. But then, 1

{

TQT JQ exp

(( fL

If I ) 118(1)1100

2) dx ::::: t;00 IJ'Y k , fLk k k

:::

and since t~ e k , the right hand side converges if fL < 1/'Ye. We conclude, in particular, that there exists constants fL and C such that if Ihl JQ f(x)dx = 0 then Ihl JQ exp(fL( Ilsgjlloo )2)dx ::::: C. While our reasoning here is correct, this last conclusion is dubious since our argument is based on the incorrect assumption that (2.0.5) is valid for p = 1. Nevertheless, despite the failure of (2.0.5) for p = 1, this last inequality is true. To correctly prove this, we begin with the following inequality. Theorem 2.0.1 8uppose {fm} is a real valued dyadic martingale on Q fa = 0 and with limit function f. Then for all A?: 0,

I~I

k

exp

(Af - ~2 8

2

(1)) dx:::::

~ ]Rn

with

(2.0.6)

1

This inequality follows from the exponential martingale as in Neveu [Ne], Stout [Sto] or Chang, Wilson and Wolff [CWW]. We present the proof here for the convenience of the reader. Proof: Without loss of generality, we may assume that Q is the unit cube Q = [0,1] x ... x [0,1]. As above, we let Fm denote the O"-field generated by the set of all dyadic cubes in ]Rn having sidelength 2- m . Fix A and for m = 1,2, ... define

and define qo = 1. It is elementary to check that qm is F m measurable and E(qm!Fm-l) = qm-l· Thus, the qm are a martingale and consequently, Q qmdx = 1 for every m. We need the following elementary inequalities. Suppose ¢ is a measurable function on a probability space with ¢ dfL = O. Then

J

J

48

2. Decomposition into Martingales

(To see the left inequality, express cp(x) as a linear combination of Ilcplloo and -llcp/loo, use the convexity of the exponential function, then integrate. The right inequality follows by comparison of Taylor series.) An application of this to Adj on Qj(x) E Fj - 1 yields E(e Adj !Fj-1)(X) :::; exp (

~2I1XQj (x)dj II~ )

.

Then, for each m, 1=

k

qm dx

~ 1Q exp (Afm(x)) exp (- ~2 fIlXQj(X)dj/l~) dx =

k

exp (Afm -

J=1

~2 S2(Jm)) dx.

Letting m ---t 00 completes the proof. Note that if we apply Theorem 2.0.1 to {- fm} and add, we obtain the same inequality with If I in place of f and with a 2 on the right hand side. Further note that if fo i:- 0, we may apply this to the martingale {fm - fo}. With these observations, we then take A = 11 8 (7)11;' and apply Chebyshev's inequality to obtain: Theorem 2.0.2 Suppose {fm} is a dyadic martingale on Q f· Suppose IIS(J)lIoo < 00. Then for all 11 > 0,

~ ]R.n

with limit function

-112 ) I{x E Q : If(x) - fol > 11}1 :::; 2exp ( 2I1S(J)II~ IQI·

(2.0.7)

By simply integrating the conclusion of Theorem 2.0.2 we obtain: Theorem 2.0.3 There are constants C 1 and C2 such that whenever {fm} is a dyadic martingale with limit f on Q and II S (J) II 00 < 00, then (2.0.8)

In Chang, Wilson and Wolff [CWW], the above martingale inequalities were used to prove a version of (2.0.8) for the Lusin area integral: Theorem 2.0.4 Suppose IIAa:flloo < 00. There are constants C1 and C2 depending only on a and n such that for all cubes Q C ]R.n, (2.0.9)

2. Decomposition into Martingales

49

The estimate (2.0.9) is reminiscent of the John-Nirenberg theorem for BMO functions. In fact, previous to this result it was already known that if IIA.(J) 1100 < 00 then I E BMO and IIIIIBMO ~ CIIA",(J)lloo. This can be shown using a simple duality argument. The John-Nirenberg inequality then yields the weaker result which has C1 11 - IQI/IIA",(J) 1100 replacing the quantity in the exponential in (2.0.9). The statement (2.0.9) is much deeper and the proof much more difficult. For this, Chang, Wilson and Wolff write I using an integral formula which expresses I in terms of its harmonic extension to the upper half space. This is known as the Calderon reproducing formula. Once I is expressed in this way, they write I as a martingale and, after some technical estimates, ultimately reduce the estimate (2.0.9) to the martingale version (2.0.8). For our results here, we will need to do more than this. We will need to show how to relate a harmonic function on the upper half space to a "truncated" version of a Calderon-type integral formula. In other words, we need to know more than just the fact that the boundary function can be reproduced using an integral formula. We will need to understand a harmonic function on lR++1 in terms of this formula. Matters are further complicated by the fact that we will often be concerned with harmonic functions u which have A",u(x) = 00 on sets of positive measure and such functions are not the harmonic extensions of integrable functions. Thus, certain tools that require integrability, most notably the Fourier transform, can no longer be used as they were in Chang, Wilson and Wolff. Unfortunately, all this requires more than just a few modifications of their results and the consequence is this chapter. We remark that ours is merely one point of departure taken from [CWW]. Numerous authors have extended this work in various ways. C. Sweezy [Sw] shows a version of Theorem 2.0.4 where A",I is defined in terms of an extension u of I which satisfies Lu = 0, where L is a certain type of second order elliptic operator. J. Pipher [Pip1] shows a double dyadic martingale version of Theorem 2.0.1 and uses this to give a bidisc version of Theorem 2.0.4. In [Pip2l, an equivalent formulation of the martingale inequality (2.0.6) is given. This takes the form of an integral inequality which does not involve exponentiation. M. Wilson [Will, [Wi2l, [Wi3] gives a detailed ~udy of th~inequality J 1I1 2 v dx ~ C J S(J)2W dx, both in the case when w = Mv, where M is some maximal operator, and in the case of arbitrary pairs of weights v and w. This chapter naturally breaks into three sections. In Section 2.1, we consider sums of "atoms" . Here an atom will simply be a compactly supported C 1 function with mean zero. We will approximate such a sum by a martingale and show how to estimate the square function of the martingale by a square function formed from these atoms. In Section 2.2, we show how to write a harmonic function as a sum of such atoms and show that the square function formed from the atoms is dominated by the Lusin area function. A combination of the results of these subsections will give our desired approximation. Finally, in Section 2.3 we present a well-known gradient estimate for harmonic functions which will allow us to control the errors in this approximation.

50

2.1

2. Decomposition into Martingales

Square function estimates for sums of atoms

We first recall our previous notation: F = {[k I 2- n , (kl + 1)2- n ) x· .. x [k n 2-n, (k n + 1)2- n ) : k 1 , ... ,kn E Z, n E {0,1,2, ... }} denotes the dyadic cubes in lR n of sidelength ::; 1 and F m denotes the cubes in F of sidelength 2- m , m = 0, 1,2, .... We now suppose that to each Q E F is associated a C 1 function gQ which satisfies: (2.1.1) and (2.1.2) For m 2: 1 define

and a martingale f

=

{fm}~=o by

fm

=

QEF 2-=+1 <:l(Q)<: 1

E(AmlFm), m 2:1 and fa

= 0.

(2.1.3)

The fact that 1 is a martingale follows immediately from (2.1.3) and (2.1.2). Of course, this is slightly inaccurate since martingales are defined on probability spaces and the 1m here are defined on all oflRn. However, if we restrict our attention to a cube of unit sidelength, then the fm are a martingale on such a cube. Since lR n is a countable union of such cubes, the martingale results we will use will be easily transferable to the fm of (2.1.3), and because of this, we will persist in this small abuse of terminology. For a precise description of how to deal with martingales in such measure spaces, see [Va]. In addition to the square function of the martingale fm, we also define

Sm(X) is itself a type of square function. Lemma 2.1.1 There are constants C 1 and C 2 depending only on n such that for all m 2: 1, (2.1.4) and

(2.1.5)

51

2.1 Square function estimates for sums of atoms

Proof: We first prove (2.1.4). Suppose x Ifm(x) - fm-1(x)1

=

E

Qm

E

Pm for each m. Then

IIQ1ml l= Am (y)dy - IQ:-11 l=-l Am-1(y) dy l

~ IIQ1ml l= Am (y)dy -

IQ:-11 k=-l Am(Y)dyl

+ IIQ:-ll k=-l Am (y)dy - IQ:-11 k=-l Am-1(y)dY ~ yn£(Qm-1) 11\7 AmIILOO(Q=_d'

!

Here, to obtain the last inequality, we have estimated the first term in the previous sum using the integral mean value theorem. The second term in that sum vanishes by (2.1.1) since for y E Qm-l, Am(Y) - Am-1(Y) = gQ=-l (y). For Y E Qm-1 we have

I\7 Am (y) I =

L

!

\7 gQ (y) !

Q,Q=_l<:::QEF

=

t;

1

XQ(y)\7gQ(y)1

2-=+1 ,,£(Q)<1 < QEF

2-=+1,,£(Q)<1 so that

QEF

2-=+1 ,,£(Q),,1 Thus, again with the notation x E Qj E Pj for each j, we have

(SUm)(X))' <; n

=n

<; n

~ 22(~-1)

~ (

i( Q,-')'

CH~Q)$, IIV

t; II\7gd~XQ(X)) 1,,£(Q),,1

go II;'XQ(X)) ,

2

2- k +

~ C+~Q'" 2(;-~;;IQ)) C+~Q)$, 2~\~!) IIVgQj[ XQ(X))

52

2. Decomposition into Martingales

2 L (k_l L 2k-1

=n m

j

k=l

= 2n

)

~

2- k

~ 2- k

00

+

+ 1 S£(Q)SI

mk-1£(Qo)11

\lgQj

m £(Qo)11

= 2n ~ k~l 2k~1 :::; 4n

£(Q) II 112 XQ(X) ) 2k - 1 \lgQ

QEF 1 S£(Q)SI

~~~~ lI\lgQI[ XQ(X)

= 2n {;~ 2k~1 m-1

L

(

J=O

112

00

\l gQj

XQj(X)

112 00

XQj (X)

£(Q) II 112 ~ ----:j\l gQj 00 XQj (x)

m-1

L

= 4n

£(Q?II\lgoll~XQ(X).

QEF 2- m + 1 S£(Q)SI

This gives (2.1.4) with constant C 1 Fm we have

Ifm(x) -

= 2y'n.

We now prove (2.1.5)0 For x E

Qm

Am(x)1 = IIQ1ml 10m Am(y)dy - Am(X)1

:::; IQ1ml 10m IAm(Y) -

Am(x)ldy

:::; y'n£(Qm)ll\l AmIIUX'(Qm) :::; v'n£(Qm-1)11\l AmIILoo(Qm_d

L

1

:::; v'n 2m - 1

II\lgollooXQ(X)

QEF 2- m + 1 S£(Qbl

<;

:::;

v'n2"~-' L+ff:(Q'" ~~~1 ) L+~(QJ<' e(Q)'"V9QI~XQ(X)) 1

v'n2~-1 (22m-1)~

(

L QEF

2-=+19(Qb 1

= 2~ v'nSm(X) which is (2.1.5).

£(Q)211\l9oll~XQ(X))

1

2

1

E

53

2.1 Square function estimates for sums of atoms

We can now explain our strategy a little more clearly. In the next section, we will show how to write a harmonic function u as a sum of atoms so that essentially, with the notation above, u(x, 2- m ) ~ Am(x) with an error bounded by Aau(x). We will also show that Sm(x) is dominated by Aau(x). These two facts combined with (2.1.5) give an approximation of u(x,2-m) by the martingale fm with an error which is O(Aau(x)). Unfortunately, this is an oversimplification of the situation and there are several technicalities with we must contend. The problem arises when we write the harmonic function u as a sum of atoms. We will not be so lucky as to obtain a decomposition where the atoms satisfy (2.1.1). Such atoms in our case will be supported on cubes slightly larger than Q. This difficulty can be overcome and the result will be that we will approximate u by a finite sum of martingales rather than just a single martingale. Conceptually, this does not really change the strategy outlined in the previous paragraph, but it does make the notation somewhat unpleasant. Thus, we will need to consider "atoms" of the type that will arise from our decomposition of harmonic functions. We now suppose that to each cube Q E F (recall F is the set of dyadic cubes in IRn of sidelength :::; 1) is associated a 0 1 function >"Q which satisfies: supp >"Q

<:;;;

Q where Q is concentric with Q with R( Q) =

2R( Q)

(2.1.6)

and

r

IlRn

>..q(x)dx

= O.

(2.1.7)

For x E IRn define FX = {Q + x: Q E F} and F;, = {Q + x : Q E Fm}. (Recall Fm is the subset of F of cubes of sidelength 2-m.) The following is from Chang, Wilson, and Wolff [CWW]. Lemma 2.1.2 There is a positive integer N, Ej of F such that {Q E F : R(Q) :::; ifQ E Ej, then

Q <:;;; Q'

Xl, ... ,XN

E IRn and disjoint subsets

~} = j0l Ej,

(2.1.8)

for a unique Q' E FXj with R(Q')

and if Ql, Q2 E Ej and Ql

#- Q2,

= 8R(Q),

then Q~

#- Q;.

(2.1.9) (2.1.10)

Proof: First note that if (2.1.9) holds and if Q E FXj, then there are at most 8n cubes J in Ej with J' = Q. Thus, if we find Ej satisfying (2.1.9), we can further decompose these Ej to find Ej satisfying (2.1.10). So it suffices to find Ej

satisfying (2.1.9).

2. Decomposition into Martingales

54

First consider the case n = 1. Then let N = 2, Xl = ~, and X2 = O. For Q = (;~1, 2';,.)' m, k E Z, m :2': 3, let Q E W if k = 0 or 1 (mod 8), and let Q E B2 otherwise. Then (2.1.9) is clear for Q E B2. Suppose now (k2~1, 2';,.) = Q E B1. We want to show that Q is contained in a cube in F X 1 with sidelength 2~' In other words, we want to show that Q contains no point of the form ~! + ~, where '71 Suppose not . Then 2"' k-1 - 2m+1 1 < 2'" 8p + "3 1 < 2m k + 2",+1 1 W h'1Ch'Imp l'1es p E tL.. ~ - 136 < p + < ~ + 116 , But this is impossible if k = 0 or 1 (mod 8). In the case n > 1, write Q E F as Q = Q1 X ... x Qn, where each Qj has the form Qj = (k2:1, ~). For a subset (J ~ {I, ... ,n} set

2n;-3

BU

=

{Q: Qi E B1 if i E (J, Qi E B2 if it/:-

(J}.

For such a (J define Xu = (Xl, .. ' ,X n ) E ~n by Xi = ~ if i E (J, Xi = 0 if i t/:- (J. Then the 2n sub collections BU and the points Xu satisfy (2.1.9). We direct the reader to Wilson [Wi3] for a refinement of this lemma. Such splittings are also reminiscent of techniques used in the study of interpolating sequences; see Garnett [Ga, p. 416]. Now fix j, 1 ~ j ~ N, and for each Q E FXj let g(j)(x) = {AQo(X)

0

Q

if Q = Q~ for some Qo E Bj otherwise.

(2.1.11)

Then (2.1.12) and

h

(2.1.13)

gg)(x)dx =0.

Thus, (2.1.12) and (2.1.13) put us back in the situation of (2.1.1) and (2.1.2). We then define

AQ(X) 2-",-29(Q)S2- 3

and martingales f(j)

= {f~·e}~=o

(2.1.14)

= QEPXj 2-",+1 S£(Q)S1

QEBj

by

(j) IFXj) m > 1 and f(j) = 0 f m(j) = E(A m m' JO .

As before we define

S(f$,{»)(x) = and

(~lld~)XQk(X)II~ )

(2.1.15)

1

'2

(2.1.16)

2.2 Decomposition of harmonic functions

55

(2.1.17) As an immediate consequence of Lemma 2.1.1 we have:

Lemma 2.1.3 There are constants C 1 and C 2 depending only on n such that for each j E {I, ... ,N} and m 2:: 1, (2.1.18) and (2.1.19) We remark here that we have continued the small abuse of terminology in calling the f;,{) of (2.1.15) martingales, even though they are defined on all of ]Rn. We have assumed that each of the ..\Q(x) in (2.1.14) is supported on Q, the cube having the same center as Q but with twice the sidelength. If we fix a dyadic cube Qo of unit sidelength and in (2.1.14) further restrict the sum to those Q E Bj such that Q has nontrivial intersection with Qo, then each of the corresponding f;,{) given by (2.1.15) are martingales supported on a cube in FXj. Furthermore, a straightforward computation shows that each of these cubes is contained in a cube having the same center as Qo but with three times the sidelength. In other words, this procedure is local: if in (2.1.14) we consider only those XiS in a certain cube, then the corresponding martingales generated by (2.1.15) are all supported in a slightly larger cube.

2.2

Decomposition of harmonic functions

In this section we develop a decomposition of harmonic functions into atoms so that we may apply the estimates of Section 2.1. We note that our decomposition is more or less the well-known Calderon reproducing formula. (See Calderon and Torchinsky [CT]. Also see Frazier, Jawerth and Weiss [FJW] for a detailed explanation of this formula.) However, most presentations of this formula require some global behavior of the functions under consideration, and since we will need local estimates, we will derive a formula suitable for our needs. Let p > 0 be given and fix a function K E coo(]Rn) which is positive, radially symmetric and decreasing, and supported in B(O, p). We also assume JlRn K(x)dx = 1. With such a K chosen we put

P(x) = -

oK LXi ax(x) - (n -l)K(x). n

i=l

t

(2.2.1)

Note that P(x) is also supported in B(O, p) and a simple integration by parts shows that JlRn P(x)dx = 1.

2. Decomposition into Martingales

56

Now set q(x) = (-XIP(X)

fJK

+ -fJ (X), ... Xl

, -XnP(X)

fJK

+ -fJ (x)) Xn

(2.2.2)

and observe that the support of q is also in B(O, p). By integrating the ith component of q first in the ith variable, it follows that { q(x)dx = (0, ... ,0).

(2.2.3)

JRn

Define Pt(x) = Cnp(f) with a similar definition for qt(x) and Kt(x). Lemma 2.2.1 If u is a harmonic function on ~~+!, and if we define

v(x, t)= (

then for

JRn

°< c < f we have

v(x,f) - v(x, c)

=

Ii JRn(

(2.2.4)

Pt(x - y)u(y, t)dy

qt(x - y) . V' yu(y, t)dydt

(2.2.5)

e;

+ (

JRn

fKt(x - y) fJfJu (y, f)dy t

{ cKe;(x - y/fJtU (y, c)dy.

JRn

We remark that since P has compact support and u(·, t) is locally integrable, the integral defining v makes sense. Proof: We first note that

~(~p(X-Y)) fJt t n t and

=

~ ~ ~ fJy· i=l

1

fJ tn P (X-y) - t - = fJt

•

(Xi-Yip(X-y)) t n+! t

(1

tn-l K (X-y)) -t,

both of which are easily proved by straightforward computation. Now apply the fundamental theorem of calculus to v(x, t) and get v(x, f) - v(x, c) =

i 1Rn Ii Ln ! C~ +I 1 Rn t -fJ fJ

t

e;

=

i

e;

y)

1 P (x -:;; -- - u(y, t)dydt t t P (x

~ y) )

u(y, t)dydt

1 (x y) -fJ fJu (y, t)dydt -:;;P -- t t t

(2.2.6)

(2.2.7)

2.2 Decomposition of harmonic functions

57

and by (2.2.6) and (2.2.7) this becomes = l ei ~ ~ e

+

IRn i=l

a

t

tn - K

IE

IE

- t-

JlRn

1~

l e1 IE

IE

+ [

JlRn

--(y t)dydt ot 2 '

eKe(X - y) ~u (y,f)dy -

= -le

au at (y, t)dydt

Xi - Yi P (x - y) au ( t)d dt t o - y, Y y,

ei -1K - IRn t n - 1 t

+ [

t

~ t n +!

IRn i=l

IE

+

y,

l eJlRn[ at (1 1 (x - y)) 1~ (x - y) 02u l

= -le

-

a_(x--y. (x-y)) -a 'n+l ' P -u(y, t)dydt

c:Ke(X - y) °atU (y, c:)dy

Xi - Yi P (x - y) au ( t)d dt t o · y, Y y,

~ t n +1

IRn i=l

IRn

[

JlRn

ut

1

t

n-l

K

(x---y) L n

t

i=l

eKe(X - y) ~u (y, e)dy -

ou 2 ~ 2 (y, t)dydt uYi [

JlRn

ut

c:Ke(X - y) °atU (y, c:)dy

Here the last equality is just a consequence of the harmonicity of u and the next to last equality is just integration by parts. Another integration by parts in the second term above gives

v(X, e) - v(x,c:)

I e1 y. (x - y) l Ln 1 (OK ~

=-

e

IRn

ei

-

e

TIl>

~n

X· -

~ 'n+!' P i=l t

i=l

[_

t n +1

+ JlRn c:Ke(x -

=

l

e

+ -

y)

-t

1 IRn

eKe(X - y) c:Ke(X

1)

(X-y)) ( -- -(y, au -t)dydt oy-'t t toy-t

au

_

[

at (y, c:)dy - JlRn c:Ke(x -

ei qt(X - y) . "!l(y, au t)dydt IRn uy

Ln

au uy,

~(y,t)dydt

a;; (y, e)dy au

-y)~(y,c:)dy

ut

which completes the proof of Lemma 2.2.1.

y)

au

at (y, c:)dy

58

2. Decomposition into Martingales

Throughout we will use the term caloric function to denote a function u(x, t) defined on jRn+l which satisfies the heat equation ~~ = ~u. Caloric functions are also sometimes called parabolic functions. Here is a corresponding formula for caloric functions.

Lemma 2.2.2 Let u(x, t) be a caloric function on jR~+l and define u(x, t) Let K t (x) be as above and set v(x, t)

=

r

ilR

= u(x, t 2 ). (2.2.8)

Kt(x - y)u(y, t)dy

n

and q(x) Then for 0

< c < if

=

( -x1K(x)

8K + 2 8Xl (x), ...

,-xnK(x)

8K ) + 2 8x n (x)

.

we have

v(x, c) - v(x, if)

=

if r E:

JJR.n

qt(X - y)'\1 yu(y, t)dydt.

(2.2.9)

The proof of this uses the same integration by parts ideas as the proof of Lemma 2.2.1 and is in fact somewhat shorter; see [BM2] for details. We remark that although we will not define the nontangential maximal function and area function for caloric functions until later, it is natural to use parabolic shaped cones when defining these. The change of variables t ------> t 2 is used to keep the same dilation structure for the kernels used for caloric functions as those used for harmonic functions. Notice too, that (2.2.9) does not involve the boundary terms that appear in (2.2.5) and this makes the proofs of the results on caloric functions easier than those for harmonic functions. We now show how to decompose harmonic functions into martingales. For Q E F (recall that F is the set of dyadic cubes in jRn of sidelength ::; 1) we define TQ

= { (y, t) : y

E Q,

£~~)

::; t ::;

£~~) }

and AQ(X)

Since supp q

<:;;;

=

JJ

TQ

qt(x - y) . '\1 yu(y, t)dydt.

B(O, p), then supp AQ

£(Q) = 2£(Q). By (2.2.3) we also have

<:;;;

Q where Q is

(2.2.10)

concentric with Q and

2.2 Decomposition of harmonic functions

59

Thus we are exactly in the situation of Section 2.1. In particular, if we define the functions gg)(x) as in (2.1.11), A~)(x) as in (2.1.14), f~) as in (2.1.15), S(J~))(x) as in (2.1.16), and S~)(x) as in (2.1.17), then Lemma 2.1.3 holds for these. For the rest of this monograph we will assume that the AQ arise from a harmonic function u as in formula (2.2.10) and that the gg), A~), etc., are defined using (2.1.11), (2.1.14)-(2.1.17) using these AQ. Consider Q E FXj and recall that gQ == 0 unless Q = Q~ for some Qo E Bj. In the latter case we have

Since 8£(Qo)

= £(Q)

we obtain

(2.2.11) where C is a constant depending on p and q. Now set

Recall Q' denotes the unique element of Define

FXj

having Q' :2

Q with £(Q') = 8£(Q).

Summing over all Q E FXj with 2- m +l ~ £(Q) ~ 1, we conclude from (2.2.11) (using the fact that gQ == 0 unless Q = Q~ for some Qo E Bj) that S~)(x) ~ CA~)u(x).

2. Decomposition into Martingales

60

Combining this estimate with lemma 2.1.3 immediately yields

Lemma 2.2.3 There are constants C 1 and C 2 depending only on n, p and q such that for each j E {l, ... ,N} and all m 2: 1, (2.2.12)

and (2.2.13) In the next chapter we will show that L:f=1 AM) (x) is approximately v(x, em) for em = 2- m - 2 /4p, which in turn we will show is approximately u(x, em). Thus, L:f=1 AM) (x) is approximately u(x, em). An easy argument will show that AM)u(x) is dominated by a truncated version of the Lusin area function at x. These facts, combined with (2.2.12) and (2.2.13), then give that u(x, em) is approximated by L:f=1 f~) with an error controlled by a truncated version of the Lusin area integral. Rather than do these final approximations now, we do them in subsequent sections; equations (2.2.12) and (2.2.13) are the most convenient point of departure for both the LIL which follows in the next chapter and the good-A inequality which follows in Chapter 4.

2.3

Controlling errors: gradient estimates

The previous discussion demonstrated how harmonic functions can be approximated by martingales, so naturally, we will need to control the errors in this approximation. This martingale approximation technique is not the only method we will use. In subsequent chapters we will present several Green's theorem arguments. These arguments always give rise to boundary terms which may also be thought of as "errors" in an approximation. These too, will need to be controlled. The sole result of this section is a well-known lemma which allows us to control these "errors" in all situations throughout this work. This lemma is used throughout this monograph and we present it here for the convenience of the reader. We first need a definition. For a > 0, h > 0 and x E lRn , recall that the truncated cone r~(x) was defined in Definition 1.8.1 by

r~(x)

= {(s, t) E lR+.+ 1 : Ix - sl < at, 0 < t < h},

(2.3.1)

Now, for any a > 0, 0 :::; k < h, and x E lR n , we define the doubly truncated cone r~(x; k) by r~(x; k) = {(s, t) E lR+.+1 : Ix - sl

< at, k < t < h}

(2.3.2)

2.3 Controlling errors: gradient estimates

When k = 0 in (2.3.2), we simply write r~(x; 0) Stein [St4, p. 207].

=

61

r~(x). The following is from

Lemma 2.3.1 Suppose a > (3 > 0, 0 :::; k < h, 0 < ry < 1 and u is harmonic on -lh r~ (x,ryk). Then: (i)

If lui:::; 1 in r~-lh(x; ryk), then tlV'u(s, t)1 :::; C in r~(x; k).

(ii)

If

fr;,,-lh(x;,k)

sl-nlV'u(s, t)l2dsdt :::; 1, then tlV'u(s, t)1 :::; C in r~(x; k).

Here C is a constant which depends only on a, {3, ry, and n. Proof: We first observe that there exists a constant C l depending only on a, {3, ry, and n, such that if (s, t) E r~(x; k) then B((s, t), CIt) ~ r~-lh(x; ryk). (Here B((s, t), CIt) is the Euclidean ball of radius CIt centered at (s, t).) Let

u(w, z)

=

1

1

jRn+l

+

-+1

((w,z) - (~'17)) r

u(~, 17)d~d17,

(2.3.3)

whenever (w, z) E r~-lh(x; ryk) and r is chosen so that the integrand in this integral has support strictly contained in r~-lh(x; ryk). Fix (s, t) E r~(x; k) and set r = C l t/2. Then (2.3.3) holds with this r for all (w, z) in a neighborhood of (s, t). Differentiate (2.3.3) with respect to wand z, evaluate at the point (s, t) and take absolute values to obtain: lV'u(s, t)1 :::; C 2 C l sup{lu(~, 17)1 : (~, 17) E B((s, t), CIt)}. This yields (i). Again consider (s, t) E r~(x; k) and set B = B((s, t), CIt). Applying the mean value theorem (Corollary 1.1.5) to the first derivatives of u yields

lV'u(s, tW :::; C 2 t- n - l llV'u(~, 17)12d~d17, where C 2 depends on nand C l . Note that for

(~,

17) E B, 17 :::; C 3 t so that

t21V'u(s, t)12 :::; C2t l - n llV'u(C 17Wd~d17 :::;

C4llV'(~, 17)1217l-nd~d17

:::; C4

r

Jr7"

-lh

(x;,k)

17l-nlV'u(~, 17)12d~d17

which finishes the proof of (ii). The analogous lemma for caloric functions, that is, solutions of the heat equation, is due to Hattemer [Hal. We will discuss this at an appropriate point later. We will also need a version ofthis lemma for the D-functional. This is Lemma 5.1.1 below.

Chapter 3 Kolmogorov's LIL for Harmonic Functions The following, in the words of K.L. Chung [Chu2, p. 231]' "is a crowning achievement in classical probability theory." Theorem 3.0.1 (Kolmogorov's LIL [Ko])

Let Sm =

m

L

X k where {Xk} is a se-

k=l

quence of independent random variables. Let Sm be the variance of Sm. Suppose that Sm ---t 00 and that

(Ko) for some sequence of constants Km

. 11m sup m----too

10. Then, almost surely,

Sm = 1. )2s;'log log Sm

(3.0.1)

This beautiful result, first proved by Khintchine [Ki] for Bernoulli random variables, grew out of the efforts of Hausdorff (1913), Hardy and Littlewood (1914) and Steinhaus (1922) to obtain the exact rate of convergence in Borel's theorem on normal numbers. Over the years there have been many efforts to obtain similar results in various settings in analysis. Some of the existing results include the LIL for lacunary series of R. Salem and A. Zygmund [SZ2], P. Erdos and I. Gal [EG], M. Weiss [We] and S. Takahashi [Ta] as well as N.G. Makarov's [Mal] result for Bloch functions. In this chapter we will prove an analogue of Theorem 3.0.1 for harmonic functions in lR~+1. As we will see in Chapter 6, this LIL implies, up to constants, the analytic LILs mentioned above. On the probability side, Kolmogorov's LIL has been extended in many directions with applications in many different fields. We refer the reader to the survey article by N.H. Bingham [Bi] which contains nearly 400 references on this subject. In 1970, W. Stout [Sto] extended Theorem 3.0.1 to the martingale setting. It is Stout's theorem, together with the invariance principle presented in Chapter 2, that we shall use for our upper bound result, Theorem 3.0.4. The proof of our lower bound for harmonic functions follows Stout's proof of the lower bound for martingales in that we use conditional Borel-Cantelli type arguments. Unfortunately, the argument is much more technical in our setting. 63 R. Bañelos et al., Probabilistic Behavior of Harmonic Functions © Birkhäuser Verlag 1999

64

3. Kolmogorov's LIL for Harmonic Functions

Let us now recall Stout's [Sto] results in the form that we will use below. As in Chapter 2, we let {fm} denote a dyadic martingale and let S(fm) be the square function of the martingale as defined in by (2.0.2) in Chapter 2. By the results in [BG 1],

{f* < oo} a~. {S (f) < oo} a~. {limit f n exists},

(3.0.2)

where by "a~'''we mean that the sets are equal up to sets of probability zero. The LIL provides detailed information on what happens on the complement of these sets. Theorem 3.0.2 (Upper-Half LIL for martingales) There is a positive constant C 1 such that

(3.0.3)

almost surely on {S (f) = oo}. Theorem 3.0.3 (Lower-Half LIL for martingales) Suppose the sequence {d m isfies

almost surely on {S(f) = oo} for some sequence of constants Km constant C2 such that · sup 11m

fm

m-----+oo JS2(fm) log log S(fm)

1 O.

}

sat-

There is a

> _ C2

(3.0.4)

almost surely on {S (f) = oo}. Actually, Stout's result is proved for the conditioned square function 1

a(fm) =

(~E(d~lFk-1))"

in place of S(fm). With this square function the limsup is almost surely equal to 1 as in (3.0.1). As mentioned earlier, a(fm) and S(fm) are pointwise comparable for dyadic martingales and hence (3.0.3) and (3.0.4) follow from Stout's result. Stout's result, first proved in [Sto] using the exponential martingale (see also Neveu [Ne, p. 154] and Chang, Wilson and Wolff [CWW]) can also be derived from the invariance principle in Philipp and Stout [PS2]. See Theorem 6.1.4 in Chapter 6 below, and Hall and Heyde [HH, p. 118] for more in this direction.

3. Kolmogorov's LIL for Harmonic Functions

65

We remark that the condition (K1) in Theorem 3.0.3 is absolutely essential as can already be seen from the sharpness of (Ko) in Kolmogorov's theorem which will be discussed below. This LIL is typical in that the upper-half requires few hypotheses, but the lower-half can only hold with some additional control of the increments. This will also be the case for our LIL for harmonic functions. Finally, in the case of continuous time martingales, we have (here the condition K1 is automatically satisfied), X*

lim sup

t

J2S;(X) loglogSt(X)

t--+oo

=1

(3.0.5)

almost surely on {S (X) = oo}. This result follows from the Brownian motion case by time change as in Durrett [Dur, p.77]. However, to illustrate the use of subgaussian bounds, we shall prove the upper half of this LIL below. We remind the reader of the definition of the doubly truncated cone: r~ (x; t) = {(y, s) E 1R~+1 : Ix - yl < as, t < s < h}. With this, we define the doubly truncated area function and doubly truncated maximal function of a harmonic function u on 1R~+1 by:

A~u(x; t) =

(

1

r

Jr~(x;t)

sl-nIV'u(y, S) 12 dYdS)

"2

and

N~u(x; t)

= sup{lu(y, s) I : (y, s)

E r~ (x;

tn.

(3.0.6)

For typographical convenience, we will write r~(x), A~u(x), and N~u(x) for r~(x; 0), A~u(x; 0) and N~u(x; 0), respectively. We recall the analysis versions (see Chapter 1, Section 8) of the martingale results (3.0.2):

{x

E IRn : A~u(x)

< oo}

a~.

{x

E IRn : N~u(x)

a~.

{x

E

IRn

:

< oo}

lim

(y,t)~(x,O)

(3.0.7)

u(y, t) exists and is finite}.

(y,t)EI'~(x)

Theorem 3.0.4 (Upper-Half LIL for harmonic functions) Fix 0 < (3 < a and o < "I < 1. There is a positive constant C~,/3,,,!,n such that if u is harmonic on IRn +1 then

+ '

lim sup lu(y,t)1 < C1 (y,t)~(x,O) J(Alu(x; "It)) 2 log log(Alu(x; "It)) - o.,/3,,,!,n (y,t)EI'~(x)

for almost every x

E

{x

E IRn

:

A~u(x) = oo}.

(3.0.8)

3. Kolmogorov's LIL for Harmonic Functions

66

Let u be harmonic on

Theorem 3.0.5 (Lower-Half 11L for harmonic functions) < t < 1, let KAu)(x; t) be defined by

lR~+1. For 0

I 2 I 2 (A;u(x;t))2 (AI ())2) (Aau(x; t)) - (Aau(x; 2t)) = Ka(u)(x; t) 1 1 (

og og

ee

+

a U X;

t

(K2)

and set Ka(u)(x) = 1 + lim sup Ka(u)(x; t). iLO

There is a positive constant C,;,n such that . 1Imsup tLO

for almost every x

E

u(x, t) > C,;,n -r~~~7 J(Az,u(x; t))21og1og(Az,u(x; t)) JKa(U) (x) {x

E

lR n : A;u(x) =

00

(3.0.9)

and Ka(u)(x) < oo}.

A few remarks are in order concerning these two theorems. If we consider an 00 and Ka(u)(x) < 00, then (K 2) shows that A;u(x) is asymptotically equal to A;u(x; "It) as t 1 o. Consequently, for such an x, we may take "I = 1 in Theorem 3.0.4. Thus, we also have an upper bound in (3.0.9) for such x's. As is well known (cf. Burkholder [Bu4]), for harmonic functions s'\lu(y, s) plays the role of the martingale difference sequence d k • Thus, if we define da(u)(x; t) = sup{sl'\lu(y, s)1 : (y, s) E f;(x, t)\f;(x, 2t)}, then a more natural analogue of (Ko) would be to require that

x for which A;u(x) =

for almost every x E {x E lRn : A;u(x) = oo}. However, this condition is actually stronger than what we require in Theorem 3.0.5. Indeed, if (K3) holds then by integrating we obtain

Our next result is an example which shows that Theorem 3.0.5 is sharp even under the stronger condition (K3). (This is possible because in this example (K3) and (K 2) are equivalent; this is a consequence of the uniformity of I'\lu I in this example.) We shall do this in the setting ofthe unit disc. First, for a> 0,0 < p < 1 and f) E [0, 27r] define the truncated cone

ra(f);p) = {rei'P: 0 < r < p,

If) - 'PI < ~(1- r)}

(3.0.10)

3.1 The proof of the upper-half

67

and the Lusin area function of the harmonic function u in the unit disc D by 1/2

Aau(B; p)

=

(

( Jr",«(};p)

lV'u(zWdxdy

and set da(u)(B; p) = sup{(1 - r)lV'u(re i'!') I : rei'!' we simply write Aau(B) for Ac.u(B; 1).

E

)

r a(B;p)}.

As before, if p = 1

Theorem 3.0.6 Let K > O. There is a harmonic function u in D with Aau(B) = for all B E [0,27r] and such that for all Po S p < 1 and all B E [0,27r],

00

lu(pe i(}) I < Ca . )(Aau (B;p))2loglog A au(B;p) - vIR Furthermore, this function satisfies

for all such p and B. Theorem 3.0.6 shows that the relationship given between the function Ka(u)(x) and the right hand side of (3.0.9) is best possible. Our harmonic function in Theorem 3.0.6 is modeled after the example of Marcinkiewicz and Zygmund [MZ] which shows the sharpness of the Kolmogorov condition (Ko) in Theorem 3.0.1 as well as the sharpness of (K 1 ) in Theorem 3.0.3. Indeed, Marcinkiewicz and Zygmund use Rademacher functions; we use lacunary series.

3.1

The proof of the upper-half

The traditional way to prove the upper bound in the Kolmogorov 1IL is via subgaussian estimates, which are usually obtained from estimates in the central limit theorem, and Borel-Cantelli arguments. For those readers not familiar with these Borel-Cantelli arguments, we present the proof of the upper half of (3.0.5). The crucial estimate for this, as well as for the good->. inequalities below, is an estimate like the following Bernstein-type inequality ([RY, p. 145]): Key Estimate 3.1.1 Let {Xd be a continuous martingale with IIS(X)lloo Then for all >. > 0, P{X*

<

00.

> >'} S Cexp (- 21IS(~)IIZx,) .

Before we discuss the proof of this estimate, let us show how it leads to the upper half of (3.0.5). Let 'T} > 1 and for k = 1,2,3, ... define the stopping times Tk =

inf{t : St(X) >

y0]k}.

68

3. Kolmogorov's LIL for Harmonic Functions

It follows then that the martingale the Key Estimate 3.1.1 imply

Xt = X tATk

has IIS(X)lIoo ::;

..;:;ik.

This, and

P{X;k > J(1 + c)21]k log 10g1]k} ::; C exp( -(1 + c) log log1]k) C (k 10g(17))!+c' for any c >

o. Thus 00

LP{X;k > J(1 + c)217 k log log 17k } <

00

k=l

and by the Borel-Cantelli lemma,

X;k ::; J(l + c)21]k 10glog1]k eventually, with probability 1. That is, for almost every w in our probability space there is a ko, which may depend on w, such that for all k > ko, X;k (w) ::; V(I+c)217kloglog17k. Pick such an w for which we also have S(X)(w) = 00. Then Tk(W) i 00 and S;k (X)(w) = 17 k . Let t be arbitrary large and pick k > ko such that Tk(W) ::; t < Tk+l(W). Then,

X;(w) < X;k+l (w) ::; -v'i'+"€V217k+lloglog17k+l = vI + cy'17J2S;k (X)(w) loglog(1]S;k (X)(w))

::; -v'i'+"€y'17J2S1(X)(w) log log 1](Sl (X) (w)) This shows that X* lim sup t < y'17-v'i'+"€, t-+oo v2S1 (X) log log St (X) almost surely. Since 1] > 1 and c > 0 are arbitrary, this gives (3.0.5). The reader can now see why we have called the above result, "Key Estimate". For continuous time martingales, such an estimate is obtained by time changing to Brownian motion and applying the reflection principle. However, there are at least two other ways to arrive at such estimates without changing to Brownian motion - at least not directly. One such way is to use the exponential martingale

and Doob's maximal inequality; we will do this argument below. Another way is to use the inequality: IIX*lIp ::; CpIIS(X)llp with Cp = O(v'P) as p ---+ 00;

3.1 The proof of the upper-half

69

this follows from B. Davis [Dav1]. This inequality for p = 2,4, ... and the power series for the exponential function combine to give an exponential square integral estimate, which by Chebychev's inequality yields the Key estimate. (This same computation was done in the introduction to Chapter 2.) Note, however, that this approach fails to give the sharp constant ~ in the exponential. In the setting of dyadic martingales, both approaches can still be used to obtain the following estimate, Key Estimate 3.1.2, which is the appropriate analogue of Key Estimate 3.1.1. Theorem 2.0.1 provides the "exponential martingale" estimate necessary for one approach and the other can be accomplished using LP estimates due to G. Wang [Wa]. Either approach will result in the following inequality. Since we have already proved Theorem 2.0.1 we will use it. Key Estimate 3.1.2 Let {in} be a real valued dyadic martingale starting at 0 with 118(J)1100 < 00. Then for all A> 0, P{f*

-A2 ) > A} :::; 2exp ( 2118(J)11~ .

Proof: Fix n and let v > 0 and A > O. Then by Doob's maximal inequality (see [Dur, p. 306]) and Theorem 2.0.1 we have:

P{ sup fm > A} l::;m::;n

=

P{ sup exp(vfm) > exp(vA)} l::;m::;n

:::; e:>' E(exp(vfn)) :::; e:>' :::; exp

exp(~2118(Jn)II~)E (exp (Vfn - ~2 (8(Jn))2))

(-VA + ~2118(Jn)II~) .

= A/118(Jn)II~. Then do the same for - fn and add. Finally let n ----t 00. With Key Estimate 3.1.2 we can mimic the above proof of the upper-half LIL for continuous martingales, (3.0.5), to provide a proof of the upper-half LIL for dyadic martingales, Theorem 3.0.2. Such a proof is presented in [CWW]. Note that the constant ~ in the exponential allows us to obtain the upper-half LIL with the best possible constant. So it seems that what is required to adapt the above proof to harmonic functions is a similar Key Estimate for harmonic functions. Such an estimate is easily obtained by applying Chebychev's inequality to the estimate of Chang, Wilson, and Wolff, Theorem 2.0.3. However, notice that in the proof of (3.0.5), we applied the Key Estimate 3.1.1 to the stopped martingales X Tk • In the case of harmonic functions, these stopping times turn out to be Lipschitz domains (see Section 4.2) and thus, what is really needed to adapt the martingale proof to the setting of harmonic functions is a Lipschitz domain version of the Chang, Wilson, Wolff result. Such a result, as we will discuss at the end of Section 4.2, is only known for Lipschitz domains in two dimensions. Thus, in lR.~ we can prove Theorem 3.0.4, Set v

3. Kolmogorov's LIL for Harmonic Functions

70

the upper-half 11L for harmonic functions, by following the proof of the upper bound in (3.0.5), but such an approach is at an impasse in higher dimensions. We will bypass this difficulty by reducing Theorem 3.0.4 to Theorem 3.0.2 using the machinery developed in Chapter 2. This is the reason why we called the results in Chapter 2 an "invariance principle". The classical invariance principles, such as those of Philipp and Stout [PSI]' [PS2], and Hall and Heyde [HH], bypass the exponential square bounds, the "subgaussian" estimates, by approximating by Brownian motion. (See Theorem 6.1.4 below.) We are now ready for the proof of Theorem 3.0.4. Using the notation of Chapter 2, we choose p = (Such precision is not really necessary for this proof. It is only important here that p be much smaller than f3 and any value of p less than or equal to 32~ will work. However, for definiteness we will set p to be

3!.;n'

this.) Let Cm set

=

T:;;'-2

for m

= 1,2, ... , and again with the notation of Chapter 2 N

Vm(x) = LA~??(x) j=1

(3.1.1)

= QEF

2-m-2~t(Q)~2-3

By Lemma 2.2.1 with C = cm and g = 2c1 we have

(3.1.2) The first integral in (3.1.2) is just over B(X,2c1P) since suppK C B(O, p). Since < f3 < a, Lemma 2.3.1 implies that

p

in

2C1K2el (x - y)1

c:;: (y, 2C1)ldy

:::; C,.,.B,')',nA!qU(X; 2/'cd [

Jan K 2q (x -

(3.1.3)

y)dy

The same argument gives (3.1.4) The estimates (3.1.2), (3.1.3), (3.1.4) and Lemma 2.2.3 combine to yield (with

3.1 The proof of the upper-half

71

the martingales {J~)} of Chapter 2) N

Iv(x, Em) -

L f~)(x)1 ~ IVm(X) -

V(X, Em)1

+ IVm(X) -

j=l

N

L f~)(x)1

(3.1.5)

j=l

+ Ca,/3,/"nA~Clu(x; 2}'Ed

~ Iv(x, 2Edl

N

+ Ca,/3,/"nA~Clu(x; }'Em) + C2 L

A~)u(x),

j=l

where as before (3.1.6) As an immediate consequence of their definitions (compare (3.0.6) and (3.1.6)), it follows that

A(j)u(x) < A 2c I E ) m /3 U(X'' m ~ A!Clu(X;}'E m).

(3.1. 7)

(We remark that it is here where we require p < Then (3.1.7) and (3.1.5) combine to give N

IV(X,E m) -

L

f~)(x)1 ~ Iv(x, 2Edl

3!..;n')

+ Ca,/3,/"nA!CI U(x; 2}'Ed

j=l

(3.1.8) Since both Iv(x,2Edl and A;CIU(X; 2},E1) are finite, Lemma 2.2.3, (3.1.7), (3.0.2), and (3.1.8) combine to show that

{x E JRn: A~u(x)

< oo} at {x

E JRn: supIV(X,Em)1 m

< oo}.

(3.1.9)

Likewise, the finiteness of Iv(x,2Edl and A;CIU(X; 2}'Ed combined with Lemma 2.2.3, (3.1.7), Theorem 3.0.2, and (3.1.8) shows that

. hmsup m->oo

IV(X,Em)1

< Ca

J(A~u(x;}'Em))2loglogA~u(x;}'Em) -

/3 /' n

",

(3.1.10)

for almost every x E {x E JRn : A~u(x) = oo}. We now show that (3.1.9) implies

{x E JRn : A~u(x)

< oo} at {x

E JRn : NJu(x)

< oo},

(3.1.11)

3. Kolmogorov's LlL for Harmonic Functions

72

which is one of the containments in (3.0.7), and that (3.1.10) implies the upper half 1IL, (3.0.8). Towards this end, let (y, to) E rb(x) and assume to::; Cl. Let m be such that c m +! < tq ::; Cm. Since "(to::; ,,(cm, we have (3.1.12) By Lemma 2.2.1,

By (3.1.3), (3.1.4) and (3.1.12), it follows that the second and third integrals above are dominated by Ca,/3",nA;U(X; "(to). We need to estimate the first integral. Applying Cauchy-Schwarz inequality we see that the first integral is dominated by

where we have used the fact that conclude that

Cm+l

< to ::;

Cm,

and p < f3 <

C\:.

Thus, we (3.1.13)

To complete the proof we need to replace vex, to) by u(y, to). The function P used in Lemma 2.2.1 may not be positive. However, it has integral 1 and it belongs to Ll. We have

Iv(x, to) - u(y, to)1 =

I~n Pta (x -

::; JlRnr IPto(x =

1

Ix-YI
jj)(u(jj, to) - u(y, to»djjl jj)llu(jj, to) - u(y, to)ldjj

IPta(X - jj)llu(jj, to) - u(y, to)ldjj

::; Ca,/3",n A ;u(x;"(to),

(3.1.14)

73

3.1 The proof of the upper-half

where once again we have used the gradient estimate provided by Lemma 2.3.1. Thus, we have Iv(x, em) - u(y, to)1 ~ Ca,,B,,,nA~u(x; 'Yto) Theorem 3.0.4, as well as (3.1.11), follow from this, (3.1.9) and (3.1.10). We note that estimates (3.1.3), (3.1.4) and (3.1.14) require p < {3, (3.1.7) requires p ~ 32~' These in fact are the only estimates that required any conditions on p and {3. This explains our choice of p. Remark 3.1.3 Theorem 3.0.4 also applies to Stein-Weiss systems of conjugate functions. Indeed, let uo, Ul, ... ,Un be harmonic functions in 1R~+ 1 = {(y, s) : y E IR n , s > O} satisfying the generalized Cauchy-Riemann equations

(C-R) where we have set Yo = s. By a slight modification of the proof of Lemma 2.3.1 we have that for any harmonic function u in 1R+.+l ,

1

r~: (x;-y't)

l ou 12 dyds ~ C

s1-n oYj

1

r~ (x;-yt)

lou

s1-n - 12 dyds AS

+ C1

where h' < h, a' < a, 'Y < "I' < h' and C and C 1 depend on these parameters and on n. Thus if UO, U1, ... ,Un satisfy (C-R), we have

for k = 0,1,2, ... ,n. With these observations we have the following corollary of Theorem 3.0.5. Corollary 3.1.4 Let uo, U1, ... ,Un be a Stein- Weiss system of conjugate harmonic functions in 1R+.+l and set

Then with a,{3 and'Y as in Theorem 3.0·4,

lim sup lu(y, t)1
for almost every x E {x E IR n : A~uo(x) = oo}.

3. Kolmogorov's LIL for Harmonic Functions

74

We now discuss a version of Theorem 3.0.4 for caloric functions. For x we define the parabolic cone in lR.~+l with vertex at x and aperture a by

Pa(x) = {(y, s)

E lR.~+1 :

E

lR.n

Ix - Yl2 < a 2 s}

and its doubly truncated version by P~(x; 1::)

= {(y, s)

E lR.~+1 :

Ix - Yl2 < a 2 s, I:: < s < h}.

If u is a caloric function, we define its parabolic nontangential maximal function and its parabolic Lusin area function by

PNau(x) = sup{lu(x,s)1 : (y,s)

E

Pa(x)}

and

PAau(x)

= ({

s-¥-IVY U(y,S) 12 dYdS) 1/2 ,

}Pa(X)

respectively. The doubly truncated versions are defined similarly and denoted by

PN~u(x;l::) and PA~u(x,I::). As before, if I:: = 0 we simply write PN~u(x) and PA~u(x). Notice we have only used the horizontal gradient in our definition for

PAa u. Theorem 3.1.5 (Upper-Half LIL for caloric functions) Fix 0

1. Then for any caloric function u,

{x

E

< (3 < a and 0 < 'Y <

lR.n : PA~u(x) < oo} a~. {x E lR.n : PNJu(x)
and lim sup lu(y, t)1
for almost every x E {x E lR.n : PA~(u)(x) = oo}. Let us now briefly explain the changes in the proof of Theorem 3.0.4 needed to obtain Theorem 3.1.5. Let u(x, t) = u(x, t 2 ) as in Lemma 2.2.2. By a simple change of variables P Aa u( x) = 2Aa u( x), and similarly for the truncated versions. We observe that in the proof of Theorem 3.0.4 the harmonicity of the function u was used only in the proof of Lemma 2.2.1 and in the application of Lemma 2.3.1, that is, to control ylV yul by the area function. Lemma 2.2.2 is the caloric function version of Lemma 2.2.1 and the requisite gradient estimates, that is, the caloric equivalent of Lemma 2.3.1, is shown in Hattemer [Ha] and Ortiz and Torchinsky

lOT].

With these observations, the proof for harmonic functions adapts to caloric functions. However, observe that since the formula in Lemma 2.2.2 does not involve derivatives in t, we do not need to consider a parabolic area function involving the full gradient of u(x, t). This is in contrast to the case of harmonic functions.

3.2 The proof of the lower-half

3.2

75

The proof of the lower-half

As mentioned above, the proof of the lower half L1L follows, in spirit, the standard methodology often used to prove laws of the iterated logarithm for dependent sequences: The upper bound is combined with conditional Borel-Cantelli type arguments. We first illustrate this by discussing, without the details or much of the technicalities, some of the key steps in Stout's proof of his L1L for martingales. As in the proof of (3.0.5), let TJ> 1. Stout [Sto, p. 280] defines stopping times, Tk

= inf{n: a(fn+1) >

H}.

(3.2.1)

Here a(fn) is the conditioned square function defined in (2.0.3); see the comments after Theorem 3.0.3. For the upper bound he shows, using the exponential martingale (the Key Estimate 3.1.2 with the right constants in his case), that for every c > 0,

L PU;k > (1 + c) J2a 00

2 (frk)

log log a 2 (frk)} <

(3.2.2)

00.

k=l

This implies, by the Borel-Cantelli lemma, that

J;k ~ (1 + c)J2a 2 (frk) logloga 2 (frk)

(3.2.3)

eventually with probability 1. Stout arrives from here at the upper-half 11L exactly as in the proof of (3.0.5). Notice that, in fact, this whole argument is exactly the same as the proof of (3.0.5). For the lower bound, Stout proves the much more difficult and delicate conditional exponential inequality. Lemma 3.2.1 ( [Sto, p. 283]) CkOk for some sequence C k k large enough,

---->

Set Ok = JTJk - TJk-l. Suppose 0 as k

----> 00,

max

rk-l :
Idml ~

Then for 'Y > 0, for all A > 0 and

(3.2.4) With this lemma in hand, Stout shows that for an appropriately truncated version f~k' for all c > 0,

~ P {f~k - f~k_l > (1- ch/20~ loglogo~l.rrk_l } =

00,

(3.2.5)

almost surely. Now the conditional Borel-Cantelli lemma (Neveu [Ne, p. 152]) implies that (3.2.6)

3. Kolmogorov's LIL for Harmonic Functions

76

infinitely often with probability 1. From here Stout obtains the same conclusion for the original sequence frk' He finally combines this with the upper bound (3.2.3) to finish the proof of the lower bound. The arguments leading to (3.2.5) (hence to (3.2.6)) are frequently called "conditional Borel-Cantelli type arguments" . The point is that one does not need independence, as in the usual Borel-Cantelli lemma, to conclude (3.2.6) from (3.2.5). Such conditional Borel-Cantelli arguments were first used, Stout notes [Sto], by P. Levy [Le2, p. 263J in proving a version of the LIL for martingales. The proof of the lower bound in the harmonic function setting, Theorem 3.0.5, is based on the following two rather technical lemmas which we will discuss after their statements. We need some notation. For any cube Q C IR n of sidelength f(Q) we let 0 denote the concentric cube of sidelength 2f(Q). Fix constants h = 100VriIa, 0 < T < 1 and 1 < K < 00 and set E

= E(K, T) = {x

E IR n : A~u(x)

= 00,

K",(u) (x; t)

< K for 0 < t < T}.

Lemma 3.2.2 There are four positive constants No(a, n), w(a, n), 8(a, n) and (K, a, n), depending only on the parameters indicated, such that for any cube Q with 2hf( Q) ::::; T if

Ro

IE n 012: (1- w)IOI,

(3.2.7)

(K loglog(eeAQ + AQ) + Ro)

(3.2.8)

and R > No where AQ = sup(A~u(x;hf(Q)))2, xEQ

then Q is the finite disjoint union of cubes {Qj} satisfying the following two properties:

(3.2.9)

For all j and all x E Qj (A~u(x; hf(Qj)))2 - (A~u(x; hf(Q))2 ::::; 2R

and: There is a subcollection {Qj;} C {Qj} such that for all ji and all x E Qj;, u(x, hf( Qj;)) - u(x, hf( Q)) and

IuQj;l2: 81QI· •

2: 8v'R

(3.2.10)

3.2 The proof of the lower-half Lemma 3.2.3 Let No, w, 8, and

suppose that

Ro be as in Lemma

77

3.2.2. Let Q be a cube and

2hi(Q) 5,7,

(3.2.11)

IQ n EI

(3.2.12)

~(1- w)IQI,

A~AQ'

B > A and R

~No ( K log 10g~e + B) + Ro )

(3.2.13) (3.2.14)

and: A

+ 2mR 5,B,

for some positive integer m.

(3.2.15)

Then there is a partition of Q into dyadic subcubes {Qj} == :F and a subcollection £ = {Q ji} c :F such that the following hold: (3.2.16)

Il) I~8mIQI· •

Qj;

(3.2.17) (3.2.18)

Furthermore, each Qji satisfies either (a)

or for all x EQj; we have

(b)

u(x, hi(QjJ)-u(x, hi(Q)) ~ 8mVR. As we will momentarily show, these two lemmas easily imply Theorem 3.0.5. Lemma 3.2.3 follows by repeated application of Lemma 3.2.2 with m = 1,2, ... , as long as A + 2mR 5, B. Thus, Lemma 3.2.2 is the key. Even if not so obvious at first reading, this lemma is the analogue in our setting (with much less precise information on the constants) of Stout's lemma above. However, we only wish to obtain a lower bound without regard to sharp constants, and consequently, do not need a lemma with the full strength of his exponential lower bound. (We do not know the best con~tants in Theorem 3.0.5, and finding them seems almost

78

3. KoImogorov's LIL for Harmonic Functions

impossible.) Our Lemma 3.2.2, translated to the martingale setting, essentially says that if T is a stopping time for Um}, fo = 0 and CIR ::; 8 2(fT) ::; C 2R, then (3.2.19) To see (3.2.19), note that since CIR ::; 8 2(h) ::; C2R the Burkholder-Gundy inequalities imply Ilhll p ~ v'R for all 1 < p < 00. In particular, IIhl12 ~ v'R, IIhl14 ~ v'R and from this it is then elementary to conclude (3.2.19). (See [Zyg2, vol. I, p. 216J.) In fact, this idea is the crux of the proof of Lemma 3.2.2. In this context, the hl(Qj) are a "stopping time" and the passage from (3.2.9) to (3.2.10) uses exactly this idea of comparing two different norms of the "stopped" function u(x, hl(Qj)) - u(x, h£(Q)). These "stopping times" turn out to be essentially the graphs of Lipschitz functions so that much of what is needed for the proof of Lemma 3.2.2 are estimates similar (although again much more technical) to those that we will do in the proofs of the good-A inequality in Section 4.2. The reader interested in the proofs can see [BKM2J. We now give the Borel-Cantelli argument to prove Theorem 3.0.5 assuming the lemmas. Let 1] and 1/ be constants much larger than 1 and for k = 1,2, ... , J is the integer set Ak = 1]k, Rk = 1/K 1]k / log k and mk = [log k /31/ KJ, where [ part function, and K is as above. Notice that mk VRk is of order 1]k log log 1]k as k gets large. (This quantity has already appeared in the proof of (3.0.5) as well as in Stout's argument above.) Fix a cube QI C IR n with 2hl(Qd ::; T. The numbers

J

(A, B, R, m)

= (Ak, A k+1, R k+1, mk+1)

satisfy (3.2.13)-(3.2.15) with A Q1 , if 1/ ~ 2No, 1] ~ 3 and k ~ kl for some kl large enough. For k 2:: k 1 , define the partition :Fk of Ql and the sub collection Ck C :Fk inductively by :Fkl = {QI} = Ckl' and given :Fk define :Fk+1 and Ck+1 by taking each cube Q E :Fk and applying one of the following cases: Case 1 If IQ n EI 2:: (1 - w)IQI, then apply Lemma 3.2.3 with A = 1]k, B = 1]k+1, R = Rk+1 and m = mk+1. We put the resulting {Qj} into :Fk+1 and the {QjJ into Ck+1. (We may apply Lemma 3.2.3 since (3.2.13) holds for Q E:Fk by (3.2.20) below.) Case 2 IflQnEI < (1-w)IQI, then let {Qj} be the 2n dyadic subcubes ofQ with l(Qj) = ~£(Q) and construct :Fk+1 and Ck+1 as follows:

(a)

If AQj ::; 1]k+1 for all j = 1,2, ... , 2n , put all of these dyadic cubes into both :Fk+1 and Ck+I.

(b)

If A Qj > 1]k+1 for some j = 1,2, ... , 2n , put Q into both :Fk+1 and Ck+1.

With this construction we have that if x E Q(k) E :Fk for a fixed x, then f(Q(k)) ----> 0 as k ----> 00. To see this, observe that if Case 1 applies then Q gets

3.2 The proof of the lower-half

79

subdivided and if not, Case 2 (a) eventually applies since 17k+1 -----t 00 and thus it also gets subdivided. Also if we abuse notation a little and use Ek + 1 to denote the set of points U Q, then using Lemma 3.2.3 QE£k+l

If Q E F k , then AQ :S 17 k . If Q E h, then

IQ n Ek+11

(3.2.20)

:::: 8rnk + 1 1QI

(3.2.21 )

If Qj E Ek+l, then one of the following is true:

(a)

(3.2.22)

For all x E Qj,

u(x, hC(Qj)) - u(x, hC(Q)) :::: 8mk+l JRk+1, where Q is the cube in Fk containing Qj, or (b)

IQj nEI < (l-w)IQjl, or

(c)

IQ n EI < (1 - w)IQI where Q is the dyadic cube of sidelength 2C(Qj) which contains Q.

Here, (3.2.22) follows by noting that if Qj arose from an application of Lemma 3.2.3, that is, via Case 1, then (a) or (b) holds, otherwise, (b) or (c) holds according as Qj arose from Case 2(b) or Case 2(a), respectively. Next, we choose v = v(a, n) so large that

8rnk

> 8(iogk)/3vK > 8(logk)/3v = k-(logl/6)/3v >

-

-

_1_.

- Vk

(3.2.23)

Notice that (3.2.21) is really a statement about conditional probabilities so we could apply the conditional Borel-Cantelli lemma as before. However, we can be more direct. With Q1 \Ek = Ek and with our above notation that Ek is the union of cubes in F k , (3.2.21) and (3.2.23) give that for k 2:: k1

Since

then for almost every x E Q1 there is an infinite sequence k1 < k2 < ... , depending on x, such that x E Eke for all C. Thus, almost every x E En Q1 has this property

3. Kolmogorov's LIL for Harmonic Functions

80

and is a point of density of E. Fix such an x. Since E c {x E IR n : A~ u( x) the upper-half LIL, Theorem 3.0.4, implies that we may also assume lim sup dO

ju(x, r)j

.J(A~u(x; r/2))210glog A~u(x; r/2)

S Ca.,n.

= oo},

(3.2.24)

But even more is true. Since x E E, K(x, t) < K for 0 < t < T so that (3.2.24) and our definition of K(x, t) gives (see the remarks immediately after Theorem 3.0.5) lim sup rlO

S Ca.,n.

ju(x,r)j

.J(A~u(x; r))210g log A~u(x; r)

(3.2.25)

In addition, since x is a point of density of E, if x E Qj E Ckt, then Qj satisfies (3.2.22), (a) with k + 1 = ke as soon as f is large enough. Hence we obtain two sequences of real numbers (depending on our point x),

and

with (3.2.26) and which satisfy by (3.2.20), (A~(u)(x; re))2

sr/t - 1

(A~(u)(x;te))2

srlt.

(3.2.27)

Thus, if f is large enough, (3.2.25)-(3.2.27) give that 6mkt yff[i:; -ju(x, re)j

u(x, te)

-;===============>-~:;=:==========:;==-.:....:. .J(A~u(x, te))2loglogA~u(x, te) y'",kt log log ",kt

C1

(lOgv'jt,k

t

-

C2y'",kt-1loglog",kr1

>--~----~~======~--------

y'",kt log log ",kt

?,C3V vK1 -C Y-:;;ff 4

>C -

5

VvK1 -_ Ca.,n ..fK'

3.3 The sharpness of the Kolmogorov condition

81

if 1J is chosen much larger than vK. (Here the constants C 1 , . .. ,C5 , Co.,n depend only on a and n.) Since Q1 with 2h£( Qd :::; T was arbitrary, we have proved that . 11m sup

u(x, t)

tlO

V(Az,u(x;t))210g10gAz,u(x;t)

Co. n :::: - ' -

VK

for almost every x E E = E(K, T). Apply this now to the sequence E(j, 11k) with j, k = 1,2, ... to complete the proof of Theorem 3.0.5. In [Jo], P. Jones proved what is a very special case of the lower bound, Theorem 3.0.5. Let F be a Bloch function in the unit disc D; that is, F is analytic in D and I!FIIB = !F(O) I + sup{(I- Izl)IF'(z)1 : zED} < 00. Suppose I!FIIB :::; 1 and that for all Zo ED, sup p(zQ,z):9/ 2

!F'(z)I(I-lzl) :::: c > 0

(3.2.28)

where p(zo, z) is the hyperbolic distance from Zo to z. Jones showed that then there is a positive constant C = C(c) such that Re F(pe iO )

. hmsup pj1

vlog

l~P log log log l~P

:::: C

(3.2.29)

for almost every () E [0, 21f]. Bloch functions trivially satisfy our condition (K3) and, as we will explain in Chapter 6, (3.2.28) implies that an appropriately truncated area function of Re F is uniformly bounded below by Clog l~P' Hence (3.2.29) will follow from Theorem 3.0.5. However, the reader interested in the full details of the proofs of Lemmas 3 .. 2.2 and 3.2.3 will find it very useful to first read Jones' paper. Indeed, these lemmas are natural extensions of his Lemmas 3 and 4 [Jo, pp. 61-63] which are technically much simpler. Remark 3.2.4 It is also most likely the case that the proofs of Lemmas 3.2.2 and 3.2.3 given in Banuelos, Klemes and Moore [BKM2] can be modified to give appropriate versions of these lemmas for caloric functions. These in turn will lead to the corresponding lower-half 1IL for caloric functions.

3.3

The sharpness of the Kolmogorov condition

In [MZ] , Marcinkiewicz and Zygmund construct a sequence of random variables to show that the condition (Ko) is best possible in order to obtain a lower bound in Kolmogorov's 1IL, Theorem 3.0.1. Their example is constructed from Rademacher functions. Motivated by their example and the well established philosophy that

3. Kolmogorov's LIL for Harmonic Functions

82

lacunary power series behave very much like independent random variables, particularly lacunary series with large gaps, we will construct an example of the form 00

u(re iO ) = L ak rqk cos(qk(J), k=l

(3.3.1)

where {ak} is a sequence of positive numbers and q is a large positive integer, both to be chosen. Not only is our example motivated by that of Marcinkiewicz and Zygmund, but also some of our computations below are inspired by theirs. Notice that the function u is the real part of the lacunary power series 00

F(z)

= Lakzqk,

zED.

(3.3.2)

k=l Finally, before we proceed with the proof, we observe that our example also shows the sharpness of the condition in the M. Weiss [We] LIL for lacunary series (this LIL will be discussed in Chapter 6) which to our surprise had not been done before. We now come to the proof of Theorem 3.0.6. This will take several steps. We will show: Step 1. With {ak} and q appropriately chosen we have

where B! = L:Z'=1IakI2 and by bm rv Cm we mean below, K is as in the statement of Theorem 3.0.6. Step 1 implies a similar result for Abel means:

lim bm/cm

=

1. Here and

m--->oo

Step 2. For all (J E [0,27r] and all p close to 1,

I ~akpqk COS(qk(J) I :::; ~JBpoglogBp where B~

= L: 00

k

a~p2q .

k=l We remark that B'tJ and B~ are traditionally defined with a factor of ~ proceeding the summation; we will also adopt that definition later in Section 3.4 and in Chapter 6. Use of the factor ~ here would just make the computations in this proof more cumbersome. Step 3. If q is large enough,

for all (J E [0, 27r].

83

3.3 The sharpness of the Kolmogorov condition Step 4. The function u satisfies condition (K4).

The next lemma constructs the sequences of am's and Bm's. Lemma 3.3.1 Let B;" = exp(Km/logm) and set am = VB;' - B;"_I and al = Then

a2 ,...., K m

B;" loglogB;,

o.

(3.3.3)

and

(3.3.4) Proof:

a;" = exp(Km/logm) - exp(K(m -l)jlog(m - 1)) =Kl

m

m-I

exp(Kx/logx) (-1 1 - (1 1 ogx ogx

1

,....,K-exp(Km/logm) logm =K B;"

logm

and log log B;, =log m

+ log K

- log log m ,...., log m.

Thus, 2 ,K. . . B;" a .,. m loglogB;,

For convenience, let us set £(m) =B;, log log Bm. Elementary algebra gives

J C(m) - J C(m _ 1) =

C(m) - C(m - 1) JC(m) + JC(m -1) C(m) - C(m - 1) ,...., 2JC(m)

using the fact that Bm ,...., B m- I .

)2) dx

84

3. Kolmogorov's LIL for Harmonic Functions

Also, £(m) - £(m - 1)

= L~l

'" L~l

{K

=

1

Co~x -

m

m-l

d -d {exp(Kx/ log x) (log X X

(lo;x)2) (lOg l:;X)

+ log K

+~ -

-loglogx)}dx

Xl~gx} exp C~:)

dx

Kexp(Kx/logx)dx '" Kexp(Km/logm)

=KB!. Thus, J£(m) - J£(m -1) '"

KB2

m

2Bm yflog log Bm

K

2yflog log Bm

v'K

1

'" -2 --a v'K m '" --a 2 m' Since am

-----+ 00

we obtain finally that 2

Lak '" v'K L (J£(k) m

k=2

m

J£(k

-1))

k=2

2 2 2 '" v'K v/ Bm log log B m ,

and we have proved (3.3.4) and hence the lemma. We have done Step 1. The estimate for Abel means, Step 2, follows from the estimate for partial sums exactly as in M. Weiss [We, pp. 267-268]. All that is required is that for k :::; m, B! -----+ 0 as m -----+ 00. This fact is clear for our ak's and Bm's. We now proceed to Step 3. This type of estimate on the square function will also be used later on in Chapter 6. The philosophy here is that for lacunary series, all norms are comparable to the L2-norm. We begin by writing the gradient of u in polar coordinates to obtain

aV

and so

3.3 The sharpness of the Kolmogorov condition

=

I

85

+ II.

The first integral can be easily computed to obtain

r

I=o:fa%q2k r 2qk - 1(I-r)dr k=l io =

a2kP2qk [q2k + 2(1 - p)q3k] > _O:B 2 2qk(2qk + 1) - 5 p.

00

0: """'

~

k=l

In the same way 00

1111

:::;2L j=l 00

j=l

k=j+1

(

J=l

1)

2 2qk 00 ak;_j k-j k=J+l q k=j+l q

L L 00

:::; 8Bp

L

00

1/2

~ (~ 2qk~_1 )1/2 ~ akP ~ k-j

:::; C-=-I yq-J.

2

k=l

j=lq

8B 2

<--p - q-l

With q chosen large enough, the estimates for I and II imply (AD: u( B; p))2 ~ which proves Step 3.

%B~

86

3. Kolmogorov's LIL for Harmonic Functions

Finally, we show that u satisfies (K4). Let 7]e i

N, with ak as constructed in Lemma 3.3.1,

Thus,

L akqk7]qk_l 00

(1 - 7]) IV'u(7]e iB ) I ::; v2(1 -7]) ::; v2aN

t

k=l

::; CaN

k=l

:k qk-N

+ v2aN

N

+ v2aN

f

k=N+1

L

:k qk-N N

(1 _ ;+1)

qk_l

q

00

ek-Nqk-N exp(-(qk _l)/qN+1)

k=N+l

by Lemma 3.3.1. However, by Step 3 we also have

(3.3.5) Since the function x 2 / log log x 2 is increasing in x, we conclude that (recall 7]e i

(dau(B; p)

)2

(Aau(B; p))2 ::; CaK 10glog((Aau(B; p))2)

and Theorem 3.0.6 has been proved.

3.4

A related LIL for the Littlewood-Paley g*-function

The proofs of the upper bound, Theorem 3.0.4 and the proof of the lower bound, Theorem 3.0.5, provide no information on the constants C;,(3",n and C~,n and it is probably hopeless with the above arguments to try to identify the best values

3.4 A related LIL for the Littlewood-Paley g.-function

87

of these. However, there is a weaker upper LIL for harmonic functions in the disc D for which sharp constants are known. For u(z) harmonic in D and 0 < p < 1, define up(z) = u(pz), zED. The Littlewood-Paley square function of up is defined by (3.4.1) This function, as we will see, is the expectation with respect to the Brownian motion starting at the origin and conditioned to exit the disc at eilJ , of the square function of the martingale obtained by composing up with Brownian motion in the disc. The following theorem is proved in Banuelos [Ba2]: Theorem 3.4.1 Let g*(p)

= Ilg*(up )(8)lloo. Suppose g*(p)

lim sup Ptl

lu(pe ilJ ) I

y!2g; (p) log log g; (p)

roo as p r 1. Then

::; 1

(3.4.2)

for almost every eiIJ E T = aD and 1 is the smallest possible constant we can have on the right hand side of (3.4.2). It is an easy exercise to show that Ac",u(8;p) ::; C a g*(up)(8) and hence, if we do not care about constants, Theorem 3.4.1 is certainly weaker than Theorem 3.0.5. Nevertheless, the proof of Theorem 3.4.1 illustrates yet another interesting application of probability to square functions and therefore we present it here. Also, as it turns out, Theorem 3.4.1 gives better information concerning the LIL for Bloch functions discussed in Chapter 6. The following proposition plays the role of the Key Estimate 3.1.1. In fact, with this proposition, the proof of Theorem 3.4.1, except for the sharpness part, is exactly the same as the proof of the upper estimate in (3.0.5) which we gave above. Throughout this section it will be convenient to use the notation dm = d8 /21f. Proposition 3.4.2 Let f be an integrable function on the unit circle with integral O. Let u(z) be the harmonic extension of f to D. Set u*(e ilJ ) = sup lu(reilJ)1 and suppose that Ilg.(u)lloo = g* < 00. Then for all t5 < 1/2, O
Proof: From the boundedness of the radial maximal function in LP, 1 we have that

< p ::;

00,

(3.4.4)

88

3. Kolmogorov's LIL for Harmonic Functions

for 2 ~ p ~ 00, with C independent of p. By (3.4.4), it is enough to show (3.4.3) with f(e i8 ) replacing u*(e i8 ). Also, since 9*(-u)(e i8 ) = 9*(u)(e i8 ), it suffices to show that m{ eiO E T : f( e i8 ) > A}

~ exp ( _ ~2~)

,

(3.4.5)

for all A > O. The estimate (3.4.5) is also sharp in the sense that for every 'Y > 0 and C-y < 00 the estimate

(1. +'Y)A 2 ) m{ei8 E T: f(e i8 ) > A} ~ C-yexp ( _ 2 9~

(3.4.6)

fails. Thus (3.4.3) does not hold for any 8 > 1/2. To prove (3.4.5), let B t be Brownian motion in the unit disc D starting from the origin and TD be its first exit time from D. Consider the martingale X t = u(Bt/vrD). By the Ito formula, its square function is given by

The connection between 9* (u) and S (X) is the following. For any positive harmonic function h in D, the Doob h-conditional Brownian motion is determined by the transition probability densities 1

D

Ph(t,z,W) = h(z)P (t,z,w)h(w) for t > 0 and z, wED, where pD(t, z, w) are the transitions of the Brownian motion B t killed when it exits D. (See Durrett [Dur], Chapter 3 for more on this.) If we take

ho(z) =

1-lz12 Iz _ ei8 12'

which is, up to a constant factor, the Poisson kernel for D with pole at ei8 , we obtain Brownian motion conditioned to exit D at e i8 . If we let E8 denote the expectation with respect to this motion starting from the origin, the expression for the transition probabilities and the fact that the Green's function for the disc is ~ log( 1;1) (we are dealing here with one-half the Laplacian) imply

89

3.4 A related LIL for the Littlewood-Paley g.-function

Using this, the fact that BTD is uniformly distributed on the circle and Jensen's inequality, we obtain

h

~2 9;(1)(8)) dm

exp ()..f(e ili ) -

= E (exp()..f(BTD) _ =E

(exp (EBTD ()"X _

~ E (EBTD =E

~2 EBt

(exp ()..X _

(exp ()..X _

(lTD IV'U(BsWdS) )

~2 S2(X)))) ~2 S2(X))))

~2 S2(X))) .

Since for every).. > 0, the process

M t = exp()..Xt

)..2

-

"2 S;(X))

is a supermartingale starting at 0 (simply apply the exp()..x t); see Durrett [Dur, p.70]), we see that

;2

Since g.(1)(8)

Ito

)..2 g;(1)(8))dm ~ 1. iTrexp()..f(eili ) - "2

~

formula to
(3.4.7)

g. for all 8, we find from (3.4.7) that

r exp()..f(eili))dm ~ exp ()..22) , 29.

iT

from which (3.4.5) follows. As we previously noted, once the "Key Estimate" given by Proposition 3.4.2 has been obtained, the proof of the LIL in Theorem 3.4.1 is the same as the proof of the upper half in (3.0.5) given above. We leave this as a very useful exercise for the reader. To prove the sharpness of (3.4.2) and (3.4.6) we use the following lemma from Banuelos [Ba2]. Lemma 3.4.3 Let u(z)

k

= 2: r q cos(qk8), z = re ili and construct the correspond00

k=l

ing function up(z) for 0 < p < 1. Given E > 0 we may take q large enough depending only on E such that for all 0 < p < 1, (3.4.8)

where B~ = ~

2: 00

k=l

p2. q

k

.

3. Kolmogorov's LIL for Harmonic Functions

90

Proof: We give the major steps of the proof, leaving a few of the easy computational details to the reader. Since

B; = ~ £lup(eiOWdm = £g;(up)(O)dm,

(3.4.9)

the left hand side of (3.4.8) immediately follows. To obtain the right hand inequality we compute much in the same way as in Step 3 of the proof of Theorem 3.0.6 (section 3.3). We have:

g;(up)(O)

(1)

_1171"{11-r2 Ireit _ eiol2 log :;

-:; -71" io =2 =2

i:1 i:1

1 1

it2 IV'up(re)1 rdrdt

(~) drdt

P(re it , eiO ) 1V'u p(reitWrlog p(reit,eiO){1

~qk(pr)qk-1pCOSqktI2

+ I ~qk(prl-1pSinqk{}rIOg (~) drdt =

2171" (1 P(reit,eiO)

i: 11 -71"

+4 =

I

io

+ II.

f

k=1

q2k(pr)2 qk -2 p2rlOg

(~) drdt r

P(re it , eiO ) k~l qk+j(pr)qk+qj-2 cos((l- qj)t) p2rlog

(~) drdt

k>i

A straightforward computation shows P(re it , eiO ) = P(re iO , eit ). Then, 1= 2

f; r

2qk - 1

L q2k p2qk io 00

k=1 00

(

0

log

U) dr = 4';2k' Also note that

1 r 2qk - 1 log -dr r

1

= 2 "" q2k lqk __

LJ k=1 =B2p

4q2k

For k > j, the function r qk - qj cos((qk - qj)t) is harmonic on a neighborhood of the disk. This, plus the observation that P(reit , eiO ) = P(re iO , eit ) and Theorem 1.2.3 yield

3.4 A related LIL for the Littlewood-Paley g,-function

91

Consequently,

IIII =

4111 k~'

j qk+ j (pr)qk+ q - 2

k>j

I:

P(re it , e iO )

cos((l- qj)t)dt p2r log

(~) drl

j=l k=j+l

=L

L

00

00

qj-k pqk+qj

j=lk=j+1

::: L L 00

00

j=l k=j+l

=

f

qj p2qj

j=l

qj_k/ qJ

f

1k

k=j+l q

B2p

(q - 1)" Given

E

> 0, we now choose

q

so that (q~l) <

E.

Then if 0 < p < 1, g;(up)(e) :::::

(1 + E)B~. With Lemma 3.4.3, we apply the Kac-Salem-Zygmund central limit theorem and the M. Weiss LIL, Theorem 6.1.2 below, to conclude the sharpness of (3.4.2) and (3.4.6). Proposition 3.4.2 is a special case of a very general result for domains in ]R.n. Let n be such a domain; assume n is regular for the Dirichlet problem and that u is harmonic in n with boundary values f. Fix Xo E n and denote by WXD the harmonic measure for an. For any x, yEn, denote the Green's function for n by Gdx, y) and its Martin kernel by Ko(x, ~), x E n, ~ E an. Define (3.4.10) We have

92

3. Kolmogorov's LIL for Harmonic Functions

Proposition 3.4.4 Suppose Ilg;O,n(u)(~)IILOO(&n)

WXO{~ E 80,: u(~) -

= g* < 00.

Then for all

u(xo) > >.}:<:; exp (- >.22) . 2g*

>. > 0, (3.4.11)

If 0, is a Lipschitz domain, such as those which arise below as stopping times, we can also obtain a version for the radial or nontangential maximal function of u as in Proposition 3.4.2. The proof of Proposition 3.4.4 is as above using the representation

where now Bs is the Brownian motion in 0" Tn is its first exit time and E~o is the expectation with respect to the Doob process starting from Xo and conditioned to exit 0, at ~. Finally, A. Baernstein (personal communication) has given an analytic proof of the inequality (3.4.7). His proof is based on the fact that the functions

are solutions to the boundary value problem

-~V + lV'ul { View = ef ,

2V

= 0 in D

which is a Dirichlet problem for a Schrodinger operator.

(3.4.12)

Chapter 4 Sharp Good- A Inequalities for A and N In this Chapter, as in Chapter 3, we will prove various sharp comparisons for A and N which are motivated by the corresponding results for martingales. First, let us recall the classical inequalities of Burkholder and Gundy [BG 1] for continuous time martingales.

°

Theorem 4.0.1 Let X t be a continuous time martingale with maximal function X* and square function S(X). Then for all < E < 1, 6> 1 and A> 0, E2

P{X* > 6A, S(X) :::; EA}:::; (6 -l)2P{X* > A}

(4.0.1)

and E2

P{S(X) > 6A, X* :::; EA} :::; 8 2 -1 P{S(X) > A}.

(4.0.2)

As they are expressed here, these are actually a refinement, due to Burkholder [Bu1], of the inequalities of [BG1]. The usefulness of such inequalities is already amply demonstrated by the following lemma, which is but one of many applications (see Section 4.3 and Section 4.4 for more) of these type of inequalities. For this lemma we consider a non-decreasing continuous function defined on [0,00] with <1>(0) = 0, not identically zero, and which satisfies the growth condition: (2A) :::; c(A) for every A > 0, where c is a fixed constant. The following is from [Bu1].

°

°

Lemma 4.0.2 Suppose that f and 9 are nonnegative measurable functions on a measure space (Y, A, J-l), and 8 > 1, < E < 1, and < 'T < 1 are real numbers

such that

J-l{g > 8A, f:::; EA}:::; 'TJ-l{g > A}

(4.0.3)

for every A > 0. Let p and v be real numbers which satisfy (8A) :::; p(A),

(E- 1A) :::; v(A)

(4.0.4)

for every A > 0. Finally, suppose P'T < 1 and fy(min{l,g})dJ-l < 00. Then { (g)dJ-l:::; ~ { (f)dJ-l. }y 1 - P'T }y 93 R. Bañelos et al., Probabilistic Behavior of Harmonic Functions © Birkhäuser Verlag 1999

(4.0.5)

4. Sharp Good-A Inequalities for A and N

94

Note that the existence of p and v satisfying (4.0.4) is a consequence of the growth condition imposed on <1>. Proof: We may assume that Jy (g)dfJ < 00. To see this simply note that if 9 satisfies (4.0.3) then so does min{n,g} for any positive integer n. Also note that since Jy (min{l, g} )dfJ < 00 then Jy (min{ n, g} )dfJ < 00 for every n. If (4.0.5) holds for min {n, g} and f for every n then it is immediate that it holds for 9 and f. Associated to is the Lebesgue-Stieltjes measure d satisfying J[a,b)d()') = (b) - (a) whenever 0 :::; a < b :::; 00. This measure is a positive IT-finite Borel measure on [0, (0). Furthermore, an elementary Fubini theorem argument shows that if h is a nonnegative measurable function on (Y, A, fJ) then jy (h)dfJ Jooo fJ{h > )'}d()'). Inequality (4.0.3) implies fJ{g

Consequently,

> 8),} = fJ{g > 8)', f:::; cA} + fJ{g > 8)', f > cA} :::; "YfJ{g > ),} + fJ{J > c),}.

i



(~) dfJ :::; 'Y

i

(g)dfJ +

i



(~) dfJ·

(4.0.6)

But Jy (g)dfJ = Jy (88- 1 g)dfJ :::; p Jy <1>( ~ )dfJ· This, plus (4.0.6) and (4.0.4) yield the conclusion of the lemma, (4.0.5). In a space of finite measure, the hypothesis Jy (min{1,g})dfJ < 00 is superfluous. In the general case this hypothesis was necessary to assure the finiteness of Jy (min{ n, g} )dfJ so that we could reduce to the assumption that Jy (g )dfJ < 00. This in turn was necessary in order to insured that we were not subtracting infinities in the last step of the proof. In the case of spaces of infinite measure, the lemma is not true without the hypothesis Jy (min{1, g} )dfJ < 00; see Journe [Jou, p. 4], or Miyachi and Yabuta [MY] for examples. In this monograph we will sometimes use this lemma to produce estimates like (4.0.5) and often this will be done on spaces of infinite measure. However, in all situations in which we use the lemma on a space of infinite measure, the inequalities we show are already known, and so we may assume the finiteness of the quantities involved. This seems to say that we will use an inequality like (4.0.5) to prove itself, but there is an advantage here: if we know that an estimate like (4.0.5) holds, then a good-)' inequality can lead back to (4.0.5), but with better constants. An alternative in the infinite measure case is to use the good-), inequality to show (4.0.5) for a set of functions for which the left hand side of (4.0.5) is finite and then take appropriate limits. It is precisely such an approach that we take in the proof of Theorem 4.4.1 below. Essentially this same idea was used in the proof of Theorem 1.7.8. If 0 < p < 00 then (),) = ),p satisfies all the conditions of Lemma 4.0.2. In this case, Theorem 4.0.1 and Lemma 4.0.2 give the following. Theorem 4.0.3 Let 0 < p < 00. There exists constants Ap and B p , depending only on p, such that if X t is a continuous time martingale starting at 0 with maximal

95

4. Sharp Good-A Inequalities for A and N

function X* and square function S(X) then

(4.0.7) and

IIS(X)II :::; apIIX*II·

(4.0.8)

If in using Lemma 4.0.2 to prove this we set 8 = 1 + ~ and c = 2~ for (4.0.7), and 8 = 1 + ~ and c = 2~ for (4.0.8), we find that Ap = O(p) and a p = O(y'P) as p ---'; 00. This is the best possible order for ap but not the best possible order for Ap; see (4.4.5) below. 2

2

Observe that, with 15 fixed, the expressions (6':'1)2 and 6L 1 in (4.0.1) and (4.0.2) go to 0 as c ---'; O. In order to obtain the best possible order of A p , and for various other applications of these inequalities, such as LIL's and estimates on various constants, it is desirable to have a better rate of decay as c ---'; O. The optimal rate of decay is given by the following result. Theorem 4.0.4 There are constants C 1 and C 2 such that for all 0 < c < 1, 1 < 15 and A> 0,

1

P{X* > I5A, S(X) :::; cA} :::; C exp ( _ (8 ~;)2) P{X* > A}

(4.0.9)

and

(4.0.10) These are the subgaussian good-A inequalities referred to in the preface. This theorem follows from more general good-A inequalities of Burkholder [Bu3, pp. 185-187] which relate maximal functions and stopping times for Brownian motion in]Rn. These appear with quantities he designates as Rn (15, c) and Ln (15, c) replacing the exponentials on the right hand side of (4.0.9) and (4.0.10) respectively, but explicit computation of these quantities yields the expressions we have stated here. To illustrate once again the vital role played by Key Estimate 3.1.1, we present Burkholder's proof of (4.0.9), breaking it into its basic parts. In fact, what Burkholder's argument really shows is that scaling and the Markov property are essentially enough for good-A inequalities. (For more on this direction, see Revuz and Yor [RY, p. 155].) The following inequality is easily derived from the Key Estimate 3.1.1 and it makes it very clear what is needed for all the good-A inequalities that follow, including those for harmonic functions.

96

4. Sharp Good-A Inequalities for A and N

Key Estimate 4.0.5 Let {Xt} be a continuous martingale with Then for all A> 0 and 8> 1, P{X* > 8A} :::; Cexp ( Proof: Let T = inf{t : Consequently,

IXtl >

(8-1)2A2)

2118(X)II~

A}. Then, {T < oo}

118(X)1100 <

00.

P{X* > A}.

= {X* > A} and IXrl = A.

P{X* > 8A} :::; P{X* > 8A, T < oo}

:::; P{sup IXt

-

Xrl > (8 - l)A,

T

< oo}

t~r

:::; P{X* > (8 -l)A}P{T < oo}

(8 - 1)2 2) * :::; Cexp ( - 2118(X)II~ A P{X > A}, where the last inequality follows from the Key Estimate 3.1.1. For the proof of (4.0.9), let T = inf{t : 8 t (X) > cAl. Applying the Key Estimate 4.0.5 to the martingale Xt = Xr/\t which has 118(X)1100 :::; cA, we obtain

-

(

P{X* > 8A, 8(X) :::; cA}:::; P{X* > 8A}:::; Cexp -

(8 2c - 1)2) P{X* - > A} 2

:::; Cexp (- (8 ~21)2) P{X* > A}, which proves (4.0.9). For dyadic martingales we have a similar estimate which follows exactly as above from the Key Estimate 3.1.2. This version is Key Estimate 4.0.6 If {fn} is a dyadic martingale with

118(f)1100 < 00,

then

The Key Estimate 4.0.6 implies the next theorem using exactly the same stopping time argument that showed that the Key Estimate 4.0.5 implies (4.0.9). Theorem 4.0.7 Let {fn} be a dyadic martingale with maximal function f* and square function 8(f). There are constants Ct, C2 such that for all 0 < c < 1 and A> 0,

P{f* > 2A, 8(f) :::; CA} :::; C 1 exp ( -

~~ )

P{f* > A}.

(4.0.11)

4. Sharp Good-A Inequalities for A and N

97

Inequality (4.0.11) is proved in [CWW] by Chang, Wilson and Wolff. For dyadic martingales in one dimension we also have P{S(f) > 2A,

1* : : ; lOA}::::; C 3 exp (- ~;)

P{S(f) > A}.

(4.0.12)

The inequalities (4.0.10) and (4.0.12) follow, with similar proofs, from the corresponding key estimates: Key Estimate 4.0.8 If X t is a continuous martingale with P{S(X) > 8>'}::::; Cexp

(

7f

-8

(8 2

1)>'2)

IIX*II~

IIX*lloo < 00.

Then

P{S(X) > >.}

and Key Estimate 4.0.9 If {In} is a dyadic martingale with

111*1100 < 00,

then

C >.2 ) P{S(f) > 2>.} ::::; C 1 exp ( -llf~lI~ P{S(X) > >'}. The inequalities in the Key Estimates 4.0.8 and 4.0.9 can be proved by showing that the condition IIX*lloo < 00 (respectively, 111*1100 < (0) implies S2(X) (respectively S2(f)) is in BMO. The John-Nirenberg theorem and a stopping time argument (see Lemma 4.2.3 for the adaptation of this stopping time argument to functions on ~n) then gives exponential inequalities involving S2(X) (respectively S2 (f)), and hence, exponential square inequalities involving S (X) (respectively, S(f)). However, it is not possible to obtain either of the Key Estimates 4.0.5 or 4.0.6 using just BMO techniques. The situation is exactly the same for harmonic functions. An analogue of the Key Estimates 4.0.8 and 4.0.9 will be obtained using BMO techniques; this is (4.2.16) below. The analogue of the Key Estimates 4.0.5 and 4.0.6 will be more difficult to obtain. In the harmonic function setting, the analogue of Theorem 4.0.1 was proved in Burkholder and Gundy [BG2]. Their inequalities, although of a slightly different form, are essentially equivalent to those of Theorem 4.0.1 with the expression 10 2 on the right hand side. The expression 10 2 was improved to 10 k for any k by R. Fefferman, Gundy, Silverstein and Stein [FGSS]. T. Murai and Uchiyama [MU] proved the following inequalities, which have exponential decay: Theorem 4.0.10 Fix 0 < f3 < 0:. There is a constant K > 1 and positive constants C 1 , C 2 , C 3 and C4 , all depending only on 0:, f3 and n, such that for all 0 < 10 < 1 and>' > 0,

I{x

E ~n : N{3u(x)

> K>', A,u(x) ::::; c>'}1

1

::::; C exp ( -

~2) I{x E ~n : N{3u(x)

> >'}I

(4.0.13)

4. Sharp Good-A Inequalities for A and N

98

and I{x E IR n : Aj3u(x) > KA, Nau(x) :::: cA}1

:::: c3 exp ( - ~;)

I{x E IR n

:

Aj3u(x) > A}I·

(4.0.14)

Notice, however, that unlike (4.0.9) or (4.0.10), (4.0.13) does not give the full subgaussian decay in c. What is needed to improve the c to c 2 is a version for harmonic functions of the Key Estimate 4.0.5. For harmonic functions on IR~+l this can readily be obtained from the result of Chang, Wilson and Wolff, Theorem 2.0.3. However, for the good-A inequality, the result in IR~+l is not sufficient. As we will discuss below, and as the methods of Section 4.2 will illustrate, what is needed is such an estimate on Lipschitz domains. This is not known at this time. As in the proof of the LIL in Chapter 3, we will overcome this difficulty by directly reducing the desired inequality to the dyadic martingale result (4.0.10) using the techniques from Chapter 2. We divide the results in this Chapter into 4 sections. In Section 4.1 we prove (4.0.13) with 1/c2 in the exponential. In Section 4.2 we prove the inequality (4.0.13) not only for IRn but also for Lipschitz domains. This is obtained in the more "traditional" way using sawtooth regions with an estimate similar to the Key Estimates 4.0.8 and 4.0.9. The Lipschitz domain result is crucial for the applications to the Chung-type LIL proved in Section 4.3. That is, the good-A (4.0.14) just on IR n , as sharp as it is, does not give this LIL. The point to be stressed here is that once we have the inequalities in Lipschitz domains we can literally copy the martingale proofs. Finally, as in [FGSS] and [MD], we were partly motivated to improve the c 2 decay in the good-A inequalities by their applications to the behavior of the LP-constants and to ratio-type inequalities for A and N. In a sense to be made precise later, the better the good-A inequality, the better the information on these quantities. These applications are presented in Section 4.4. It should be mentioned here that just as with most of the results in this monograph, ratio inequalities were first proved for martingales. (See Garsia [Gar] or M. Barlow and M. Yor [BY1], [BY2].)

4.1

Sharp control of N by A

Theorem 4.1.1 Fix 0 < f3 < a. There are positive constants K, C 1 and C2 , with K > 1 and all depending only on a, f3 and n, such that whenever u is harmonic on IR~+l, 0 < c < 1 and A > 0 I{x E IR n

:

Nj3u(x) > KA, Aau(x) :::: cA}1

:::: C 1 exp ( -

~~) I{x E IRn : Nj3u(x) > A}I·

4.1 Sharp control of N by A

99

We first state and prove a lemma from Burkholder and Gundy [BG2]. We will use this not only in the proof of Theorem 4.1.1 but in several other places in the text. Lemma 4.1.2 Suppose, > f3 > 0 and u is continuous on 1R~+I. Then for all A > 0,

Here C depends only on" f3, and n. Proof: For (x,y) E 1R~+1 set B(x,y) = {s E IR n : Ix - sl < y}. If a point (x,y) has lu(x,y)1 > A then B(x,f3y) ~ {s: N{3u(s) > A}. Set E = {s E IR n : N{3u(s) > A}. Suppose x E IR n has Nyu(x) > A. Then there exists (s,y) E 1R~+1 with lu(s,y)1 > A and x E B(s"y). Note that then B(s,f3y) ~ E. Thus, MXE(X) > 1 { XE(t)dt - IB(x, (r + f3)y) I JB(x,b+[3)Y)

> IB(s, f3y) n B(x, (r + f3)y) I IB(x, (r + f3)y) I

-

IB(s,f3y)1 IB(x, (r + f3)y) I

= Set c =

('!f3)n.

(~) n. Then by Theorem 1.4.3 (the Hardy-Littlewood maximal theo-

rem),

1 e

I{Nyu(x) > A}I ::; I{MXE(X) > e}1 ::; -llxEilI

=

1 e

-lEI.

The following lemma localizes Theorem 4.1.1 to a cube. We recall that for h > 0, N$u(x) denotes the truncated (at height h) nontangential maximal function. Lemma 4.1.3 Suppose f3 < a. There are positive constants K, C 1 , and C2 , with K > 1 and all depending only on a, f3 and n, such that if Q is a cube in IR n with center q, h = 2£(Q)v'nlf3, and u is harmonic in 1R~+l with u(q, h) = 0, then for all A > 0 and 0 < c: < 1,

I{x E Q : N$u(x)

> KA, Aau(x) ::; cA}1 ::; C1exp ( -

~~)

IQI·

Proof of Theorem 4.1.1: Let us first show how this lemma implies the theorem. Let 'Y be very large and let R = {x E IR n : N"Iu( x) > A}. By the previous lemma,

4. Sharp Good-A Inequalities for A and N

100

IRI ::::: I{x E ~n : Nf3u(x) > A}I with constants depending only on f3,'Y and n. In particular, we may assume IRI < 00. Let {Qj} be a Whitney decomposition of R. Then for each j we can pick a point Xj E RC such that IXj - xl < CR.(Qj) for every x E Qj; here C is a constant which depends only on n. For each j we let qj denote the center of Qj and let hj = 2R.(Qjhrn/f3 be as in the statement of Lemma 4.1.3. Also, for each j we set T;j(x) = rf3(x) - rf3h j (x); this is the "top" part of a cone. If "( is large enough depending only on f3 and C, T;J (x) ~ r ')'(Xj) whenever x E Qj and thus, for every x E Qj, (4.1.1)

Let x E Qj have Nf3u(x) > KA and A,u(x) ::; cA. Then since N')'u(xj) ::; A, (4.1.1) gives KA

< N~ju(x).

(4.1.2)

Since we also have lu(%,hj)l::; A, it follows from (4.1.2) that

Therefore if we choose K = that

(K + 1), where K

is as in Lemma 4.1.3, we conclude

{x E Qj : Nf3u(x) > KA, Aau(x) ::; cA} ~

{x

E

h·

-

Qj : N/ (Iu - u(qj, hj)I)(x) > KA, Aau(x) ::; cA}.

But, by lemma 4.1.3, for each j

Summing over Whitney cubes proves Theorem 4.1.1. Proof of Lemma 4.1.3: We may assume that Q is the cube [-!, !In = Qo and = O. In this case, h = 2vn/ f3. We continue to use the notation of Chapter 2 and the proof of the upper-half LIL, Theorem 3.0.4: set p = f3 / (32vn) (recall that the kernels K, P, and q used in Section 2.2 were all supported in B(O, p» and cm = (2- m - 2 R.(Q»/4p. Thus, with this notation, h = 2c1. Suppose Xo E {x E Qo : N3u(x) > KA, A",u(x) ::; cA}. Then there exist (y, t) E r~(xo) such that q

\u(y, t)\ > KA and an m such that any 0 < 'Y < 1,

cm+!

<

t ::; Cm. By (3.1.13) and (3.1.14), for

\u(y, t) - v(xo, cm)1 ::; Ca,f3,')',nA~u(xo; "(t) ::; Ca,f3,')',n cA

(4.1.3)

101

4.1 Sharp control of N by A

and thus, fixing 'Y

=

~, say,

(4.1.4) where C depends only on a,(3 and n. By (3.1.5) we also have N

N

Iv(xo,cm)1 ~ Iv(xo,cm) - Lf~)(xo)1 j=l

+ I Lf~)(xo)1 j=l

(4.1.5)

N

~ Iv(xo,2cdl+ CA a u (xO)+ Llf~)(xo)l. j=l Arguing as in the proof of (3.1.14) and recalling U(0,2cl) that

Ln = ILn

Iv(xo, 2cl)1 =

~

I

P2el

= 0), we obtain

(xo - iJ)u(ij, 2c 1 )diJ l

P2e1 (xo - iJ)(u(iJ, 2cl) - u(O, 2c 1 ))diJ

(4.1.6)

l

Ca,,a,y,nAau(XO) ~ Cc>..

Here for the penultimate inequality we have used the fact that if Ixo - iJl < 2cIP, then (iJ,2cl) E f,a(x) and the fact that (0,2cl) E f,a(x) (both of which are immediate from our choice of P = (3/(32.jn) ) and applied Lemma 2.3.1. From (4.1.4), (4.1.5) and (4.1.6) it follows that N

L If~)(xo)1 2 (k - Cc)>.,

(4.1.7)

j=l where C = Ca,,a,n' By (4.1.7), there exists a 1

~ j ~

(f(j))*(xo)

2

N, which may depend on Xo, such that

(k ~

Cc) >. > 2>',

(4.1.8)

if k is chosen large enough depending only on Nand Cc. Since 0 < c < 1, k depends only on N and C and hence only on a, (3 and n. Also, since Aau(xo) ~ c>', Lemma 2.2.3 gives that S(f(j))(xo) ~ C'c>.,

for all 1 ~ j ~ N. With

k

(4.1.9)

and C' as above we conclude that

{x E Qo: Ngu(x) > k>., Aau(x) ~ c>'} N

~ U{x E Q: (f(j))*(x) > 2>', S(f(j))(x) ~ Cc>.}. j=l

(4.1.10)

102

4. Sharp Good-A Inequalities for A and N

Because we have restricted our attention to Qo, we can consider that each of the martingales f(j) is supported on a slightly larger cube containing Qo and that all of these larger cubes are in turn contained in a cube which has the same center as Qo but with three times the sidelength. (See the comments immediately after Lemma 2.1.3.) We now apply (4.0.11) of Theorem 4.0.7 to each of the terms on the right hand side of (4.1.10) to complete the proof of Lemma 4.1.3. We remark that in Lemma 4.1.3 it is not really necessary that h have the precise quantity stated in the hypothesis and, in fact, the result is valid with any larger value of h. (Although changing h would change the values of K, C 1 and C2 .) What was important to the proof was that we chose h = 2c1 where C1 = 1/(32p). Recall that in the proof of Theorem 3.0.4 we set a definite value of p = f3/(32fo) but also remarked that any smaller value would have worked. A smaller choice of p then merely results in a larger value of h. Just as in the 1IL of Chapter 3, we also have a version for caloric functions. Theorem 4.1.4 Let u be a caloric function in IR~+1 and 0 < f3 < Ct. There are positive constants K, C 1 and C 2 , with K > 1 and all depending only on Ct, f3 and n, such that for all 0 < C < 1 and A > 0

(4.1.11)

The changes in the proof of Theorem 4.1.1 needed to obtain the caloric case are very minor. Put u(x, t) = u(x, t 2 ) as in lemma 2.2.2. Then since PNf3u(x) = Nf3u(x) and PA",u(x) = A",u(x), (4.1.11) follows from:

:s: cA}1 ~~) I{x E IR n : Nf3u(x) > A}I·

I{x E IR n : Nf3u(x) > KA, A",u(x)

:s: C1 exp ( -

(4.1.12)

This can be proved exactly as Theorem 4.1.1. In fact, everything we have done in that proof applies equally well to U.

4.2

Sharp control of A by N

In this section we will prove a version of the Murai-Uchiyama good-A inequality, (4.0.14), for Lipschitz domains. To do this, we follow a strategy common to many similar results. For example, in IRn, the good-A inequality controlling A by N can be shown by considering a "sawtooth" region over {x E IRn : N",u(x) > CA}, that is, by considering E = {x : N",u(x) > CA}, W = UXEEc r ",(x), and then estimating a version of the area integral defined on aw. Note that with this strategy it results that aw is the graph of a Lipschitz function. However, if instead, E were a set on a Lipschitz graph then the corresponding aw would still be the graph of a

4.2 Sharp control of A by N

103

Lipschitz function. Consequently, since our estimation of A on oW will make use of techniques from the study of Lipschitz domains, it causes us no extra effort to show the good-,\ inequalities in the more general setting of Lipschitz domains. These good-'\ inequalities have the added benefit that they lead immediately to LIL's and this is the main reason why we present them in this generality. The techniques we will use in this section have been used by several authors. The idea of working on a "sawtooth" domain goes back to Privalov [Prj (see also Zygmund [Zy2, vol. II, p. 200]). The proofs of the good-,\ inequalities of Burkholder and Gundy [BG2]' and the subsequent improvements by Fefferman, Gundy, Silverstein and Stein [FGSS], and Murai and Uchiyama [MU] (together with their martingale versions) provide not only an outline for our proof, but indispensable techniques. These works are themselves descendants of the earlier work of Privalov, Marcinkiewicz and Zygmund, Spencer, Calderon and Stein, who showed that, except for sets of Lebesgue measure zero, the nontangential maximal function and Lusin area function are finite on the same sets. (See Section 1.8.) The work of Dahlberg [Da1], [Da2] will provide us with many of the necessary tools for estimates on Lipschitz domains. We recall that ¢ : IR.n ----+ IR. is called a Lipschitz function if there exists a constant M such that I¢(x) - ¢(y)1 ::; Mix - yl for all x, y E IR. n . The smallest such M for which this remains valid for all x and y will be called the Lipschitz constant of ¢. Consider a Lipschitz graph {(x, ¢(x)) : x E IR.n}. For points P in this graph, we define r a(P) to be the cone r a(O) translated so that its vertex is at P. That is, if P = (x, ¢(x)) then

r a(P)

=

{(s, t) : Is - xl < a(t - ¢(x))}.

We similarly define the truncated cones r~(p)

= {(x, t) : Is -

xl < a(t - ¢(x)), t < ¢(x)

+ h}.

Consider a function u which is harmonic on an unbounded Lipschitz domain D = {(x, y) : y > ¢( x)}. Consider any a > 0 for which there exists a' > a such that r a' (P) <;:;; D for all P E aD. (The allowable size of such a depends only on the Lipschitz constant of ¢.) We then define the nontangential maximal function and Lusin area function of u respectively by:

Nau(P)

=

sup{lu(x, y)1 : (x, y) E

r a(P)},

(4.2.1) (4.2.2)

for P = (xo, ¢(xo)) E aD. Our main result is:

4. Sharp Good-A Inequalities for A and N

104

Theorem 4.2.1 Suppose 0 < {J < and C 2 , all depending only on o < C < 1, A > 0 then

Q,

Q. There is a constant K > 1 and constants C 1 {J, n and the Lipschitz constant of ¢ such that if

IT{P E aD: A,au(P) > KA, Ncr.u(P)::; CA} ::;

Here IT represents surface measure on aD.

The proof of this is long and much of the work will be done via a sequence of lemmas. These lemmas will also be useful for our proofs, in Chapter 5, of good-A inequalities involving the D-functional, so we will write these in sufficient generality to be applicable here and in that chapter. As we have noted, the proof of this theorem involves the construction of a "sawtooth" region and estimates of the area function on the boundary of this region. Specifically, we will estimate the BMO norm of the square of the area function on the boundary of this region. We will first need to collect some general facts about BMO. For the basics on BMO, see, for example, Garnett [Gal, or Bass [Bas2] for a probabilistic point of view. Lemma 4.2.2 Let f : ]Rn ---t ]R be measumble. Suppose that there exist an s, 0 < s ::; ~ and a AO > 0 such that for every cube Q ~ ]Rn there is a constant aQ such that I{x E Q : If(x) - aQI > Ao}1 < slQI· Then f E BMO and IlfllBMO ::; CAo where C depends only on n. This is proved in Stromberg [Str]. Its proof resembles that of the John-Nirenberg theorem. The following lemma will ultimately be applied to the square of the area function, thus producing the harmonic function analogue of Key Estimates 4.0.8 and 4.0.9 above. This essentially appears in Murai and Uchiyama [MUl, but we shall give a proof since we require a slightly stronger result than what is obtained there. Lemma 4.2.3 Suppose IlfllBMO ::; 1. Then for all A > 0, I{x

E]Rn : If(x)1

> 2A}1 ::; C1 exp(-C2 A) I{x

E]Rn : If(x)1

> A}I,

where C 1 , C 2 are positive constants which depend only on n. Proof: Fix A. We may assume that I{x E ]Rn : If(x)1 > A}I < 00. Let {Qi} be the maximal dyadic cubes in ]Rn such that I{x E Qi : If(x)1 > A}I > ~IQil. Then {x E ]Rn : If(x)1 > A} ~ UQi almost everywhere. Since IlfllBMO ::; 1, IfQi - f Qi I ::; 2n for every i ; here fQi denotes the average of f over the cube Qi (similarly for f Qi ) and Qi is the unique dyadic cube containing Qi which has twice the sidelength. The maximality of the Qi and the fact that IlfllBMO ::; 1 implies that for each i, If 2 + A. Thus IfQi I ::; A + "I, where "I = 2n + 2. The John-Nirenberg theorem applied to each Qi then yields

Q.I ::;

I{x E Qi : If(x)1 > 2A}1 ::; I{x E Qi : If(x) - fQ.I > A - "I} I

4.2 Sharp control of A by N

105

Summing over i gives the conclusion of the lemma. We will also need to collect some facts about Lipschitz domains. Since we will want BMO estimates on such domains, we will need to make local estimates. This will be accomplished using auxiliary domains. These domains will be bounded Lipschitz domains constructed above "cubes" on the original unbounded Lipschitz domains. We say that a bounded domain 0 ~ JR~+1 is a starlike Lipschitz domain if it satisfies the following two properties. First, we require that the boundary of 0 is locally Lipschitz, that is, for each P E 00 there is a coordinate system (z, t), z E JRn, t E JR, and a Lipschitz function 't/J such that for some neighborhood V of P, V n 0 = {(z, t) : t > 't/J(z)}. We require the existence of an M so that all such 't/J have Lipschitz constant at most M. Second, we assume that there exists a point P* EO such that for every point P E 00, {tP + (1 - t)P* : 0 ::::; t ::::; I} ~ O. In this case we say that 0 is a Lipschitz domain which is starlike with respect to P*. Now consider such a bounded starlike Lipschitz domain 0 which has the additional property that there exists (x' > 0 such that for every P E 00 the cone r~, (P) with aperture (x', vertical axis P P* and height h = IP - P* I is contained in O. If (X < (x' and P E 00 we consider the cone r~(p) with aperture (x, vertical axis PP* and height h = IP - P*I, and for U harmonic on 0 we then define Nau(P) and Aau(P) as before except now we use the cones r~(p). That is, we define Nau(P)

= sUp{IU(8, t)1 : (8, t) E r~(p)}

and (4.2.3) where, in the definition of Aau(P) we have written 't/J for the Lipschitz function defining the boundary in a neighborhood of P = (xo, 't/J(xo)). This Aau(P) is, up to multiplicative constants which depend only on the Lipschitz constant of the domain, equivalent to the coordinate free formula (4.2.4) where for any P E 0 we set d(P) = dist(P, (0). Because these two versions of Aa u are equivalent up to constants, we will be able to do much of our estimation using the version of (4.2.4) even though we are ultimately interested in a version like (4.2.3). We now consider Gn(P, P*), the Green's function for 0 with pole at P* which, for brevity, we will denote as G(P) throughout this section, and let w denote

4. Sharp Good-.\ Inequalities for A and N

106

harmonic measure on an taken with respect to the point P*. For P E 0,\ {P*} we let P be the point of intersection of an with {r(P - P*) + P* : r > O} and for P E an we define fl (P, r) = an n {P' : IP' - PI < r}. Because of our requirement that r~, (P) ~ 0, for every P E an, there exists a constant ro > 0 such that B(P*,ro) = {P' E ]Rn+1 : IP' - P*I < ro} ~ n. In fact, there exists a constant C(o;') depending only on a' such that ro ~ C(o;')diam(n). Set 0,* = n\B(p*, ro). The following appears in Dahlberg [Da1], [Da2]. Lemma 4.2.4 There is a constant k > 0 such that for all P E 0,* we have

(4.2.5)

The constant k depends only on the Lipschitz constant of 0" the aperture 0;', and the dimension n. Let (J denote surface measure on an. Dahlberg [Da1] has also shown that w and (J are mutually absolutely continuous and that in addition they satisfy an Aoo condition. More precisely, There are constants b > 0, a > 0, C > 0 depending only on the Lipschitz constant ofn such that if E ~ fl = fl(P,r) ~ an then

Lemma 4.2.5

w(E) < C (dE))b and (J(E) < C (W(E))a w(fl) (J(fl) (J(fl) w(fl)

(4.2.6)

We can now describe the construction of our auxiliary domains which will allow us to estimate the BMO norm of the area function on the boundary of a Lipschitz domain. If D is a Lipschitz domain, then we will call a set Q ~ aD a cube if Q = {(x, 1jJ(x)) : x E Q'} where Q' is a cube in]Rn and 1jJ(x) is the Lipschitz function defining D (either globally or locally). We now consider an unbounded Lipschitz domain D = {(x,y) : x E ]Rn, y E]R, Y > 1jJ(x)} where 1jJ is Lipschitz and we fix Q ~ aD. For u harmonic on D we consider Ai3u(P), defined as in (4.2.2), with P E Q. We will separately consider the contribution to Ai3u(P) from the part of the cone r i3 (P) which is "close" to Q and the part of the cone r i3 (P) which is "far" from Q. To estimate the part which is "close" we will construct an auxiliary domain n. Choose "(' and "( so that "(' > "( > (3 and so that r "I' (P) ~ D for every P E aD. As above we write Q = {(x,1jJ(x)) : x E Q' ~ ]Rn} and let us denote by Xo the center of Q' so that (xo,1jJ(xo)) is a "center" for Q ~ aD. There exists R > 0 with the following properties:

+ Re(Q')).

Then for every P E Q, {tP

+ (1 -

t)P* :

(i)

Set P* = (xo, 1jJ(xo) 0:::; t:::; I} ~ D.

(ii)

For every P E Q, the cone r~(p) with aperture ,,(, vertical axis P P*, vertex at P and height h = IP - P* I is completely contained in the cone r'Y"(p), where "(" = =y'~1.

4.2 Sharp control of A by N

(iii)

107

For every P E Q, the cone r2(p) with vertical axis {(x, 1/J(x) + 8) : 8 > O} and height h = 1/J(xo) + R£(Q') -1/J(x) is completely contained in the cone r, (P) with aperture ,,(, vertex at P and vertical axis P P* .

We assume that R is the smallest such constant for which (i), (ii) and (iii) holds for all cubes Q <::;; aD. The constant R then depends only on (3, ,,(, "(' and the Lipschitz constant of 1/J. But ,,(, "(' can be chosen to depend only on (3 and the Lipschitz constant of 1/J; hence R depends only on the Lipschitz constant of 1/J and the aperture (3. For P = (x,1/J(x)) E Q, let r,l(p) denote the cone with vertex P, vertical axis {(X,1/J(X) +8): 8 > O} and set

n =

U r,l(p) n {(x,y) : y < 1/J(xo) + 2R£(Q')}. PEQ

The domain n is Lipschitz and starlike with respect to the point P*. Furthermore, there exists an KO > 0, depending only on (3 and the Lipschitz constant of 1/J, such that for every PEon there exists a K ~ KO so that the cone r~(p) with vertex at P, height h = IP - P* I, aperture K, and vertical axis P P* is completely contained in a larger cone which is contained in n. In fact, for P E Q <::;; an, we may take r~(p) to be the cone r~(p) given by (ii) above. For P E Q set r1(p) = r2(p) where r2(p) is given by (iii) above and set r 2 (p) = rj3(p)\rdP). For j = 1,2, P = (x, 1/J(x)), we set

For typographical convenience, we shall henceforth write A~u(P) for (Aj3u(p))2 and similarly write AJu(p) for (A j u(p))2, j = 1,2. Then with this notation and the previous definitions, we have A~u(P) = Aiu(p) + A~u(P). The basic idea of splitting the area function into two pieces for the proofs of good-A inequalities goes back to Burkholder and Gundy [BG2]. The splitting of Aj3u which we have just introduced is essentially found in Murai and Uchiyama [MU]. Kurtz [Ku] shows that under appropriate hypotheses, IIAj3fIIBMO :::; CllfIIBMO. He uses a similar splitting of Aj3f. Dahlberg [Da2] contains a similar idea. We will first estimate A1u(P). The following lemma is based on the work of Dahlberg [Da2]. It follows from his Lemma 3. However, the present lemma is just a simpler version of Dahlberg's and will be sufficient for our purposes. The proof is similar to Dahlberg's lemma, but a little easier. This argument is somewhat reminiscent of the Green's theorem arguments used in the proofs of Theorems 1.1.3 and 1.2.3.

4. Sharp Good-A Inequalities for A and N

108

Lemma 4.2.6 There is a constant C, depending only on (3 and the Lipschitz constant of'ljJ, such that if lui::; 1 on D and Q ~ aD is any cube, then

k

Aiu(P)dw(P) ::; C,

where C is independent of Q. Proof: We consider domains no ~ n, E > 0 which have no ~ n and such that for every E > 0, P* E no, no ~ n and no has smooth boundary. (See Stein [St4, p. 206] for this construction.) Let Go denote the Green's function for no with pole at P*. We then apply Green's theorem to the functions (u(p))2 - (u(p*))2 and Go(P) on no. Since (u(p))2 - (u(p*))2 vanishes at P = P*, standard arguments (see, for example, the proofs of Theorems 1.1.3 and 1.2.3) yield

Since u ::; 1 on D "2

As

E

~

0,

no, this implies

no ~ n,

Go(P)

~

JJ

0

G(P) and consequently, we obtain

IVu(PWG(P)dP ::; 1.

For PEn, pI E Q set X(P, PI) = 1 if PErl (PI), 0 otherwise. Then, using the Green's function estimate of Lemma 4.2.4 we obtain:

JJ IVu(PWG(P)dP ?: C JJoIVu(P)12d(p)1-nw(A(p, d(P))dP ~ C JJoIVu(P) 12 d(p)1-n kx(P, z)dw(z)dP C 10 JJoIVu(P) 12 d(p)1-n x (p, z)dPdw(z) ?: C 10 Aiu(z)dw(z),

1 ?:

0

=

which completes the proof of the lemma. The next result controls the contribution to A{3u from the "tops" of the cones, that is, we estimate A 2u. The result is simply that A 2u(P) does not vary much as P ranges over Q. This is similar to lemma 3.1 of Murai and Uchiyama [MU]. This lemma will be useful not only here but also in subsequent sections.

4.2 Sharp control of A by N

109

Lemma 4.2.7 With the notation as above, suppose additionally that for every P E D, dist(P,8D)\V'u(P)\ ::; 1. If there exists a Po E 8D such that A,Bu(Po) < 00, then for every PI, P2 E Q, (4.2.7)

Here, the constant 0 depends only on (3 and the Lipschitz constant of 'IjJ. Proof: Since dist(P,8D)\V'u(P)\ ::; 1 on D, then if (s,t) E r,B(x,'IjJ(x)), (t'IjJ(x))\V'u(s, t)\ ::::; 0 where 0 depends only on (3 and the Lipschitz constant of 'IjJ. Set PI = (x, 'IjJ(x)) , P2 = (y,'IjJ(y)). Then

IJr[

2 (p,)

1

\V'u(s, tW(t - 'IjJ(x))l- ndsdt - [

Jr

::; G(A) \V'u(s, t)\

2

(p,)

\V'u(s, t)\2(t - 'IjJ(y))I-ndsdtl

2\(t - 'IjJ(y))n-l - (t - 'IjJ(x))n-l\ (t- 'IjJ()) x n -1(t- 'IjJ()) y n -1 dsdt

(4.2.8)

For any two sets E and F, denote by EIIF their symmetric difference. We have for fixed t,

\(r 2 (Pdllr 2 (P2 ))

n {(s, t -

'IjJ(y)) : s E ~n}\ ::; O(t - 'IjJ(y))n-l\P1 - P2 \.

Therefore,

(4.2.9)

4. Sharp Good-A Inequalities for A and N

110

Estimates (4.2.8) and (4.2.9) give the result assuming that one of A 2 u(Pi ), i = 1,2 is finite. But these estimates plus the assumption A,Bu(Po) < 00 imply A 2 u(P) < 00 for all P E Q. Hence, we obtain the conclusion of the lemma. We can now obtain the necessary BMO estimates of area functions on Lipschitz domains. Notice that this proposition, together with Lemma 4.2.3, gives the analogue for harmonic functions of Key Estimates 4.0.8 and 4.0.9. Proposition 4.2.8 Suppose /3 > 0 and u is harmonic on an unbounded Lipschitz domain: D = {(x,y) : x E JRn, y E JR, y > ,¢(x)} where,¢ is Lipschitz. Suppose that for every P E D, lu(P)1 1. Suppose also that there exists a point Po E aD at which A,Bu(Po) < 00. Then IIA~u(x, ,¢(x))IIBMO S C, where C depends only on /3, the Lipschitz constant of'¢ and the dimension n.

s

Proof: Fix a cube Q ~ aD. Define the auxiliary domain corresponding Al u and A 2 u. Lemma 4.2.6 implies w{P E Q:

A~u(P)

> A} S

~

S

Cw~Q)

n

as above and the

,

where w is harmonic measure on n taken with respect to P* and where for the last inequality we have used Lemma 2.1 in Hunt and Wheeden [HWJ. Since wand (1 satisfy the ADO condition (4.2.6) on an, then (1{P E Q : Aiu(p) > A} S C~~Q) for some b > O. Since '¢ is Lipschitz,

I{x E Q': A~u(x,'¢(x)) > A}I S

ClQ'I Ab.

(4.2.10)

Since lu(P)1 S 1, the gradient estimate, Lemma 2.3.1, and Lemma 4.2.7 imply that (4.2.11) whenever x, y E Q'. Let Xo be the center of Q'. Then (4.2.10) and (4.2.11) imply

I{x

E

Q' :IA~u(x, ,¢(x)) - A~u(xo, ,¢(xo)) I > A}I

S I{x

E Q':

A~u(x,'¢(x)) > A -

*

C*}I S

(A~~~)b·

If AO is chosen so that (>'O~C.)b = then Lemma 4.2.2 implies that A~u(x, ,¢(x)) is in BMO and IIA~u(x, ,¢(x))IIBMO S C, where C depends only on the choice of AO, and hence only on the Lipschitz constant of ,¢, the aperture /3, and the dimension n. Before finally proving Theorem 4.2.1 we need one more lemma. This lemma will allow us to compare area functions defined on the boundary of a sawtooth region to an area function defined on the original domain.

111

4.2 Sharp control of A by N

Lemma 4.2.9 Let u be a harmonic function defined on the cone r,8(O, 0). Suppose v < (3 and (x, d) E r v(O, 0). Then [ lV'u(s, t)12(t - d)l-ndsdt :::; L [ lV'u(s, tWt1-ndsdt, Jr{3(x,d) Jr{3(o,o) where L is a constant which depends only on (3 and v. In particular, if x may simply take v = ~ and then L depends only on (3.

(4.2.12)

=0

we

We remark that this is obviously true with L = 1 when n = 1. Proof: For (s,t) E r,8(x,d) let B«s,t), r) be a ball with center (s,t) and radius r = eod where eo < 1 is chosen so that B«s, t), r) <;:; r,8(O,O) for all such (s, t). Note that eo depends only on (3 and v. The subharmonicity of lV'ul 2 then yields [ lV'u(s, tW(t - d)l-n dsdt Jr{3(x,d) :::; [ IB/ )1 [ lV'u(w, zWdwdz (t - d)l-n dsdt Jr{3(x,d) s, t iB«s,t),r)

:::;

d~l

[ [ XB«w,z),r)(s, t)(t - d)l-ndsdtlV'u(w, z)1 2 dwdz. (4.2.13) Jr{3(o,o) Jr{3(x,d)

We estimate the inner integral. For z > 3d, [ XB«w,z),r)(s, t)(t - d?-n dsdt :::; Cz1-nIB«w, z), r)1 :::; Jr{3(x,d)

C~+lzl-n. (4.2.14)

For

z:::; 3d, [ XB«w,z),r)(s, t)(t - d)l-n dsdt ir{3(x,d)

:::; [

1 )dl

Jr {3 (x,d)n{ (8,t):t«3+eo)d}

:::; c

(2+e o

o

Isl<,8t

(t - d)l-ndsdt (4.2.15)

tl-ndsdt

=Cd2 :::; CdnHz 1 - n .

Simply combine (4.2.13), (4.2.14) and (4.2.15) to obtain the conclusion of the lemma. We are finally in a position to prove Theorem 4.2.1. Proof of Theorem 4.2.1: Fix e, A and assume that cr{P E aD : A,8U(P) > A} is finite. Let E = {P E aD : Nau(P) > eA} and set D' = UPEEc r a(P). We

4. Sharp Good-A Inequalities for A and N

112

may also assume E C -I- 0 so that D' -I- 0. Then D' is a Lipschitz domain with constant determined by a, say D' = {(x,y) : y > ¢(x)} where ¢ is Lipschitz. On D ' , lui ~ cA. So by Proposition 4.2.8, IIA~u(x,¢(x))IIBMO ~ C(cA)2 where C depends only on the Lipschitz constant of ¢, the aperture {3 and the dimension n. Then, by Lemma 4.2.3, for all 'TJ > 0,

Choosing

'TJ

= A2 yields

I{x E lRn

:

Af3u(x, ¢(x)) > V2A}1 ::; C 1 exp

(-c~2 )

I{x E lRn

:

Af3u(x, ¢(x)) > A}I.

(4.2.16)

Set K = V2L where L is the constant appearing in Lemma 4.2.9. Then, using (4.2.16) and Lemma 4.2.9 we obtain:

a{P E aD : Af3U(P) > KA, Nau(P) ~ cA} = a{P E E C : Af3u(P) > KA} ~ a{P E aD' : Af3U(P) > KA} ~ CI{x E]Rn: Af3u(x,¢(x)) > KA}I

~ C1 exp ( -c~2 ) I{x E lR ~

-C2 ) a{P E C 1 exp ( ~

n :

Af3u(x, ¢(x)) >

~}I

aD : Af3(P) > A}.

This completes the proof of Theorem 4.2.1. For caloric functions the corresponding good-A inequality for the case D lR~+l was proved in M. Kaneko [KaJ.

=

Theorem 4.2.10 Let U be a caloric function in lR++l and 0 < {3 < a. There are constants K, C~ and C~, with K > 1 and all depending on a, (3 and n only, such that for all 0 < c < 1 and A > 0, I{x E lRn

:

PAf3u(x) > KA, PNo.u(x) ::; cA}1 ::;

C~ exp (- ~i) I{x E lR

n :

PAf3u(x) > A}I·

Actually, this is just a special case of Kaneko's result. Kaneko considers certain second order differential operators L and functions u for which Lu is a positive Borel measure, and shows a good-A inequality between an area function and a

4.2 Sharp control of A by N

113

nontangential maximal function defined for these. In the case when u is harmonic, these are just the usual area function and nontangential function of u. In the particular case here when u is a caloric function, the change of variables u(x, t) = u(x, t 2 ) yields a function u(x, t) which satisfies Lu = 0 where L = t..x - ~t-1 %t and the area function and nontangential maximal function he considers reduce to the usual parabolic area and parabolic nontangential maximal function. His results also include the area function of subharmonic functions studied by T. McConnell [Mc]. We have not checked whether Theorem 4.2.10 also holds for Lipschitz domains. (Recall, with the change of variables u(x, t) = u(x, t 2 ) one obtains P A,Bu(x) = A,Bu(x) and P NO/.u(x) = NO/.u so that parabolic regions never enter into the proof. It is thus perfectly reasonable to ask for such results for Lipschitz domains.) The reader may also wish to consult R. Gundy and 1. Iribarren [GI] where certain types of good-A inequalities are proved for functions which are not solutions to partial differential operators. However, the inequalities in [GI] do not have the sharpness of those presented here and it is not known whether their inequalities can be improved to inequalities having gaussian behavior. We now elaborate some more on the status of Theorem 4.1.1 on Lipschitz domains. As we have already noted, the missing ingredient to extend this result to Lipschitz domains is the following open problem which is the exact analogue for harmonic functions of our Key Estimate 3.1.1 for martingales. Problem 4.2.11 Let D be a Lipschitz domain, D = {(x, y) : y > if> (x)} , where if> : IRn - t IR is Lipschitz. Suppose u is harmonic in D with boundary values f and that IIAO/.uIILOO(oD) = IIAO/.ull oo < 00. Prove that for any cube Q caD,

for all A > 0 where 0 1 and O2 are constants. One possible approach to this gives rise to another problem. With D as above, we define : 1R~+1 - t D by (x,y) = (x,y + if>(x)). If u is harmonic on D, then u 0

114

4. Sharp Good-A Inequalities for A and N

the boundary of the conformal mapping from D onto IR~ and applying the ChangWilson-Wolff result, Theorem 2.0.3, for the half space. With this result the above arguments give Theorem 4.2.12 Suppose D = {(x,y) : y > ¢(x)} C 1R2 where ¢ is Lipschitz with constant M. Let 0 < (3 < a with 1 < a < 11M. There are constants C 1 and C 2 such that for all A > 0 and 0 < 10 < 1, cr{P E aD : Nf3U(P) > 2A, Aau(P) ~ lOA}

~ C 1 exp ( - ~~) cr{P E aD : Nf3U(P) > A}. Remark 4.2.13 Theorem 4.2.12, combined with the Borel-Cantelli argument of the upper-half of the Kolmogorov LIL, Theorem 3.0.1, can be used to prove Theorem 3.0.4 in two dimensions. This technique will be clearly illustrated in Section 4.3 below where we prove a Chung-type LIL. Recall that in Section 3.4, we considered another area function, the LittlewoodPaley g* function, and showed, just as we did for the Lusin area function, that if a function has this area function bounded, then it is exponential square integrable. Along these lines, one open problem which has been of some interest concerns the Littlewood-Paley g-function. By Theorem 1.7.5, for 1 < p < 00, if f E V', then Ilg(f)llp ~ Ilfllp" There are also BMO estimates for the operator f --+ g(f). (See Torchinsky [To], or Journe [Jou] for these.) But it is not known if any subgaussiantype estimates hold for g. Naturally, since g(u) is pointwise smaller than Aau (see the remark immediately following Definition 1.7.6), it should be more difficult to use g( u) to control u or any maximal function of u. The obvious point to begin such an investigation would be with the work of Chang, Wilson and Wolff discussed in Chapter 2. We recall their theorem (Theorem 2.0.4): Suppose a > O. There exists constants C 1 and C 2 , depending only on a and n, such that if f has Aaf E U X ), then for any cube Q ~ IRn, (4.2.17)

Question 4.2.14 Does such a theorem hold with g(f) in place of Aaf? That is, are there constants C 1 and C 2 depending only on n such that if f has g(f) E £00, then for every cube Q ~ IRn, (4.2.18) To appreciate the complications involved when considering the g-function, let us revisit some of the estimates of Chapter 2. We recall the estimate of Theorem 2.0.1:

4.2 Sharp control of A by N

115

If {In} is a dyadic martingale on Q ~ ~n with fo = 0 and with limit function f, then for all A > 0,

I~I 10 exp (Af(X) - ~2 (Sf(x))2) dx :::; 1.

(4.2.19)

This can be used to show an analogue of itself for harmonic functions. Consider a harmonic function u on ~~+1. Suppose Q ~ ~n is a cube, and suppose that on Q, u(x, y) has a non-tangential limit, call it u(x). Let q be the center of Q and let h be as in Lemma 4.1.2. We claim: There exists constants C 1 and C2 , depending only on 0: and n, such that for every A > 0, (4.2.20) This gives an analogue for harmonic functions of the inequality (4.2.19). The proof of (4.2.20) is an application of the ideas in the proof of Lemma 4.1.3. Using the notation of that proof, we note that limm-+oo v(x, Em) = u(x) for almost every x and since v(x, Em) can be approximated by a sum of martingales with the square function of each of these martingales pointwise dominated by Ca,nAau, we can pass from the martingale estimate (4.2.19) to the harmonic function estimate (4.2.20). We leave the details of this to the reader. We now apply the same observations that allowed us to deduce Theorem 2.0.2 from Theorem 2.0.1. We replace u by -u in (4.2.20); the resulting inequality combined with (4.2.20) yields a similar inequality which has lu(x) - u(q, h)1 in place of u(x) -u(q, h). Then, if Aau E L oo , we substitute A = A' /2C11IAaull~ and rearrange to obtain:

This yields the estimate

which upon integrating yields (4.2.21 ) This is essentially (4.2.17). Since 'P(x) = exp(x 2 ) is increasing and convex, Jensen's inequality applied to (4.2.21) shows that IUQ - u(q, h)1 :::; ClIAauiloo and this and (4.2.21) then gives (4.2.17). Thus, (4.2.17) is a consequence of (4.2.20). Later, in Chapter 5, Lemma 5.2.9 will give another proof of the estimate (4.2.20). This proof will show that (4.2.20) follows from the good-A inequality of

116

4. Sharp Good-A Inequalities for A and N

Theorem 4.1.1. In lR~, Theorem 4.1.1 can be shown by using (4.2.17) together with the conformal mapping arguments of Problem 4.2.11 to produce a version of (4.2.17) on Lipschitz domains and, just as in the proofs of this section, this provides the necessary "stopping time" estimates for Theorem 4.1.1. Thus, in lR~, (4.2.17) implies (4.2.20) and from the discussion above, these are equivalent. As we have noted, we do not know if an estimate like (4.2.17) remains valid on Lipschitz domains in higher dimensions and this was precisely what prevented us from proving Theorem 4.1.1 directly using the martingale stopping time arguments. Our invariance principle was a way to circumvent this difficulty. Thus, in higher dimensions we do not know how to show that (4.2.17) implies (4.2.20). Nevertheless, heuristically, we can say that subgaussian estimates like (4.2.17) combined with stopping time arguments give good-A inequalities like Theorem 4.1.1 and by Lemma 5.2.9 these then imply an estimate like (4.2.20). However, here the subgaussian estimates and stopping time arguments are done in the setting of martingales and merely transferred over to harmonic functions. So, at the very least, we can claim that in higher dimensions, (4.2.17) "philosophically" implies (4.2.20). Given the relationship between (4.2.17) and (4.2.20), it seems clear that to prove (4.2.18), we should try to answer the following question: Suppose u is harmonic on lR~+1, Q ~ lRn is a cube and u(x, y) has a nontangentiallimit, call it u(x). Suppose also that q is the center of Q and h is as in Lemma 4.1.3. Does there exist constants 0 1 and O2 , depending only on n, so that (4.2.22) The answer is no. By a construction of Bagemihl and Seidel. [BS], there exists a harmonic function u(x, y) on lR~ which has g(u)(x) < 00 for almost every x but for which limy!o u(x, y) = 00 for almost every x. (See Stein [St4, p. 238] for the exact details on this application of the work of Bagemihl and Seidel.) Consider a cube Q and the functions ue(x,y) = u(x,y + e). If (4.2.22) were true, then in particular

for every t > 0 with 0 1 and O2 independent of t. Since g(ue)(x) ::; g(u)(x) for every t > 0, Fatou's lemma implies

But the left-hand side is infinite and hence (4.2.22) is false. However, this does not necessarily mean that (4.2.18) is false. For the Lusin area function in lR~ we showed above that (4.2.17) and (4.2.20) are equivalent. For the g-function, (4.2.22), although false, would imply (4.2.18) if it were true.

4.2 Sharp control of A by N

117

This follows from the same arguments used to show that (4.2.20) implies (4.2.17). But we cannot follow the line of reasoning used to show (4.2.17) implies (4.2.20) to produce a proof that (4.2.18) implies (4.2.22). One of the problems here is that there are simply no natural "stopping times" which we can use. Thus, there is no reason to believe that (4.2.18) and (4.2.22) are equivalent, hence the failure of (4.2.22) does not seem to necessarily indicate the failure of (4.2.18). Another possible approach towards a proof of (4.2.18) involves a reduction of the problem to an investigation of the constants in (4.2.17). Let Vn denote the volume of the unit ball in lRn. We observe that if, say u(x, y) is harmonic on lR~+1 and continuous on lR~+\ then limaLo Vn1a n A;u(x) = (g(u)(x))2. In equation (4.2.17), the constant C 1 depends on a. Let us suppose that we could show C 1 = o(an) asa 1 o. We could then pass to the limit in (4.2.17). This gives (4.2.18) in the case u is harmonic on lR~+l and continuous on lR~+l. For arbitrary u harmonic on lR~+1, (4.2.18) would follow from this case by considering the functions ue(x, y) = u(x, y + c). Unfortunately, a careful detailed examination of the proof of (4.2.17) yields at best C 1 = 0 (a 3n ) as a 1 O. Thus, for this approach, what is needed is a different proof of (4.2.17) which gives better control of the constant C 1 as a 1 O. Also relevant to the study of the g-function are the examples of P. Jones (personal communication) which further emphasize the difficulty of this endeavor. Jones has shown that there exist harmonic functions u and /1 > /2 such that A)'2 u E Loo but An u ~ L oo . These examples can readily be modified (see Moore [Mol]) to give an example of a harmonic function u for which A,u E Loo but A,IU ~ L OO for every /' > /, and an example of a harmonic function u for which g(u) E Loo but A,u ~ Loo for every / > O. In separate but related direction, let us consider a lacunary sequence of positive integers nj; no = 0, n1 = 1, with n~+l > J1 > 1 for every j. For f E L1([_7l', 7l')) with Fourier coefficients ak, we let Sn(lJ) = L~=-n akeik() denote the nth partial sum ofthe Fourier series of f and we set tljf(B) = Snj (B) - Snj_l (B), j = 1,2, ... , and tlo(B) = Sno(B). We then define

Qf(9)

~ (~I~;f(9)1')

, 1

A celebrated theorem of Littlewood and Paley asserts that for 1 < p < 00, IIQfll p ~ Ilfllp' The function Qf(B) is reminiscent of the martingale square function or the Lusin area function and hence, it is natural to ask if any of the above subgaussian estimates hold for Qf. Here, as in the previous problem, it seems that this investigation should begin with an analogue of Theorem 2.0.4. We can ask:

118

4. Sharp Good-A Inequalities for A and N

Question 4.2.15 Does there exist constants C 1 and C 2 such that whenever f E L1([-n, n)), (4.2.23) Here we would want C1 and C2 to be independent of f, but do allow these to depend on the sequence {nj }~o, preferably only on the lacunarity constant J.L. If f is a lacunary series of the form 2::;:0 ajeinj(J where the nj are as above, then for each j, !1 j f(B) = aj and hence,

The Khintchine inequality for lacunary series, Zygmund [Zy2, vol. I, Theorem 8.20], gives that

(4.2.24) where Cp depends on p and J.L and Cp = O(.jP) as p ---- 00. We now repeat the argument which appeared in the introduction to Chapter 2. Considering p = 2,4, ... in (4.2.24) and summing the power series for eX, we obtain (4.2.23) for the lacunary series f. Of course, many other more sophisticated subgaussian-type estimates are known for lacunary series, among these, the central limit theorem of Salem and Zygmund and the LIL's of Salem and Zygmund, Erdos and GaI, and M. Weiss, all of which will be discussed in Chapter 6. The above argument remains valid in the general case. That is, (4.2.23) follows from the Littlewood-Paley inequality Ilfllp ::; CpllQfllp, if we can prove this with Cp = 0(.jP) as p ---- 00. Computations with the central limit theorem for lacunary series show that Cp = 0(.jP) is the best attainable. For general f E L1([-n,n)), Pichorides [Pi] has shown this inequality with Cp = O(plogp) as p ---- 00 and this seems to be the best known estimate now. Moore [Mol] has shown the inequality II flip ::; C.jPQf where Qf is the variant of Qf defined by Qf = (2::;:0 II!1j flloo 2)1/2. But Qf is larger than Qf so this seems to be of little help in answering Question 4.2.15. Question 4.2.16 Consider a function h which satisfies conditions (i)-(iii) stated at the beginning of Chapter 1, Section 7. As in Theorem 1.7.2, this can be used to define an "area function", which was there denoted as Tf. Theorem 1.7.2 together with Lemma 1.7.3 show that for 1 < p < 00, IITfllp ~ IIfllp, Consider another function ¢ which satisfies the conditions of Theorem 1.4.7 and use it to create a "nontangential maximal function" as in Definition 1.4.6. Call this N f. Thus,

4.3 Application I. A Chung-type LIL for harmonic functions

119

for 1 < p < 00, liTflip ~ liN flip· Are there any good-A inequalities relating this area function and this nontangential maximal function? If we do not assume any relation between hand ¢, this seems almost intractable. Essentially we have discussed the only situations where this is known: when ¢ is either the Poisson kernel or the Gauss kernel and h is (respectively) a derivative of the kernel. However, Gundy and Irribarren [GI] have made some progress on this question. They have shown good-A inequalities relating a nontangential maximal function and an area function; their maximal function is created using a ¢ which satisfies a certain relation-a "dilation equation", but their area function is created somewhat differently than we have done here. We end this section with some further remarks concerning the boundedness properties of square functions on the space BMO of ~n. If we let Aaf denote the square function of the harmonic function u in ~~+1 with boundary values f, it follows from Proposition 4.2.8 that if f E Loo, then A~J E BMO. As we have already mentioned, several authors, among them Kurtz [Ku] and Qian Tau [Qi], have shown that Aa : BMO ---7 BMO. The arguments we presented above can be modified to obtain the stronger result that BMO ---7 BMO, with similar results for VMO, the space of functions in ~n of vanishing mean oscillation. There are also bilinear versions of these results constructed for various other types of area functions. For this, we refer the reader to Banuelos and Brossard [BB] and to the subsequent papers related to this topic, [Che1], [Che2], and [Che3].

A; :

4.3

Application I. A Chung-type LIL for harmonic functions

The following 1IL was first proved by K.L. Chung [Chu1] for independent random variables. Theorem 4.3.1 Let X t be a continuous time martingale and recall that X; sup IXsl and St(X) = (X);/2. We have O<s:C;t

1· . f (lOglOg S t(X))1/2 X * - ~ i~~ S;(X) t - y8

(4.3.1)

almost surely on {S (X) = oo}.

To state our result for harmonic functions, we denote by r a(x, t) the cone of aperture a and vertex at (x, t) E ~~+l. For t < 1 the truncated version will be similarly denoted by r;(x, t). Notice that here that these truncated cones are a little different than those in Chapter 3. We make this change simply because these new cones are more adaptable to the Lipschitz methods we will use. The results for both types of truncations are equivalent. Using these cones, we define, as in (4.2.1) and (4.2.2)

Nau(x, t) = sup{lu(y, 8)1 : (y, s)

E

r a(x, tn

4. Sharp Good-A Inequalities for A and N

120

and A"u(x, t)

=

(

r

Jr",(x,t)

(s - t)l-nlV'u(y, sWdyds )

l~

,

with a similar definition for the truncated operators N~u(x, t) and A~u(x, t), t < 1. As above, if t = 0 we simply write A~u(x) and N~u(x). The following theorem is motivated by Theorem 4.3.1. Theorem 4.3.2 (Lower-Half Chung-type LIL for harmonic functions) Let u be harmonic in lR~+1. Fix 0 < (3 < a and suppose there are points (xo, to), (Xl, tl) in lR~+1 such that A,B'u(xo, to) < 00 and NaIU(Xl' tl) < 00 for some (3' > (3 and a' > a. Then 1·

. f (IOgIOg(A.a u (x,t)))1/2 N ( ) > C (A .aU ())2 aU X, t X, t

(4.3.2)

1m In tlO

for almost every X E {x E lRn a, (3, a', (3' and n.

:

Aau(x) = oo}. The constant C depends only on

Problem 4.3.3 Prove the upper upper-half Chung-type LIL for harmonic functions. (The full Chung-type LIL does hold for lacunary series as (6.1.10) in Chapter 6 shows.) Proof of Lower-Half of Theorem 4.3.1: In the introductory chapter we claimed that once we have sharp good-A inequalities on Lipschitz domains we can literally copy the martingale proofs over to the harmonic function setting. To illustrate this point more precisely, we first prove that the lower bound in Theorem 4.3.1 follows from the corresponding good-A inequality (4.0.10) in Theorem 4.0.4. We will then "copy" this proof to show that Theorem 4.3.2 follows from the corresponding goodA inequality, Theorem 4.2.1. Towards this end, fix D > 1, ", > 1 and v > 1. For k large, k 2: [(log",)-l] + 1 = k* will do, set Ak = Rand

Define the stopping times (4.3.3) Note that since the function f(r)

=

(IOg[Ogr) 1/2 is increasing :s: Tk

Tk+1.

the martingale inequality (4.0.10) we obtain

P{STk(X) > DAk, X;k

:s: ckAk}:S: C

2

exp(-vloglog",k)

C2

Applying

4.3 Application I. A Chung-type LIL for harmonic functions

L

121

00

P{STk (X) > OAk} <

00.

k=k*

The Borel-Cantelli lemma now implies that (with i.o. meaning "infinitely often")

or

P { STk (X) ::::: OAk eventually}

=

1.

That is, for almost every w in our probability space there is a ko = ko(w) such that STk(X)(W) ::::: OAk for all k > ko. Since {X* = oo} a~. {S(X) = oo}, we can pick such an w for which S(X)(w) = X*(w) = 00 and therefore X;k (w) = EkAk, by the continuity of the paths of X t . Since X*(w) = 00, we also have that Tk(W) i 00. For t sufficiently large we can then pick k > ko (w) such that Tk (w) < t ::::: Tk+ 1 (w). Then

We again use the fact that the function f(r) clude

=

( log ro r ) g

1/2

is increasing to con-

Thus, we have

..

li~~f

(loglogS;(X)) 1/2 * S;(X) Xt

7r

::::

V8

(0 2 _1)1/2 Oy'rIV ,

(4.3.4)

almost surely on {S (X) = oo}. Since this holds for any 8 > 1, 7] > 1 and v > 1, the lower bound in (4.3.1) follows.

Proof of Theorem 4.3.2: We may assume that Ixl : : : 1, (xo, to) = (0,1) and A,61U(0, 1) = 1. Since (3 < (3', then A,6u(x,t) < 00 for all x E]Rn and t > 0; this is by Lemma 4.2.9 (or at least the ideas in the proof of that lemma). Since NalU(X1' td < 00 and a < a', we also have that Nau(x, t) < 00 whenever t > o. Now let 7] > 1 and set Ak = and for k:::: [(1og7])-l] + 1 = k*, set

H

Ek -

-

(

1 ) v log log 7]k

1/2

4. Sharp Good- A Inequalities for A and N

122

where 1/ is a constant we will specify momentarily. (In the proof ofthe lower-half of 4.3.1, the E:k'S were carefully chosen so as to obtain the precise lower bound 7f/VS. This precision was possible because we know the exact values of the constants in the good-A inequality (4.0.10). We now intend to use the good-A inequality of Theorem 4.2.1 and since the constants there are not precise, there is no point in being so careful here.) Now define the analogue of the stopping times in (4.3.3) by

The functions ¢k are Lipschitz and since f(r) = COg rO g r)1/2 is increasing, ¢k+l(X) :S ¢k(X), Also, Na.u(x, t) < 00 for every x E ]R.n and t > 0 implies that ¢k(X) ! 0 as k i 00 for each Ixl:S 1. Let (3 < "( < min ((3' , a). Set

u The Dk'S are Lipschitz domains as in Section 4.2 with Lipschitz constant depending only on "( = "((a, (3, (3'). For Ixi :S 1, the boundary of Dk is just the graph of ¢k and we can extend ¢k to all of]R.n by defining it to be the boundary of Dk for Ixi 2': 1. Thus, with this, Dk = {(x, t) : x E ]R.n, t > ¢k(X)}. Since (3 < "( < (3', there is an M depending only on "( and (3' such that for all x 2': M, r,8(x, ¢k(X)) C r,811 (0,1) C r,81 (0,1) where (3" = {Jltr. Thus, by Lemma 4.2.9, A,8u(x, ¢k(X)) :S L if Ixi 2': M. Hence, for k large enough, depending only on L, we have

for some C depending on M. Set "(' = {J~1. Then (3 < "(' < "( < a and r "'(' (z) ~ ~ Dk whenever Z E 8D k . Thus, by Theorem 4.2.1, for all such large k,

r ",((z)

O'{z E 8Dk : A,8u(z) >KAk' Na.u(z) :S E:kAk} :S O'{z E 8Dk : A,8u(z) > KAk, N"'(lu(z) :S E:kAd C :S C 1 exp(-C2 1/loglogryk) = k2 ' if 1/

= 2/C2 • As in the case of martingales,

for all

Z

E 8D k and it follows that

L 00

k=k*

I{x E]R.n : Ixi :S 1 and A,8u(x, ¢k(X)) > KAk}1 < 00.

The Borel-Cantelli Lemma now shows that A,8U(X,¢k(X)) :S KAk eventually for almost every x. That is, for almost every x with Ixi :S 1, there exists ko = ko(x)

4.3 Application I. A Chung-type LIL for harmonic functions

123

such that A,aU(X, ko. Pick such an x for which both A,au(x) and Nau(x) are infinite. Let t be very small and choose k > ko such that
A,au(x, t) :::; LA,au(x,
=

KLVryk+1.

As in the martingale case we have

Nau(x, t) ~

(

k

1 171 k 1/ og ogry

) 1/2

2

( ~ VIi 1

)1~ (A,au(x, t)/ .,ftiKL) log log (A,au(x, t)/.,ftiK L)2

which implies that 1 . . f (log log(A,au(x, t))2) 1/2 N ( ) 11m In "'U x,t > - - ltO (A,au(x, t))2 - .JVTJK L

for almost every x E {x E]Rn : Ixl :::; 1 and A,au(x) = oo}. Since we have chosen 1/ = and ry > 1 is arbitrary, this completes the proof of Theorem 4.3.2.

52'

Corollary 4.3.4 Under the assumptions of Theorem 4.3.2 we have liminf tlO

for almost every x

E

{x

(

loglog(A1u(x, t)) ) (AI,au ())2 x, t

E]Rn :

1/2

1

N",u(x, t) ~ C

(4.3.5)

Abu(x) = oo}.

Proof: The hypothesis A,6'u(xo, to) < 00 implies that the contribution to A,au(x, t) from the "top parts" of cones is finite; this essentially follows from Lemma 4.2.9. Similarly, the contributions to N",u from the "top parts" are irrelevant. In other words, the liminf's in the non-truncated version of Theorem 4.3.2 and in the truncated version, (4.3.5), are the same a.e. on the set {x E]Rn: Abu(x) = oo}. We remark that this corollary is true without the hypotheses involving the points (xo, Yo) and (Xl, Yl). These were necessary for Theorem 4.3.2 which we proved first simply because we had proved the necessary good-A inequalities to do so. By using the techniques of Section 4.2 it is possible to prove similar sharp good-A inequalities on bounded Lipschitz domains; these can be used, by following the proof of Theorem 4.3.2, to provide a direct proof of (4.3.5). We leave the details to the reader. Finally, there is also another upper LIL similar to Theorem 3.0.4, with the roles of A and N reversed. This follows from Theorem 4.2.1. The proof is similar to that of Theorem 4.3.2 and Corollary 4.3.4 and we also leave this to the reader.

124

4. Sharp Good-A Inequalities for A and N

Theorem 4.3.5 Under the assumptions of Theorem 4.3.2 we have

lim sup flO

A1u(x, t)

for almost every x E {x E lRn Q, (3, Q', (3' and n.

4.4


J(N,iu(x, t))2loglogN,iu(x, t) :

N~ u( x) =

oo}. The constant C depends only on

Application II. The Burkholder-Gundy -theorem, sharp V-constants and ratio inequalities

In this section we will show how the good-A inequalities of Theorems 4.1.1 and 4.2.1 can be used to produce various comparisons of functionals of the nontangential maximal function and Lusin area function. We first consider the theorem of Burkholder and Gundy, Theorem 1.7.9. For this we consider a nondecreasing continuous function cI> defined on [0,00] with cI>(0) = 0, cI> not identically zero, and which satisfies the growth condition: cI>(2A) ~ Ccp cI>(A) for every A > where Ccp is a fixed constant. Here is a more thorough version of Theorem 1.7.9:

°

Theorem 4.4.1 Suppose then

(i)

Q,

(3 >

°

and cI> is as above. If u is harmonic on lR++l,

J~n cI>(Aau(x))dx ~ C J~n cI>(N~u(x))dx.

If the left hand side of (i) is finite, then limy-->oo u(x, y) exists and is finite and constant for x E lRn. If u is normalized so that this limit is zero, then the converse inequality holds:

(ii)

J~n cI>(N~u(x))dx ~ C J~n cI>(Aau(x))dx.

Here the constant C depends only on

Q,

(3, n and the growth constant Ccp.

This seems to follow almost immediately from Lemma 4.0.2 and the good-A inequalities of Theorems 4.1.1 and 4.2.1. However, a closer examination reveals that there are a few technicalities. In Theorem 4.4.1, Q and (3 are arbitrary, but each good-A inequality required that one be bigger than the other. This is easy to overcome: since I{x E lRn : N,,/u(x) > A}I ~ I{x E lRn : Nou(x) > A}I for every A > 0, with the constants in the equivalence depending only on 'Y and 8 (Lemma 4.1.2), in our proof we may suppose in (i) that (3 > Q and in (ii) that Q > (3. Also recall that to obtain an estimate like (i) via a good-A inequality and Lemma 4.0.2, it is necessary to have the apriori estimate J~n cI>(min{1, Aau(x)})dx < 00, with a similar estimate involving N~u(x) necessary for (ii). To circumvent this difficulty, we will apply this good-A inequality technique to a related harmonic function which does satisfy the requisite apriori estimates. Then we will take limits to obtain (i) and (ii). This limiting argument is much

4.4 Application II. The Burkholder-Gundy -theorem

125

like the proof of Theorem 1.7.8 and much like a limiting argument in Fefferman and Stein [FS; Theorem 8]. The argument will also borrow much from Burkholder and Gundy's original proof of Theorem 4.4.1. The original proof of this theorem used localized versions of the good-A inequalities which resulted in slightly fewer complications.

Proof of Theorem 4.4.1: (i) We assume a < (3, and we may assume the right hand side of (i) is finite. Fix 0 < c: < L < 00 and set v(x, y) = u(x, y + c:) - u(x, y + L). For convenience we set Xo = Y (and likewise So = t). Then

A~v(x) =

1 t I::. r",(x) j=O

(s,

J

t)1

2

t1-ndsdt (4.4.1)

For (s, t) E r Q(x) and j fixed,

8U

18s/ s , t + c:) -

t'

t

8u t + L) I = (L - c:) I8iBs/s, 8 2u , I, 8s/s, t)

t

where is between + c: and + L. In particular, (s, t') E r Q(x). Since a < (3, there exists a constant e = eQ ,{3 such that the ball B = B((s, t'), et') is contained in r{3(x). Let r.p be a radial function supported on B(O,et') ~ ffi.n+l with

r

iIR n +1

r.p(x)dx = 1.

Such a r.p can be chosen so that also

whenever that

lal

= 2, where 0 depends on e and

n. From Theorem 1.1.4 it follows

Thus,

18~~:j (S,t')1 :s (LI~~~j (1J-(S,t'))1 2d1J ) 2

(Llu(1J) 12 d1J )

:s OIBI (t')-2(n+3) (N{3u(x) )21BI = O(N u(x))2(t')-4 {3

< O(N{3U(X))2

-

(t+c:)4

126

4. Sharp Good-A Inequalities for A and N

This and (4.4.1) yields

J J

By assumption,


00.


00.

Rn

This implies that

Rn

Then by the good-A inequality, Theorem 4.2.1, and Lemma 4.0.2,

J


J


(Note that the C here does not depend on cor L.) Since Nf3v(x) :S 2Nf3u(x) we have

r
iRn

iRn

(4.4.2)

Now

and by Lemma 2.3.1,

for (8, t) E r o:(x). Since we have assumed that the right hand side of (i) is finite, Nf3u(x) < 00 for almost every x. Thus, for all such x, Igsu (8, t + L)I-t 0 as L - t 00 for all (8, t) E r 0: (x), j E {O,... ,n}. Then, by the faC't that

which is (i).

4.4 Application II. The Burkholder-Gundy -theorem

127

To prove (ii), we may assume a > (3 and JlRn (Ac,u(x))dx < 0< s < L < 00 and set v(x,y) = u(x,y+s) - u(x,y+L). Suppose (so, to) E r ,e(x). Then, using Lemma 2.3.1,

[v(so, to)[ = IlL

00.

~~ (so, t + to)dtl ::; c lL A",u(x) (t: to)

Thus, N,ev(x) ::; c£,LA",u(x) and consequently, JlRn (N,ev(x))dx the good->. inequality, Theorem 4.1.1, and Lemma 4.0.2,

<

Again, fix

dt. 00.

Then by

Note that, as before, the constant C does not depend on s or L. By the triangle inequality and Lemma 4.2.9, A",v(x) ::; CA",u(x) where C depends only on a and n. Thus, (4.4.3) For 0< L < 00 set WL(X, y) = u(x, y)-u(x, y+L). Elementary considerations show that liminf£-+o N,ev(x) ::::: N,ewdx) for every x, so by Fatou's lemma and (4.4.3), (4.4.4) where C depends on a, (3 and n, but not on L. We claim limy-+ CXJ u(O, y) exists. To see this, let s > 0. Choose M so that [B(O, M)[(s) is larger than the right hand side of (4.4.4). Then, for every L, there exists x (= XL) in B(O, M) at which N,eWL(X) < s. But if y ::::: ~, (0, y) E r,e(x) and hence s > [wdO, y)[ = [u(O, y) - u(O, y + L)[, for all L > 0. Therefore, limy-+ CXJ u(O, y) exists and is finite. Fix x and suppose Xo E ]Rn. Suppose y is large enough so that [xo[ < (3y and [xo - x[ < (3y. Then both (0, y) and (x, y) are in r,e(xo) so that

[u(O,y) -u(x,y)[::; [x[sup{['Vu(t,y)[: (t,y) E r,e(xo)}::; C ElA",u(xo) y

by Lemma 2.3.1. Choosing Xo so that A",u(xo) < 00 we see that limy-+ CXJ u(x, y) = limy-+oo u( 0, y), so that this limit is constant for x E ]Rn. If we assume that this limit is zero, it is easy to see N,eu(x) ::; liminfN,ewdx). This, Fatou's lemma and (4.4.4) yields (ii). L-+oo An important special case of this theorem is the case (>') = >,P, P > 0, which relates the LP norms of the nontangential maximal function and Lusin area function. What is of further interest in this case is to find the best constants

128

4. Sharp Good-'\ Inequalities for A and N

in these LP inequalities. In general, sharper constants can be obtained by using good-'\ inequalities with better decay. We shall now discuss this. The good-A inequality

I{x E]Rn : A,au(x)

> 8A, Nau(x) :::; EA}I :::; Ca,,a,n

(82E~ 1) I{x E ]Rn : A,au(x) > A}I

of Burkholder and Gundy [BG2] and Lemma 4.0.2 shows that IIA,aullp :::; Ca,,a,nJP IINauilp for 0 < p < 00 (Burkholder [Bu4, p. 295], or see the discussion immediately after Theorem 4.0.3). The order of the constant on the right hand side, O(JP), is the smallest possible order of magnitude in this LP-inequality. A similar good-A inequality in [BG2] with the roles of A and N reversed yields IIN,aullp :::; CpllAaullp with only Cp = O(p) as p --+ 00. In fact, even the exponential decay given by Murai and Uchiyama in (4.0.13) does not improve this order in p. We now would like to show how to use the subgaussian decay in the good-A inequalities of Theorem 4.l.1 to show that IIN,aullp :::; Ca,,a,nJPIIAaullp for 1 :::; p < 00, which is again the best possible order. In fact, we shall prove more. Once again, our results are motivated by those first proved for martingales and we recall these next. In [Dav1], B. Davis found the best possible values for the constants a p and Ap in the following LP-inequalities for continuous martingales:

IIXllp :::; ApIIS(X)llp { and IIS(X)llp :::; apllXllp,

(4.4.5)

for 1 < p < 00, where we write X for the limit of X t as t --; 00. The constants are zeros of parabolic cylinder functions and confluent hypergeometric functions. They are both of order JP as p --+ 00. In [Wa], G. Wang proves that for 3 :::; p < 00 these constants are also best possible for discrete martingales Un} (with the more traditional square function S(f) defined by (2.0.4)) whose martingale difference sequence {d n } is conditionally symmetric. This means that for all n and real numbers T, P{dn+l > T Id1 , ... ,dn } = P{dn+ 1 < -T Id1 , ... ,dn } almost surely. These martingales include the one dimensional dyadic martingales. We should mention here, however, that for arbitrary martingales Un}, the best constant Cp in the inequality I flip :::; CpIIS(f)llp for 2:::; p < 00 (again with the square function S(f) of 2.0.4) is p - 1. This was shown by Pittinger [Pit] for integers p 2 3 and by Burkholder [Bu5] for all 2 :::; p < 00. Thus, for general martingales, the constants can be much larger. In [BY2], Barlow and Yor introduced the operators

4.4 Application II. The Burkholder-Gundy -theorem

129

o < a ::; 1, and proved that with Ga,p = Oa(JP), as p

----+ 00.

Motivated by this we have

Theorem 4.4.2 Suppose u is harmonic in lR~+l, not identically zero, 0 and 0 < p < 00. If u(x, t) ----+ 0 as t ----+ 00 then

o < a ::; 1

< (3 < 0:, (4.4.6)

In general, (4.4.7) Furthermore, both G and G' are Oa,cx,(3,nC/p) as P best possible.

----+ 00

and this order in p is

The additional normalization in (4.4.6) that u vanish at 00 is necessary as can be seen by considering constant functions. As mentioned at the beginning of this Chapter, the desire to improve the rate of decay in c of the good-A inequalities arose partly in efforts to prove ratio inequalities between A and N. (Ratio inequalities were first proved for martingales by A. Garsia in [Gar].) The following ratio inequalities sharpen those in R. Fefferman, Gundy, Silverstein and Stein [FGSS] and Murai and Uchiyama [MU]. Theorem 4.4.3 Suppose u is harmonic in lR.~+ 1 and 0 < (3 < 0:. There exists constants G 1 and G2 , depending only on 0:, (3 and n, such that for any 0 < p < 00,

(4.4.8) and (4.4.9) We now proceed with the proof of Theorem 4.4.2. Since the proofs of (4.4.6) and (4.4.7) are the same, we just prove (4.4.6). Let M be a large real number to be specified later. We have

4. Sharp Good-A Inequalities for A and N

130

= I

+ II,

where _

Eij - {2

i

2i -

2i-j

j - 1

< N{3u ::; 2i+1 ,~< Aa u ::; M}·

Estimating the double sum and applying Theorem 4.1.1 with A = 2i I K and c K2- j 1M we obtain I::;

LL

2i+l ( 2 -j-l M i

)P

=

(2i-j)ap

M

IEijl

iEZjEl\!

As functions of j, 2(1-a)Jp is increasing and exp( C2~22 3) is decreasing. We estimate the last sum by an integral and successively set w = 22x and v = (C2M2/4K2)w to find "

=

2

2"

1

K 2) (1-a)p/2 °O _1_ ( _4__ v((1-a)p/2)-le- v dv 2 log 2 C2M2 C2M2 / K2

Substituting this last estimate back into the estimate for I gives

4.4 Application II. The Burkholder-Gundy -theorem

This gives that for any 0 < p <

00

131

and any 0 < a ::::: 1,

The last inequality already proves (4.4.6) with some constant C. To obtain the desired information on the constant, we estimate C(p, a, K, M). Assume first a = 1. We have

(4.4.10) Now, if we take M = C 3y'P and choose C 3 as to make the right hand side of (4.4.10) ~ (see the comments immediately after Lemma 4.0.2) we find that (4.4.11) Next, we check that for p ~ 2, we can take C 3 = C 4 independent of p and therefore (4.4.11) proves (4.4.6) with the correct behavior in C when a = 1. If 0< a < 1, we choose p large enough so that pa ~ 2 and obtain from (4.4.11) that (4.4.12) with C 4 depending only on

0:,

f3 and n. From this we find that

C 1 23p - ap KP22ap+12PCap( lM'i)a p 4 Y ap (1

) /2 C 2 -a p ap2log

2

2C1 (23-aK4aC:t)P r :::::(2log2)ap C~1-a)/2 2PCHy'P)P

r (AautPdx

JIRn

1

00

C2C~p/ K2

(1-

a

V

((1-a)p/2)-1

r

e

-v

d

v

1

(A

IRn

) ( lM'i)a p (A )ap d yap JIRn aU X+

-2- P

aU

)aPd

X

132

4. Sharp Good-A Inequalities for A and N

For fixed 0 < a < 1,

r

(1- a

-2- p

) lip ~ ~

(1 _a

) ';a

-2- P

,

as p ---> 00. Therefore if C(a, 0:,(3, n,p) denotes the constant on the right hand side of the last inequality we obtain

(C(a,o:,(3,n,p))l/p~ C;~;:,n(apt/2

I-a

C;a p)-2 +Cy'P

as p ---> 00 and we have proved (4.4.6) with the correct behavior of the constant in p. The fact that the behavior of the constants in (4.4.6) and (4.4.7) (for a = 1) cannot be better than y'P, as p ---> 00, follows from the fact that the behavior of the LP-constants for the Riesz transforms cannot be better than p, as p ---> 00. However, in the unit disc one can even obtain more information on these constants using the central limit theorem for lacunary series and computations similar to those in the proof of Theorem 3.0.6. For more on this, see [BM3], [Ba2] and [Mol]. Proof of Theorem 4.4.3.' Again, the proofs of (4.4.8) and (4.4.9) are the same so we just prove (4.4.8). Define the sets Eij as in the proof of Theorem 4.4.2 with M = 1. The good-A inequality, Theorem 4.1.1, implies

IEijl:::; cexP (-C'2 2j

)1 {x E]Rn: NfJu > ~} I,

where K is the constant appearing in that theorem. Thus,

:::; C

in

(NfJu)Pdx +

~

=

I

L L exp (Cl2;~~;~l)) 2(i+ l )PIE

ij l

iEZjEN

+ II.

i

II:::; Cp LLexp((16Cl - C')2 2j ) 2ip {x E]Rn: NfJu

>

~} I

tEZ JEN

:::; Cp

r (NfJu)Pdx,

J~n

if C 1 is chosen small enough relative to C'. This completes the proof of Theorem 4.4.3.

133

4.4 Application II. The Burkholder-Gundy -theorem

In the same way, the good-A inequalities for caloric functions, Theorems 4.1.4 and 4.2.10 give: Theorem 4.4.4 Suppose u is a caloric function in lR~+1 and 0 < (3 < any 0 < a S; 1 and 0 < p < 00,

PNfju I p S; CII(PA",u) a lip II (PA.u)1-a

Ct.

Then for

(4.4.13)

and (4.4.14)

with C and C f both Oa,,,,,fj,n(.jP) as p ~ and C4 such that

Ln

exp (C1

00.

Also, there are constants C1, C 2 , C3

(;~::) 2) (PNfju)Pdx S; C

2

Ln

(PNfju)Pdx

(4.4.15)

and (4.4.16) Once again, we could ask for more information than just the best order of magnitude in the above constants. Our methods above do not give any finer information. In this direction, let us consider the g*-function in the unit disc defined by (3.4.1). Since, as we showed in Section 3.4, this function is the conditional expectation of the martingale square function, Jensen's inequality and Davis' result, (4.4.5) above, give Theorem 4.4.5 Let a p and Ap be the constants in (4.4.5). Then:

For 2 S; p <

00,

(4.4.17)

and: For 1 < p S; 2, (4.4.18) The g*-function is not bounded on V for 1 < p < 2. However, since A",f(()) S; C",g*(f)(()), one does have Ilfllp S; Cpllg*(f)llp for 1 < p < 00. We do not know how to prove (4.4.18) with the Davis constant Ap for the full range 1 < p < 00. (Although, see Banuelos [Ba2] for the case p = 4.) The following problem is natural.

134

4. Sharp Good-A Inequalities for A and N

Problem 4.4.6 What are the best constants in the inequalities (4.4.17) and (4.4.18)?

The inequalities (4.4.17) and (4.4.18) also hold for the version of g* in ~n and even in more general settings. For a further probabilistic study of the g*-function, we refer the reader to P. A. Meyer [Mel], [Me2] and N. Th. Varopoulos [Va]. There are many other applications of the above good-A inequalities. Throughout, our good-A inequalities have been stated using Lebesgue measure, but it is possible to use these to produce good-A inequalities where Lebesgue measure is replaced by any measure satisfying an Aoo condition with respect to Lebesgue measure. This then produces weighted inequalities involving the nontangential maximal function and Lusin area function (Gundy and Wheeden [GW]). The subgaussian good-A inequalities also hold for vector valued martingales. These give information on the LP-constants for the vector of Riesz transforms; for this, see Banuelos [Bal]. Finally, there are also other directions and applications of the good-A inequalities. See, for example, Coifman and Fefferman [CF] or Torchinsky [To] for inequalities between singular integrals and maximal functions, and Burkholder [Bu2] for applications to one-sided nontangential maximal functions.

Chapter 5 Good- A Inequalities for the Density of the Area Integral In this section we will discuss the density of the area integral, also known as the Dfunctional. It is a harmonic analysis analogue of local time. We will consider what is known as the maximal density, and state and prove each of the four possible good-A inequalities involving the maximal density and either the nontangential maximal function or Lusin area function. We first wish to recall and amplify our discussion in the introduction. As we noted, in addition to X* and S(X), there is a third important random variable associated to a continuous martingale X t starting at O. This is defined by considering the measure f. L on JR given by f..L(E) = 00 XE(Xt)d(X)t where d(X)t is the Riemann-Stieltjes measure on [0,00) associated to the non-decreasing function (X}t. A theorem of P. Levy [Lei] asserts that f. L is absolutely continuous with respect to Lebesgue measure and we write L(a) for its density. The random variable L(a) is called the local time of the martingale and the random variable L* = sup{L(a); a E JR} is called the maximal local time. Note that by definition, 00 XE(Xt)d(X)t = IE L(a)da for every Borel set E ~ JR and from this it follows that for any nonnegative Borel function f on JR,

10

10

(5.0.1) If we take

f == 1 in (5.0.1) and note that L(a) = 0 if a rJ. [-X*,X*] we obtain: S(X)2

=

r d(X)t = 1-x· r L(a)da::; 2L*X*. 10 x'

00

(5.0.2) 1

1

We can use Cauchy-Schwarz to deduce that IIS(X)llp ::; J2IIL*IIJIIX*IIJ. This combined with the fact that IIS(X) lip ~ IIX* lip, 0 < p < 00, (the BurkholderGundy inequalities) shows that IIS(X)llp ::; CpIIL*llp, 0 < p < 00. The reverse inequality is more difficult and was shown by Barlow and Yor [BYl], [BY2]. In fact, even more is true. For each pair, L* and X*, and L* and S(X), there are two good-A inequalities relating the pair. The following theorem is due to Bass [BasI] and independently, to Davis [Dav2]. We remark that the statements of their results 135 R. Bañelos et al., Probabilistic Behavior of Harmonic Functions © Birkhäuser Verlag 1999

136

5. Good-A Inequalities for the Density of the Area Integral

are different than what appears here, but a careful analysis of their methods yields the sharp versions below. These together with the earlier good-oX inequalities of Burkholder and Gundy [BG1] give all six possible good-oX inequalities relating pairs chosen from X*, S(X), and L*. Theorem 5.0.1 There are constants C 1 and C 2 such that for all 0 < c < 1 and oX> 0,

> 2oX,

L* :<:; coX} :<:; C 1 exp (-~) P{X*

> oX}

(a)

P{X*

(b)

P{S(X»2oX, L*:<:;CoX}:<:;C1exp(-~)P{S(X»oX}

(c)

P{L*

> 2oX, X* :<:; coX} :<:; C 1 exp (-~) P{L* > oX}

(d)

P{L*

> 2oX,

S(X) :<:; coX} :<:; C 1 exp (-~) P{L*

> oX}.

Our immediate goal in this chapter is a version of Theorem 5.0.1 for harmonic functions. Surprisingly, even though L and L * had been studied extensively for a number of years, it was not until 1983 that R. Gundy [Gu1] proposed harmonic analysis analogues. Let u(x, t) be a harmonic function on lR++1 and for r E lR consider the subharmonic function (u(s, t) - r)+. Its distributional Laplacian is a nonnegative Radon measure on lR++1, which we denote by ~(u(s, t) - r)+(dsdt). We then define Dau(x; r)

=

f

Jra(x)

t 1-

n

~(u(s, t) -

r)+(dsdt)

(5.0.3)

and Dau(x)

= supDau(x;r). rEIR

Equivalently, we may define

where (J"r (dsdt) denotes surface measure on the set {( s, t) : u( s, t) = r}. Relatively simple computations using the coarea formula from geometric measure theory and the definition of distributional Laplacian show that for all x this definition of Dau(x; r) and the version in (5.0.3) agree for almost every r. (See Gundy [Gu2, p. 6], or Brossard [Br1, p. 299], for these arguments.) In [Br1], Brossard shows that in fact, these two versions agree for all r. This result is much !llore difficult. Dau(x;r) is called the density of the area integral and Dau(x) is (';dl('d the maximal density. The density of the area integral is the analogue of the local time of a martingale and the maximal density is the analogue of the maximal local time. These analogies are evident if we consider the "change of variables" formula of

5. Good->' Inequalities for the Density of the Area Integral

137

Gundy and Silverstein [GS]: If 'ljJ(s, t) is a nonnegative Borel function on JR.'~+1 and if f is a Borel function on JR., then

jJer

IR n+ 1 +

=

'ljJ(s, t)f(u(s, t))IVu(s, t)1 2 dsdt

rr JIR

JIRr't+1

(5.0.4)

'ljJ(s, t)f(r)D..(u(s, t) - r)+(dsdt)dr.

If we set 'ljJ(s,t) = t1-nXra(x)(s,t) then we obtain a formula analogous to the "occupation time formula" (see [RY, p. 209]) of (5.0.1). If, in addition, we set f == 1 we obtain:

r

Jra(x)

IVu(s, t)1 2 t 1- ndsdt =

Note that D..(u(s, t) - r)+ this last formula as

rr

JIRJra(x)

tl-nD..(u(s, t) - r)+(dsdt)dr.

= 0 on r a(X) unless r

:s: Nau(x).

We may then rewrite

(5.0.5) This is a perfect analogue of (5.0.2) and clearly justifies calling Dau(x; r) the "density of the area integral." Reasoning as in the probability case gives IIAauilp :s: CpllDaullp, 0 < p < 00. As before, the reverse inequality is more difficult. This was done by Gundy [Gu1] when n = 2 and by Gundy and Silverstein [GS] in higher dimensions; see also Gundy [Gu2]. The proof in Gundy [Gu1] uses the representation of the density as the conditional expectation of the local time of the martingale obtained by composing the harmonic function with Brownian motion. This representation, which we will not discuss in this monograph, has turned out to be very useful in the study of the D-functional. This point is certainly demonstrated in [Br], [BC1] and [BC2], for example. As in the case of martingales, more is true. For each pair, Dau and Nau, and Dau and Aau, there are two good->. inequalities relating the pair. These inequalities, first proved in Banuelos and Moore [BM4], provide the harmonic analysis analogue of the results of Bass and Davis. In Section 4.2 we proved a good->. inequality between Nau(x) and A,au(x) for a> (3. We did this in the more general setting of Lipschitz domains simply because it caused us no extra effort to do so and because our estimates then lead to LIL's. We do the same here. Recall our definitions: for P = (x, Y) E JR.~+l, we set r a(x, y) = r a(P) = {(s, t) : Ix-sl < a(t and define

yn

Nau(x, y) = Nau(P) = sup{lu(s, t)1 : (s, t) E r a(pn and

5. Good-A Inequalities for the Density of the Area Integral

138

Likewise, we will now define

Dau((x, y); r) = Dau(P; r) =

r

JraCP)

(t - y?-n f},(u(s, t) - r)+(dsdt)

(5.0.6)

and

DaU(X,y) = Dau(P) = supDau(P;r). rEIR

The following is the analogue for harmonic functions of (a) and (b) in Theorem 5.0.1. ------; IR be a Lipschitz function and let D = {(x,y) : x E IR n , y E 1R, y > ¢(x)}. Suppose u is harmonic in D and 0 < (3 < a < where M is the Lipschitz constant of ¢. There are constants K 1 , K 2 , G1 , G2 , G3 and G4 , with K1 and K2 > 1 and all depending only on a, (3, nand M, such that if A > 0, 0 < E < 1 then

Theorem 5.0.2 Let ¢ : IR n

(a)

I{x

E

ir

IR n : N(3u(x,¢(x)) > K 1 A, Dau(x,¢(x)) :::; EA}I :::; G1 exp ( -

(b)

~2)

I{x E IRn

:

N(3u(x, ¢(x)) > A}I

I{x E IRn : A(3u(x, ¢(x)) > K 2 A, Dau(x, ¢(x)) :::; EA}I :::; G3 exp ( -

~; )

I{x E IRn

:

A(3u(x, ¢(x)) > A}I.

Note that (b) involves a subgaussian estimate. Just as before, this estimate will lead to an LIL. The estimate in (a) is not of subgaussian type, however, as we will later show, it is, in some sense, the sharpest possible. (This will be made precise.) The inequality in (b) is in this same sense the sharpest possible. All of this will be discussed in the third section of this chapter. Our second theorem is similar to Theorem 5.0.2 but with the roles of the functions reversed. For this theorem it will be necessary to work with "smoother" versions of the density and maximal density. Let cp(x) be a smooth, positive, radially symmetric function supported on a ball B(O, (3). Then set

Du(x; r) =

r

JIR"j+l

tCPt(x - s)f},(u(s, t) - r)+(dsdt)

Du(x) = supDu(x;r)

r E IR (5.0.7)

rEIR

where, as usual, we have set CPt(x) = t~CP (~). Both Du(x;r) and Du(x) depend on the choice of cP and this should probably be indicated by a notation such as D
5. GOOd-A Inequalities for the Density of the Area Integral

139

discussion of the following theorem, Theorem 5.0.3, and simply write Du(x; r) and Du(x). In practice, versions (5.0.7) and (5.0.3) behave similarly since given any (3 > 0 and any 0 < p < (3 we can always find a Coo function

0, 0 < C < 1,

(a)

if (3 < a then

I{x

E]Rn :

Du(x) > K 1 A, Nau(x) ~ cA}1

~ C 1 exp ( - ~2 ) (b)

if (3

I{x

E ]Rn :

Du(x) > All,

< (2048n)-l a then I{x

E ]Rn :

Du(x) > K 2 A, Aau(x)

~

cA}1

~ C 3 exp (- ~:) I{x E]Rn : Du(x) > A}I. The unfortunate aspect of (b) is the restriction on (3 and a. This result should no doubt be true only under the assumption (3 < a, but we have been unable to prove it and we leave it as an open question. If we assume only that (3 < a, then in Banuelos and Moore [BM4], it is shown that (b) holds with the expression C 3 exp (-~) replacing the expression C3 exp (-~). In fact, this is shown in the more general setting of Lipschitz domains. The proof uses the representation of the density function in terms of the conditional expectation of the local time and the results of Barlow and Yor. Also [BM4] contains a different proof of a version of (b) which has as hypothesis (3 < a. However, the conclusion ofthis result is merely the same good-A inequality with the expression C3 exp( - gCf;3) replacing the expression Furthermore, the proof does not seem to be adaptable to the setting C3 exp ( of Lipschitz domains. In keeping with the general theme of this monograph, we find it more desirable to prove a subgaussian estimate like Theorem 5.0.3(b) rather than these other results, despite the annoying and probably unnecessary restriction on (3 and a. Another advantage of Theorem 5.0.3(b) as written is that it is in some sense the sharpest attainable. This will be made precise and discussed later in the third section of this chapter. Theorems 5.0.3(a) and (b) as written should probably be true in the setting of Lipschitz domains, but our proofs do not seem to be adaptable to that situation. We leave this as another open problem.

*).

140

5. Good-A Inequalities for the Density of the Area Integral

This chapter will be divided into 4 sections. In Section 5.1 we prove Theorem 5.0.2 and in Section 5.2 we will prove Theorem 5.0.3. In Section 5.3 we will show some corollaries, show the sharpness of our results and present some further discussion. Finally, in Section 5.4 we present an application of the techniques of sections Section 5.1 and Section 5.2 to the study of the space L log L within Hl. These results of Brossard and Chevalier [BC2] further reinforce the notion of the D- functional as an analogue oflocal time. Much of this chapter, in particular, Theorems 5.0.2 and 5.0.3(a) is taken from Banuelos and Moore [BM4], which in turn, is based on earlier work of Gundy and Silverstein [GS] (see also Gundy [Gu2]). The proof of Theorem 5.0.3(b) is from [M02].

5.1

Sharp control of A and N by D

Our proof of Theorem 5.0.2 follows exactly the strategy of Section 4.2; in both (a) and (b) we will build a "sawtooth" region over {x E ~n : Dau(x, ¢(x)) > c'\}. That is, we set E = {x E ~n : Dau(x, ¢(x)) > c'\} and D' = UXEEc f a(x, ¢(x)). Then aD' is the graph of a Lipschitz function, call it 1jJ(x). For Theorem 5.0.2(a) we will estimate IIN,au(x,1jJ(x))IIBMO and for Theorem 5.0.2(b) we will estimate IIA~u(x,1jJ(x))IIBMO. In both cases these BMO estimates will be obtained as in Section 4.2; for each cube Q ~ aD' we will form an auxiliary domain n above Q and consider the contributions to either N,au(x,1jJ(x)) or A~u(x, 1jJ(x)) from both the part of the cones f,a(x,1jJ(x)) inside n (the part "close by Q") and the part outside n (the part "far away from Q"). Exactly as in Section 4.2, the "close by Q" part will be estimated using Green's theorem arguments on n and the "far away" part will be controlled using gradient estimates. With these BMO estimates in hand, the good-'\ inequalities will follow exactly as in Section 4.2. To get this program under way, we first show a lemma that provides the necessary gradient estimates to control boundary terms and other errors that will arise in our approximations. This is an analogue for the D-functional of the gradient estimates of Lemma 2.3.1 and will be just as indispensable. Unfortunately, its proof is somewhat longer.

Lemma 5.1.1 Suppose a > f. There is a constant C, depending only on a, 'Y and n such that ifu is harmonic on fa(x) and if (s,t) E f,(x) then

tIV'u(s, t)1 :::; CDau(x). Proof: Fix (8, t) = Po E f,(x). We may assume that u(Po) = 0; otherwise consider u - u(Po). Choose 10 > 0 so that B(Po, 4/0) ~ f a(X); if '0 is chosen as large as possible then '0 ~ Ct where C depends only on a and 'Y. For j = 1,2,3,4 set B j = B(Po,j,o) and set M j = sup{lu(z,y)1 : (z,y) E B j }. By the subharmonicity of IV'ul and the Gundy-Silverstein formula (5.0.4) we have:

t2IV'u(8,t)12:::; C

r IV'u(z,y)1 yl-ndzdy

lE2

2

5.1 Sharp control of A and N by D

-s.CJ M 2

{

-M2 JB2

141

~(u(z,y)-r)+yl-ndzdydr

-s. CM2 Da u (X). Thus, (5.1.1) using similar reasoning we can also conclude that if (z, y) E B2 then YIV'u(z, y)1 -s. C.;M3JD a u(x). Since u(s, t) = 0 and for (z, y) E B 2, Y ~ 2Cro, it then follows that (5.1.2) Now consider B4 and apply Green's theorem to lu(P)-rl, P E B 4, r E lR and G(P,Po) = IP - pol1-n - (4ro)1-n. Technically, we must approximate lu(P) - rl by smooth functions of u(P) - r and then take limits. To do this, we consider a Coo function a(r) which has fIR a(r)dr = 2, a( -r) = a(r), supp a ~ [-c:, c:] and set a,,(r) = ~a(~). Let b,,(r) be smooth functions satisfying b~(r) = a,,(r) and b,,(O) = b~(O) = O. Then b,,(r) i Irl as c: ! O. We now apply Green's theorem. We

remark that here we are using essentially the same argument as in the proof of the mean value property, Theorem 1.1.3. We obtain:

{

JB4

~(b,,(u(P)-r))G(P,Po)dP=

C:ro JaB4 ( b,,(u(P)-r))d(J(P)-C~b,,(u(Po)-r).

However, by assumption, u(Po) = o. Also, ~(b,,(u(P)-r)) = a,,(u(P)-r)IV'u(P)12, so that by the formula of Gundy and Silverstein, (5.0.4), this last equation becomes:

( a,,(s) {

JIR

JB4

~(u(P) -

r - s)+G(P, Po)dPds =

C:ro JaB4 ( be(u(P) - r))d(J(p) - C~be(-r).

The arguments in Brossard [Brl, Lemma 2] show that as a function of r, fB4 ~(u(P) - r)+G(P, Po)dP is continuous. We then let c: --t 0 in the above expression to obtain:

2 {

JB4

~(u(P) -

r)+G(P, Po)dP

=

C: (

ro JaB4

lu(P) - r)ld(J(P) -

C~lrl.

We remark that since ~lu(P) - rl = 2~(u(P) - r)+ this last equation is exactly what we would have obtained had we formally applied Green's theorem to lu(P)-rl and G(P, Po) on B 4. (See Gundy [Gul], [Gu2], or Gundy and Silverstein [GS] for similar applications of Green's theorem.) Rearranging this last equation gives: (5.1.3)

5. GOOd-A Inequalities for the Density of the Area Integral

142

[

lB4

~(u(P) -

r)+G(P, Po)dP

S [

~(u(P) -

r)+G(P, Po)dP +

S [

~(u(P) -

r)+G(P, Po)dP + C [

S [

~(u(P) -

r)+G(P, Po)dP + CDo:u(x).

lBI

l~ lBI

[

lB4\BI

~(u(P) -

l~\~

r)+

~(u(P) -

~l

ro

dP

r)+ pi- n dP

This and (5.1.3) gives:

~ [ lulda S ro laB4 Choosing r

Clrl + [ ~(u(P) lBI

r)+G(P,Po) dP

(5.1.4)

+ CDo:u(x).

= M2 in (5.1.4) yields: (5.1.5)

;f;

Simple estimates for the Poisson kernel show that for P E B 3 , lu(P)1 J8B4 lulda. This and (5.1.5) show that:

s

(5.1.6) Substituting (5.1.6) into (5.1.2) we have

Consequently, M2 S CDo:u(x) and this and (5.1.1) complete the proof of the lemma. In Section 4.2, it was necessary to compare A,eu(x, Yl) and A,eu(x, Y2) for Y2 > Yl; this was done in Lemma 4.2.9. Here it will be necessary to compare Do:u(x, Yl) and Do:u(x, Y2) for Y2 > Yl· For n = 1 it is clear that Do:u(x, Y2) S Do:u(x, Yl), but for n ;::: 2 this is no longer obvious. The following lemma will allow us to make the necessary comparisons. However, it will be necessary to.show this in greater generality than we did when we showed the corresponding lemma for area functions. Nevertheless, the proof of this lemma for the D-functional will be essentially the same as the proof of Lemma 4.2.9. Let r(p) be a cone in lR~+1, either infinite or truncated, with vertex P. We do not assume r(p) has axis parallel to {(O,y) : Y > O}.

5.1 Sharp control of A and N by D

143

For u harmonic on r(p) we define Du(P;r)

=

Du(P)

=

r

ir(p)

d((s,t),p)l-nA(u(s,t) -r)+(dsdt)

(5.1.7)

supDu(P;r). rEIR

Note that if r(p) = r",(x,y), then Du(P;r) as defined in (5.1.7) and D",u((x,y);r) as defined in (5.0.6) are equivalent up to constants depending on a and n. Lemma 5.1.2 Suppose u is harmonic on r",(x,y), a> p and r(p) ~ rp(x,y). Then Du(P) ~ LD",u(x, y), where L is a constant which depends only on a, p and

n. Proof: First we note that there exists a constant Co, depending only on a and p such that B(P, 2Cod(P, (x, y))) ~ rp(x, y) where p = For j = 1,2, set B j = B(P, jCod(P, (x, y))). . If (s, t) E r(p)\B l then d((s, t), (x, y)) ~ (1 + 0 )d((s, t), P) so that

Pt"'.

6

r

ir(p)\Bl ~ ( ~

d((s,t), p)l-nA(u(s,t) -r)+(dsdt)

1 l+C

)l-ni

o

d((s,t), (x,y))l-nA(u(s,t)-r)+(dsdt)

(5.1.8)

r(p)\Bl

CD",u(x, y).

Let G(Q, P), Q, P E lR.n + l be the Green's function for B2 with pole at P. Then for (s, t) E r(p) n B l , d((s, t), p)l-n ~ G((s, t), P). This and Green's theorem then yields:

r

d((s,t), p)l-nA(u(s,t) -r)+(dsdt)

ircP)nBl

~C

r

iB2

G((s,t),P)A(u(s,t)-r)+dsdt

= CJ(ffB2 )

(5.1.9)

kB2 ((u(s, t) - r)+ - (u(P) - r)+)dCJ(s, t).

Here we note that technically, to apply Green's theorem we must approximate u(s, t) - r by a smooth function of u(s, t), then apply Green's theorem, and then pass to a limit. The details of this argument are exactly the same as in a similar application of Green's theorem in the proof of Lemma 5.1.1. As in that case, we ultimately obtain the same formula we would have obtained had we formally applied Green's theorem.

5. Good-A Inequalities for the Density of the Area Integral

144

Now B2 <::;; rp(x, y) so by Lemma 5.1.1, (t - y)I'Vu(s, t)1 :::; CDau(x, y) whenever (s, t) E B 2. Since the radius of B2 = 2Cod(P, (x, V)) :::; C(t - y) whenever (s, t) E B 2 , (5.1.9) implies

r

Jr(p)nB l

d((s, t), p)l-n ~(u(s, t) - r)+(dsdt) ::; CDau(x, V).

(5.1.10)

The lemma then follows from (5.1.8) and (5.1.10). We now consider a Lipschitz domain D = ((x,y) : y > 7jJ(x)} where 7jJ is a Lipschitz function. As before, we call Q <::;; aD a cube if Q = {(x,7jJ(x)) : x E Q'} where Q' <::;; ]Rn is a cube. We now fix such a cube Q. We also fix (Y > /3 and set "(' = 2ai{3, "( = ai2{3 so that (Y > "(' > "( > /3, and assume that there exists an (Y' > (Y so that r a' (P) <::;; D for every P E aD. Above Q we construct a domain D as before (section 4.2):

D=

(u

r,,(p)) n {(x,y) : y < 7jJ(xo)

+ 2RC(Q' )} ,

PEQ

where Xo is the center of Q' and R depends only on (Y, /3 and the Lipschitz constant of 7jJ. The domain D is starlike with respect to the point P* = (xo, 7jJ(xo) + RC( QI)). (See Section 4.2; the notation here is unchanged from that section.) As before, for each P E Q we have r{3(p) = r1(p) U r 2 (p), a disjoint union, where r1(p) = r{3(p) n {(s, t) : t < 7jJ(xo) + RC(Q')}. We define A1u and A 2 u as before, that is, Al u(P) is defined using the usual definition of A{3u but with the integration taken only over r1(p) and similarly for A 2 u(P). Likewise we define N1u(P) and N 2 u(P).

Recall that for each P E aD there is a cone r(p) with vertex at P, height h = IP - P* I and vertical axis P P* which is contained in D. These cones have aperture K,(P) ::::: K,o where K,o is a constant depending only on (Y, /3, n and the Lipschitz constant of 7jJ. Furthermore, we choose K,(P) so that for some r > 1 (depending only on (Y, /3, n and the Lipschitz constant of 7jJ), the cone with slightly larger aperture rK,(P) is also contained in D. For P E Q ~ aD we may take these cones to have aperture "( and we recall that R is chosen so that for such P E Q, r1(p) ~ r(p) ~ r,"(p) where "(" = J~l'. Lemma5.1.3 Suppose Dau(x, 7jJ(x)) ::; 1 for every x E ]Rn. SupposeQ = {(x,7jJ(x)): x E Q'} ~ aD is a cube and D and P* E D are associated to Q as in the preced-

ing discussion. There is a constant C, depending only on constant of 7jJ, such that (a)

IJ'I

1Q , Aiu(x, 7jJ(x))dx ::; C and

(b)

IJ'I

1Q , N1U(X, 7jJ(x))dx :::; C whenever u(P*) =

o.

(Y,

/3, n and the Lipschitz

5.1 Sharp control of A and N by D

145

Proof: For P E 80 we define Du(P), Au(P) and Nu(P) using the cones f(P): the 1

function Du(P) was defined in (5.1.7), Au(P) = (Ir(p) IVu(S)1 2 d(S, p)l-n dS )

'2

and Nu(P) = sup{lu(S)I: S E f(P)}. Lemma 5.1.2 and our construction of 0 implies that Du(P) :::; C for every P E 80. Assume u(P*) = O. Then by (5.0.5), 1

r

1

2

r

0-(80) Jan Au(P) dCJ(P) :::; C CJ(80) Jan Nu(P)dCJ(P) :::; C :::; C

(o-(~O) lan NU(P)2 dCJ (P)) (CJ(~O) lan AU(P)2 dCJ (P))

1

'2 1

'2 ,

where the last inequality is from Dahlberg [Da2]. Thus, 1

r

2

CJ(80) Jan Au(P) do-(P) :::; C.

(5.1.11)

Since f(P) ~ fl(P) whenever P E Q, CJ(80) ,::;; IQ'I, and 'IjJ is Lipschitz, (5.1.11) implies the conclusion (a) under the assumption u(P*) = O. But since both Al u(P) and Dau(x, 'IjJ(x)) are unchanged if we alter u by a constant, (a) remains valid without the assumption u(P*) = 0 and (a) is proved. To show (b) we again assume that u(P*) = 0 and note that the aforementioned result of Dahlberg and (5.1.11) imply a(~n) fan NIU(p)2dCJ(P) :::; C. Again, since fl(P) <;;; f(P) for every P E Q, CJ(80) ,::;; IQ'I, and'IjJ is Lipschitz, we obtain (b ). The following proposition, together with Lemma 4.2.3, will again provide the necessary analogue of the Key Estimates 4.0.8 and 4.0.9. This should be compared to Propositions 4.2.8 and Propositions 5.1.8 and 5.2.5 which follow below.

Suppose 0: > (3 and u is harmonic on the unbounded Lipschitz domain D = {(x,y) : x E JR. n , y E JR., y > 'IjJ(x)}. Suppose that for every P E 8D, Dau(P) :::; 1 and that there exists Po E 8D such that A{3u(Po) < 00. Then IIA~u(x, 'IjJ(x))IIBMO :::; c, where C depends only on 0:, (3, n and the Lipschitz constant of 'IjJ.

Proposition 5.1.4

Proof: Fix a cube Q = {(x, 'IjJ(x)) : x E Q'} <;;; 8D and consider Alu and A 2u corresponding to this Q. Lemma 5.1.1 and Lemma 4.2.7 imply (5.1.12) whenever give:

Xl, X2 E

Q'. Let

Xo

be the center of Q. Then (5.1.12) and Lemma 5.1.3(a)

146

5. Good-A Inequalities for the Density ofthe Area Integral

:::;

I~'I

k,

Afu(x,7/J(x))dx +

I~'I

k, IA~u(x,7/J(x))

-

A~u(xo,7/J(xo))ldx:::; e

which completes the proof of the proposition. This proposition will almost immediately imply the good->. inequality in Theorem 5.0.2(b). Before doing this, however, we first would like to show an analogous proposition for N[3u(x,7/J(x)). To do this we will need a few lemmas. The first of these resembles Lemma 4.2.7 which was the analogous lemma for A 2 u. Lemma 5.1.5 Suppose u is harmonic on the unbounded Lipschitz domain D = {(x, y) : y > 7/J(x)} and that additionally, dist(P, aD) lV'u(P) I :::; 1 for every P E D. Suppose Q = {(x,7/J(x)) : x E Q'} ~ aD is a cube and N2U is associated to Q as defined above. If there exists a Po E aD such that N[3(Po) < 00, then for every

P 1 ,P2 E Q,

IN2u(P1) - N2u(P2)I :::;

e

IP1 - P21 £(Q') .

Here the constant e depends only on (3, n and the Lipschitz constant of 7/J. Proof: Set P1 = (x1,7/J(xd), P2 = (X2,7/J(X2)). Suppose (Sl,t1) E r 2(X1,7/J(xd). Simple computations then show that if we define (S2, t2) = (Sl +X2 -xl, max{t1 + 7/J(X2) -7/J(xd, td) then (S2, t2) E r 2(X2, 7/J(X2)). Note that if P = (s, t) has sEQ' and t > 7/J(xo) + m(Q') then dist(P,aD) ?: e£(Q'). Thus, IU(Sl, t2) - U(S2, t2)1 :::; I(sl, t1) - (S2, t2)1 sup{lV'u(P)I : P E (Sl, h)(S2, t2)}

< eIF1 - P2 1 -

£(Q').

Thus, N 2u(Pd :::; e 1~(Q~21 +N2u(P2). Note that N[3u(Po) < 00 and a similar computation involving the sup of lui over a top part of r [3 (Po) shows that N 2u(P) < 00 for every P E aD. The lemma then follows. The next lemma is a simple result whose usefulness will soon become apparent. Lemma 5.1.6 Suppose f and 9 are real valued functions defined on a cube Q and suppose a, bE R Then for every >. > 0,

{x E Q: Imax{f(x),g(x)} -max{a,b}1 > >.} ~ {x E Q : If(x) - al > >.} U {x E Q : Ig(x) - bl > >.}. Proof: We can assume a > b. Then {XEQ: Imax{f(x),g(x)}-al >>.} ~{x E Q : f(x) ?: g(x), If(x) - al > ,x}

~ ]R.n

5.1 Sharp control of A and N by D

147

U {x E Q : f(x) < g(x), g(x) - a> A} U {x E Q : f(x) < g(x), g(x) - a < -A} ~{x E Q : f(x) ~ g(x), If(x) - al > A} U {x E Q : g(x) - b > A} U {x E Q : f(x) - a < -A} ~{x E Q : If(x) - al > A} U {x E Q : Ig(x) - bl > A}. Lemma 5.1.6 immediately implies:

Lemma 5.1.7 Suppose f and 9 are real valued measurable functions defined on a cube Q ~ ~n and suppose a, b E ~. If

k I~I k

I~I and

If(x) - aldx:<:::; C,

Ig(x) - bldx:<:::; C

then

I~I

k

I max{J(x), g(x)} - max{a,b}ldx:<:::; 2C.

Once again, we have the Key Estimate in the following: Proposition 5.1.8 Suppose a > {3 and U is harmonic on the unbounded Lipschitz domain D = {(x, y) : x E ~n, y E ~, Y > 1P(x)}. Suppose that for every P E aD, Dau(P) :<: :; 1 and that there exists Po E aD such that N{3u(Po) < 00. Then IIN{3u(x, 1P(x))IIBMO :<: :; C, where C depends only on a, {3, n and the Lipschitz constant of 1P.

Proof: Fix a cube Q = {(x, 1P(x)) : x E Q/} ~ aD and consider Nl U and N 2 u corresponding to this Q. Lemma 5.1.3(b) implies that if we set v = U - u(P*), then Ibl JQ N 1 v(x,1P(x))dx:<:::; C. But IN1 u(x,1P(x)) -lu(P*)II:<:::; N 1 v(x,1P(x)) by the triangle inequality so consequently,

I~I

k

IN1 u(x,1P(x)) -lu(P*)lldx:S C.

Lemmas 5.1.1 and 5.1.5 imply

where, as before, Xo denotes the center of Q'. The proposition then follows immediately from Lemma 5.1.7.

148

5. Good-A Inequalities for the Density of the Area Integral

Finally we conclude this section with the proof of Theorem 5.0.2. The proofs of (a) and (b) are essentially the same; we will do (a) and leave (b) to the reader. In fact, these follow exactly the proof of the good-A inequality of Theorem 4.2.1; because of this, we will be brief. To prove Theorem 5.0.2(a), fix c, A. We may assume O"{P E aD : Nf3U(P) > A} < 00. Let E = {P E aD : Dau(P) > cA} and set D' = UPEEc r a(P). We may also assume E C =I- 0 so that then D' =I- 0. Then D' is a Lipschitz domain with constant determined by O!, say D' = {(x,y) : y > 7{i (x)} , where 7{i is Lipschitz. By Lemma 5.1.2 and Proposition 5.1.8, IINf3u(x, 7{i(x))IIBMO :::; GcA, and thus, by Lemma 4.2.3, for all TJ > 0, I{x E lRn : Nf3u(x, 7{i(x)) > 2TJ}1

(-~;TJ)

:::; G1 exp

In particular, this holds with

I{x E lR n

:

Nf3u(x,7{i(x)) > TJ}I.

TJ = A so that then:

O"{P E aD: Nf3u(P) > 2A, Dau(P) :::; cA}

= o-{P

E EC

:::; CI{x E lRn

N{3u(P) > 2A}

:

:

N{3u(x,7{i(x)) > 2A}1

:::; G1 exp (- ~2) I{x E lRn :::; G1 exp ( -

5.2

:

Nf3u(x, 7{i(x))

~2) O"{P E aD : Nf3U(P)

> A}I

> A}.

Sharp control of D by A and N

In this section we will prove the good-A inequalities in Theorem 5.0.3. The proof of this theorem has ideas in common with the proof of the good-A inequalities in Theorem 5.0.2, and the proofs of several of our other theorems. We will still use gradient estimates to estimate contributions from the "far away" parts of cones. For part (a) we will use Green's theorem to estimate the contributions from the "close by" parts of cones and for part (b) we will estimate these "close by" parts using "invariance principle" techniques similar to those used in the proof of Lemma 4.1.3. The problem is that these arguments readily produce estimates for Du(x; r) for each fixed r, but what we want, of course, is an estimate for Du(x) = sUPrEIR Du(x; r). This will make our task more difficult as we will need to estimate Du(x; r) as both x and r vary. We recall that in Theorem 5.0.3 we are considering the "smoother" versions of the density and maximal density given by (5.0.7). Throughout the course of the proof, we will need to consider various versions of these defined on sub domains

5.2 Sharp control of D by A and N

of lR~+l. For k > 0 and a sub domain n

D~u(x;r)= D~u(x)

=

r

c

lR~+l, we set

tCPt(x-s)t!.(u(s,t)-r)+(dsdt)

inn {t<:;.k}

149

rElR

(5.2.1)

supD~u(x; r). rEIR

If n = lR~+l we will drop the subscript n, and likewise if k = 00 we will not write it. With D~u(x; r) as in (5.2.1) set D~u(x; r) = Dnu(x; r) - D~u(x; r) and D~u(x) = sUPrEIR D~u(x; r). These are the contributions to Dnu(x; r) and Dnu(x) from the "top" part of the cone. Of course, they depend on the choice of k, but to avoid cumbersome notation we will not decorate these with a k. Besides, we will use this notation sparingly and the k will always be clear from the context. Our first lemma shows for the D-functional what Lemma 4.2.7 shows for the area function and Lemma 5.1.5 shows for the nontangential maximal function.

Lemma 5.2.1 Suppose n c lR~+l is a subdomain, k > 0 and D~u(x; r) and D~u(x) are defined as above. Suppose also tlV'u(s, t)1 ::; 1 for every (s, t) E where ::J n is a domain which has the property that there exists a Co > 0 such that ((v,w) : dist((v,w),n) < Cow} c Then there exists a constant C = C(Co, n) such that if x, y E lRn ,

n

n

n.

(a)

ID~u(x; a) - D~u(y; a)1 ::;

(b)

ID~u(x) - D~u(y)1 ::;

Clxk-yl,

a E lR

Clxk-yl

Proof: Fix x, y E lR n and set

We recall that supp cP

R

= (r{3(x) u r{3(Y)) n n n {t 2: k}.

c::;;;

B(O, (3). Then

ID~(x; a) - D~ (y; a)1 =

Il

t(cpt(x - s) - CPt(Y - s))t!.(u(s, t) - a)+(dsdt)

::; Clx -

I

yll Cnt!.(u(s, t) - a)+(dsdt).

!.

We may assume that Co < Then we can find a Coo function ¢(s, t) such that o ::; ¢(s, t) ::; 1 for all (s, t) E lR~+l, ¢(s, t) = 1 on R, supp ¢ c::;;; {(v, w) : dist((v,w),R) < Cow} c::;;; and 1V'¢(s,t)l::; Note that for t < (1- Co)k, I{s : (s, t) E supp¢}1 = 0 and for t 2: (1- Co)k,

n

f.

I{s: (s, t) E supp¢}1 ::; I{s: dist((s,t),r{3(x)) < Cot}1

+ I{s: dist((s,t),r{3(Y))::; Cot}l::; Ctn.

5. Good-A Inequalities for the Density of the Area Integral

150

Therefore,

r rn~(U(s,t)-a)+dsdt::;J

In

: ; oj

: ; 01 : ; 01

rn¢(s,t)~(u(s,t)-a)+dsdt

supp¢

(t-nIV¢(s,t)1 +rn-I¢(S,t)) IVu(s,t)ldsdt

supp¢

r n - 2 dsdt

supp¢

o

00

r

n - 2 1{s:

(s,t) E supp¢}ldt

(I-Ca)k

::; k' Then (a) follows; (b) follows by taking supremums. For certain subdomains W ~ ~~+1 we will need to create a slightly different version of D{vu(x; r) which approximates it, but is easier to estimate. This new version will be obtained from D{vu(x; r) via numerous integrations by parts. So at this point it is convenient to state and prove a technical lemma that will allow us to control the boundary terms that arise from these integrations by parts. Lemma 5.2.2 Suppose (3 < /, E ~ ~n and W = UXEE f /,(x). Suppose also that p is a function supported on B(O, (3). Let h > and set f(x) = f~(x) n W. There is a constant 0, depending only on (3, /, n such that for x E ~n,

°

r

I Jar(x) rnp

(~) du(s, t)1 t

Here u denotes surface measure on

::; Ollplloo.

of (x).

Proof: Clearly 8r(x) ~ {(s, h) : Ix - sl ::; (3h} U(r f3(x) noW) U{ (s, t) : Ix - sl = (3t}. Since p is supported on B(O, (3), the integral of Pt(x-s) vanishes on the third set. The integral of Pt(x - s) over the first set is clearly bounded by Ilplloo. To control the integral over the second set we note that since oW is the graph of a Lipschitz function with Lipschitz constant ~, elementary geometric arguments show that u(f f3(x) noW) ::; O(inf{ t : (s, t) E r f3(x) noW})n where 0 depends on 'Y, (3 and n. Then,

r rnp (~) du(s, t)1 I Jrf3(x)naw t ::; Ilplloou(f f3(x) n W) (inf{ t : (s, t) E f f3(x) n W} )-n ::; Ollplloo , which finishes the proof of the lemma.

5.2 Sharp control of D by A and N

151

Now consider a set E ~ IR n , suppose a> "/ > {3 and set W = UXEE f '"'((x). For h > 0 consider D{Vu(x; r) and D{Vu(x) as defined in (5.2.1), set W' = UXEE f ",(x) and define N~hw'u(x) = sup{lu(s, t)1 : (s, t) E f;h(x) n W'}. We now create a new version of D{V'u( x; r) which approximates it. Lemma 5.2.3 With the notation as in the previous paragraph, there exists a vector

valued function (x) = (l(X), ... ,n+1(x)), with the following properties: (a)

Each i(X) is supported on B(O,{3), has mean value zero, and is smooth.

(b)

If we set

J

D{Vu(x;a)=

t(x-s)·V(u(s,t)-a)+dsdt

r3(x)nw then for

lal ::; N~~w'u(x)

we have

ID~u(x; a) - D~u(x; a)1 ::; CN~~wlu(x). Here C depends only on a, ,,/, {3, n and our original choice of cp. Proof: For typographical convenience, we will write f(x) for f~(x)nW throughout this proof. Then by the divergence theorem,

D~u(x; r) = =

1

-1

r(x)

tCPt(X -

r(x)

+

r

s)~(u(s, t) -

a)+dsdt

V (tCPt(x - s))· V(u(s, t) - a)+dsdt

Jaqx)

tCPt(X-S):! (u(s,t)-a)+d(](s,t) un

= I + II. Here there are several technicalities in this application of the divergence theorem, since neither the functions involved nor the regions involved are smooth. To overcome these, we first note that the region f(x) is of the kind considered by Stein [St4, p. 206]. Stein shows that there exist smooth regions V8 such that V8 ~ f(x) for every 8> 0 and V6 f(x) as 8 ! O. We may then apply the divergence theorem on each V6 as before. (See the proofs of Lemmas 5.1.1 and 5.1.2.) That is, we apply the divergence theorem to smooth approximations of (u(s, t) - a)+ and then take limits. We actually then obtain a similar formula on each V6 and subsequently let

r

8! O.

To estimate II, note that for (s, t) E 8f(x), It tn (u( s, t) - a)+ I ::; tIVu(s, t) I ::; CN~~wlu(x) where the last inequality is essentially just Lemma 2.3.1. (Lemma

152

5. Good-A Inequalities for the Density of the Area Integral

2.3.1 does not apply directly, but simply note that in this situation, there exists a constant Co, depending only on a,{3,"(, such that if (s,t) E af(x), then B((s, t), Cot) <;;; f;h(x) n W'. This was the essential ingredient in the proof of Lemma 2.3.1. That proof can now be readily adapted to the present case.) Then

by Lemma 5.2.2. Also,

r -r

t ~
-r


1=-

Jr(x)

Jr(x)

Jr(x)

V's(t
ut

s)~ (u(s, t) ut

s)~ (u(s,t) ut

a)+dsdt

- a)+dsdt

= Ia +h +Ic. For i

= 1, ... ,n set ki(s) = Si
Then by the divergence theorem we have:

If

lal :::; N~hwlu(x) then by Lemma 5.2.2, ,

where C depends on a, {3, ,,(, n and our original choice of <po We now set

5.2 Sharp control of D by A and N

=

r

Jr(x)

153

-{Vs(tipt(x - s))· Vs(u(s, t) - a)+ - taa 'Pt(x - s)aa (u(x, t) - a)+ t t

}

a

n

- t;(kiMx - s) aS i (u(s, t) - a)+ dsdt. The above computations show that for lal ~ N~hWlu(x), ID{Vu(x; a) - D{Vu(x; a)1 ~ CN~~wlu(x). Elementary computations with'the chain rule show that if we set

then we may write D{Vu(x; a) more succinctly as:

J

D{Vu(x;a) =

IPt(x-s)·V(u(s,t)-a)+dsdt.

r(x)

The proof of the lemma is finished if we note that since 'P is radially symmetric, then each of the coordinate functions of IP has mean value zero on JRn. We now show a lemma that will allow us to estimate D{Vu(x; a) as both x and a vary. We first make a definition. For i5 w u(x; a) as defined in Lemma 5.2.3, I ~ JR, and 1 ~ p < 00 we set

Br(x) =

(11 I r r

i5{Vu(x; a) - ~(Vu(x; b) I dadb) la - bl 2 P

1

P

(5.2.2)

The following is an almost immediate consequence of a lemma of Garsia, Rodemich and Rumsey [GRR]. (See Gundy and Silverstein [GS] for a short discussion of the application of the Garsia, Rodemich, Rumsey lemma to the study of the Dfunctional. See also Barlow and Yor [BY2] for more on the use of this lemma.) Lemma 5.2.4 With the notation as in Lemma 5.2.3, suppose I = [e, d] ~ JR and Br(x) is as in (5.2.2). Suppose Q ~ JRn is a cube and that for some p > 4, Ihl i Q 1i5{Vu(x; a) - i5{Vu(x; b)IPdx < Cia - bl ~ where C is a constant indepen-

dent of a and b. Then for almost every x E Q, sup ID{Vu(x; a)1 ~ 8(4~)III!-% Br(x) aEr

+ CN~hWlu(x) + 1i5{Vu(x; d)l. '

Proof: Definition (5.2.2) and the hypothesis, combined with Fubini's theorem shows that Br(x) < 00 for almost every x E Q. Furthermore, Kolmogorov's continuity theorem implies the existence of a measurable function f(x, a), x E Q, a E I which is continuous in a for every x and such that for each a E JR, f(x, a) = i5{Vu(x; a)

154

5. Good-A Inequalities for the Density of the Area Integral

for almost every x. A simple Fubini argument shows that if we define ih(x) as in (5.2.2) with f(x, a) in place of i5{vu(x; a) then BJ(x) = ih(x) a.e. on Q. Since f(x, a) is continuous as a function of a, the Garsia, Rodemich, Rumsey inequality 1 1 1 states that If(x,a) - f(x,b)1 8(4 v )la - bl"2- v B J(x) for every a, bE R Thus, there exists a null set A such that if x E Q\A then

s

A

whenever a E I is rational. So if x E Q\A, a is rational, and Lemma 5.2.3 implies that

lal s N~hwlu(x) then '

But this is true even if lal > N~hWlu(x) since then left hand side is O. Since D{vu(x; a) is lower-semicontinuous'as a function of a (see Gundy and Silverstein [GS]), the conclusion follows. We have now finished the preliminaries that will be used in both the proofs of the good-A inequalities in Theorem 5.0.3 and begin the proof of Theorem 5.0.3(a). We let A > 0, 0 < c: < 1 be fixed. We set 'Y = ,B~"', E = {x E IRn : N",u(x) S C:A} and define W = UXEEf-y(x) and W' = UXEEf",(X). We may assume E #- 0 so that W #- 0. Then

lui S C:A on W tl\7u(s, t)1 S CC:A for (s, t)

E

W.

(5.2.3)

The first of these statements is obvious and the second follows from the gradient estimates of Lemma 2.3.1. For x E IRn , a E IR we now consider Dwu(x;a) and Dwu(x) as defined in (5.2.1). As before, we have the corresponding Key Estimate:

If W is as above, and Dwu(x) is defined by (5.2.1), then IIDwullBMO S CC:A where C = C(a,{J,n,
Proposition 5.2.5

Proof: Fix a cube Q ~ IRn; we want to show that there exists a constant aQ such that

I~I 10 IDwu(x) -

aQldx < Cc:A.

= f(Q), consider D{vu(x; a), D{vu(x) as defined in (5.2.1), and Dwu(x; a) = Dwu(x; a) - D{vu(x; a) and let Dwu(x) = sUPaEIR Dwu(x; a). Then,

Set h

lal > C:A Dwu(x; a) = 0 if lal > C:A Dwu(x; a) = D{vu(x; a) + Dwu(x; a) Dwu(x) S D{vu(x) + DWu(x).

(i) D{vu(x; a) = 0 if

(ii) (iii)

(5.2.4)

5.2 Sharp control of D by A and N

155

Lemma 5.2.1 provides a way to estimate D'{;.ru(x; a) and D'{;.ru(x). Our next lemma will estimate D{vu(x; a). Lemma 5.2.6 Suppose

Here C

=

1~p <

00.

Set 0

=

(UXEQ r~(x)) n W. Then if lal ~ EA

C(p, 0., f3, n, 'P).

Proof: Let Qbe the cube in]Rn concentric with Q and with sidelength (1 +4(3)£(Q). Then if (s, t) E 0, x ~ Q, 'Pt(x - s) = O. Let J be any sub cube of Q, set k = £(J) and form V = (U xEJ r~(x)) nO. Then by Green's theorem:

rI JIR'-i-+ln{(s,t):t~k} r t'Pt (x - s )Xn( s, t)fl.( u(s, t) - a)+ dSdtldX

JJ

Iv ~ lav :n ~

tfl.(u(s, t) - a)+dsdt t

c

~

(u(s, t) - a)+

(5.2.5)

+ (u(s, t) -

a)+1 ;~ IdO'(S, t)

C(EA)IJI

where the last inequality follows from (5.2.3) and the fact that 0'(8V) ~ CIJI. (Again, to apply Green's theorem we first need to smooth 8V and the function (u-a)+, then apply Green's theorem, and pass to a limit. These type of arguments are presented in Chapter 1 and in Section 5.1.) Let Xo be the center of J and set Dru(x; a) = Dnu(x; a) - D~u(x; a). Then Lemma 5.2.1 implies IDru(x; a) Dru(xo; a)1 ~ CEA for every x E J. This, (5.2.5), and the triangle inequality combine to give:

_111 J

Thus,

rr

I JJ JlR ++ n

t'Pt(x - s)Xn(s, t)fl.(u(s, t) - a)+dsdt - Dru(xo; a)ldx 1

J

~ CEA.

t'Pt(x - s)xn(s, t)fl.(u(s, t) - a)+dsdt

1R'-i-+1

is in BMO on Qwith BMO norm less than CEA. Since (5.2.5) holds for J = Qthe conclusion of the lemma follows. This last lemma provides an estimate for D{vu(x; a), but we really need to estimate D{vu(x). This will be done by use of Lemma 5.2.4; our next lemma merely puts us in a position to apply it.

156

5. Good-A Inequalities for the Density of the Area Integral

Lemma 5.2.7 For 2 < p <

h

00,

lal < EA,

Ibl < cA,

ID{'vu(x; a) - D{'vu(x; b)IPdx :::; ClQlla -

bl~ (cA)~.

Proof: Suppose g(x) is a function supported on Q with Ilgllq As before, set n = (UxEQ r~(x)) n W. Then

=

I, where ~ +

*=

1.

h

(D{'vu(x; a) - D{'vu(x; b))g(x)dx

r r xn(s, t)[t(x - s)· (V'(u(s, t) - a)+ - V'(u(s, t) - b)+)]dsdtg(x)dx = r (t * g(s)) . (V'(u(s, t) - a)+ - V'(u(s, t) - b)+)Xn(s, t)dsdt JlR n+ r r ipt(x-s)(t * g(s)) . (V'(u(s, t) - a)+ - V'(u(s, t) - b)+)Xn(s, t)dsdtdx JlRnJIR~+l =

JQ JIR~+1 +

1

=

1

(JlRrn

tipt(x - s)Xn(s, t)IV'((u(s, t) - a)+ - (u(s, t) _ b)+WdSdt) 2 dx

+1

+

(I.. (1.+" t~,(x =

1

- ')Xn(', t)IV'( (u( " t) - a)+ - (U(8, t)

- W)I'dSdt) ! dx )

,

I . II.

The first quantity I :::; Cqllgll q Silverstein formula, (5.0.4),

r

JlR +n+ =

= Cq,

by Theorem 1.7.2. Also, by the Gundy-

tipt(x - s)Xn(s, t)IV'((u(s, t) - a)+ - (u(s, t) - b)+Wdsdt

1

Ib JIR~+l r a

tipt(x - s)Xn(s, t)~(u(s, t) - r)+dsdtdr.

157

5.2 Sharp control of D by A and N

Then Jensen's inequality and Lemma 5.2.6 give:

II <; ~

(f..la - bl ~-l l 1

1

1

(1.:;+,'
1

1

1

la - bl"2- p la - bl P C(cA)"2IQI P

and this gives Lemma 5.2.7. We can now complete the proof of the BMO estimate, Proposition 5.2.5. First note that Lemma 5.2.7 allows us to apply Lemma 5.2.4 with I = [-cA,cA]. Because i5{vu(x; CA) = 0 and N~hWlu(x) ~ cA we conclude that for almost every ,

x E Q, ID{vu(x) I ~ C(CA)~-~ Br(x) + CcA. Fix p > 4, integrate the pth power of this over Q and use Lemma 5.2.7 and Fubini's theorem to obtain:

Choose p = 5, say, then by Jensen's inequality, 1 TQT Now let

I~I

k

Xo

Q

(5.2.6)

h Dwu(x)dx ~ Cd.

be the center of Q. Then

IDwu(x) -

k ~ I~I k ~ I~I

1

D~u(xo)ldx

IDwu(x) -

D~u(x)ldx + I~I

kID~u(x)

-

D~u(xo)ldx

D{vu(x)dx + CcA

~

CcA,

where the second to the last inequality follows from (5.2.3)(iii) and Lemma 5.2.1 and the last inequality follows from (5.2.6). Theorem 5.0.3(a) now follows from Proposition 5.2.5 using the usual strategy. By Lemma 4.2.3, IIDwullBMO < CcA implies for every 'T} > 0,

-C2 'T}) I{x E]Rn : Dwu(x) > 2'T}}1 ~ C 1 exp ( ~ I{x E]Rn : Dwu(x) > 'T}}I. Then, using this with 'T}

= A we obtain

I{x E]Rn : Du(x)

> 2A, Nau(x) =

~

cA}1

I{x E E: Du(x)

> 2A}1

158

5. Good-A Inequalities for the Density of the Area Integral

= I{x E E ::; I{x

: Dwu(x) > 2A}1

E]Rn :

Dwu(x) > 2A}1

::; 01

-02 ) I{x exp ( -c-

E]Rn :

Dwu(x) > A}I

::; 01

-02 ) I{x exp ( -c-

E]Rn :

Du(x) > A}I·

We now proceed with the proof of Theorem 5.0.3(b). We will show:

Proposition 5.2.8 Assume a and (3 are as in the statement of Theorem 5.0.3(b). Then here exist constants 01 and O2 depending only on a, (3, cp and n such that if Q is any cube in]Rn and if h = 2yn£(Q)(3-1 then

r

1 i exp [h IQI D U(X) Q

-

01 (Aau(x))

2

]dx::; O2.

We first make a few remarks. Quite possibly Aau(x) = 00 for some x E Q. For such x the integrand of the above integral is to be interpreted using the convention e- oo = 0 and 00 ·0 = O. A theorem of Brossard [Brl] states that, except for sets of Lebesgue measure zero, whenever Aau(x) < 00 we also have Dhu(x) < 00. Thus, the integrand could never be of the form 00 . 0 on a set of positive measure. If Aau(x) = 00 almost everywhere on Q the result is trivial so we will assume Aau(x) < 00 for all x in a set of positive measure in Q. Consider an a' < a such that also (3 < (2048n)-1o:' and set uc;(x,y) = u(x,y+c). Then since A:>u(x) < 00 for some x, this and a' < a, c > 0 shows that Aa1uc;(x) ::; 0' for all x E Q. (N aturally, 0' depends on a', c.) Suppose that we show

for some 01 and O2 independent of 0'. We then note that this remains valid with a replacing a', and then use Lemma 4.2.9 to obtain

where 01, O2 are independent of c, and L is the constant in Lemma 4.2.9. Now let c ----+ O. We conclude that to prove the Proposition we may assume Aau(x) ::; 0' for all x E Q, if, of course, we obtain 01 and O2 independent of 0'. We also note that our limiting procedure is consistent with our conventions on arithmetic and that the validity of the Proposition for Uc; and this limiting procedure gives a proof of part of Brossard's theorem: Aau(x) < 00 implies Dhu(x) < 00 for almost every x. (To see this simply restrict the integration in the last integral to the set {x E Q : Aau(x) < oo}. This forces liminfc;---->o Dhuc(x) < 00 for a.e. x in this set and subsequently forces Dhu(x) < 00 for a.e. x in this set.)

5.2 Sharp control of D by A and N

159

Before proving the proposition we first show how Theorem 5.0.3(b) follows from it. Fix A. Note that if I{x E IRn : Du(x) > A}I = 00 we are trivially done so we assume the contrary. We then choose dyadic cubes Q ~ IRn such that 1

I{x E Q : Du(x) > A}I > "2IQI and such that if Q is the dyadic cube in IR n with i(Q) = 2£(Q) then

I{x

E

1 -

Q : Du(x) > A}I :::; "2 IQI .

We will show that for such Q, -C2 ) I{x E Q : Du(x) > KA, A",u(x) :::; cA}1 :::; C1 1QI exp ( """"€2

.

Summing over Q gives the result. Fix such a Q and an c with 0 < c < 1; we may also assume that there exists an Xo E Q such that A",u(xo) :::; cA. Also we may pick Xl E Q such that DU(X1) < A. Our choice of (3 insures that r/3(x)\r~(x) ~ r~ (xo) whenever X E Q. (This just requires 4(3 < a.) As before, we set DT u(x; a) = Du(x; a) - Dhu(x; a), DT u(x) = sUPaEIR DT u(x; a). Then Lemma 5.2.1 implies

IDT u(x) - DT u(xdl :::; CcA for all X E Q. Thus, IDTu(x)1 :::; CcA + A for all x E Q. Set KI = K - (C + 1). Then

I{x E Q : Du(x) > KA, A",u(x) Set

E = {x Replace u by

c

2 A-lU

E

< cA}1 :::; I{x E Q: Dhu(x) > KIA, A",u(x) :::; cA}I·

Q : Dhu(x) > K 1 A, A",u(x) :::; cAl.

in Proposition 5.2.8; this gives

and the result follows by taking K 1 , and hence K, large enough. Thus, we have reduced matters to the proposition. Before proceeding with this we show a lemma that will be used several times throughout the proof of the proposition.

160

5. Good-A Inequalities for the Density of the Area Integral

Lemma 5.2.9 With the notation as above, let q denote the center of Q and suppose u(q,2h) = O. (Here h = 2.Jii,i(Q)f3-1 as in Proposition 5.2.8.} Then there exists constants C 1 and C 2 , which depend only on 01,13 and n, such that

1 r IQT 1Q exp[N22h,Bu(x) -

2

C 1(Aa u (x)) ]dx ~ C2 .

(5.2.7)

Proof: We use Lemma 4.1.3. In the present situation, however, we have 213 in place of the 13 which occurs there, so that the corresponding h in the statement of Lemma 4.1.3 is h = 2y'rif(Q)/2f3 which is smaller than the 2h = 4y'rif(Q)/f3 we are using in the statement of this lemma. But, as we have noted, Lemma 4.1.3 remains valid for any larger value of h than stated there. (See the remarks after the proof of Lemma 4.1.3.) Thus, for all A > 0,0 < c < 1,

I{x E Q: N?gu(x) > K)", Aau(x)

~ c)..}I~ C3 1QI exp ( - ~:).

(5.2.8)

For k,j E {O, 1,2, ... } set Ekj

= {x E Q

: k ~ N:;gu(x)

< k + 1,

j ~ Aau(x)

< j + 1}.

To complete the proof of the lemma, we merely break up the integral in (5.2.7) into a sum of integrals over the E kj , noting that we really only need to consider those E kj for which j2 is (approximately) less than k. Then use (5.2.8) and note that if C 1 is chosen large enough (depending on K and C4) the resulting sum converges. We leave the details to the reader. We remark that here we have really shown that an inequality like (5.2.7) is valid whenever the aperture used in the definition of the nontangential maximal function is less than the aperture used in the definition of the Lusin area function. We did not write the lemma in this generality simply because we will only use it in the context of the proof of Proposition 5.2.8. This Lemma and proof should also be compared to the discussion in Problem 4.2.14 where (5.2.7) is shown in another way. As in the proof of Theorem 5.0.3(a), it is fairly easy to obtain estimates for j)hu(x; a) and we do this in the next lemma. Here we connect with the ideas of Chapters 2 and 3 and exploit the particular form of j)hu(x; a). Lemma 5.2.10 There are constants C 1 and C 2 independent of a E IR such that

Proof: Recall that j5hu (x; a)

=

r

1r~(x)

t(x - s) . ~(u(s, t) - a)+dsdt

5.2 Sharp control of D by A and N

161

where c1? = (c1?1, .•. ,c1?n+1) and each c1?i is supported on B(O,{3), has mean value zero, and is smooth. As in Chapter 2, we write jjhu(x; a), x E Q as the sum of martingales. An examination of the proof there shows that the square function of each of these martingales is dominated by

(See also (3.1.7).) By our choice of {3, 32..fii{3 < Q so this last expression is dominated by Aau(x). The result then follows from the corresponding theorem for martingales, Theorem 2.0.1. This proof should be compared to the proof of inequality (4.2.20) which is essentially the same. Of course, we really need estimates for Dhu(x). To obtain these, we will break up Dhu(x) into "smaller" maximal densities. For j an integer set

Dju(x)

=

sup{Dhu(x; a): a E [j,j + I)}.

Then

Dhu(x) ::; sup{Dju(x):

Ijl::; Ni$u(x) + I}.

We have Lemma 5.2.11 There are C 1 and C 2 independent of j such that

i

Before proving Lemma 5.2.11 we show how it implies Proposition 5.2.8. For set Ei = {x E Q: i::; Ni$u(x) < i + I}. Then

= 0,1,2,3,

I~I

k

::; L

1 -IQI

00

exp[Dhu(x) - C 1 (Aa u(x))2]dx

1

i=O

i+2

::; ~ .L 00

.=03=-.-2

0::

exp[ sup Dju(x) - C 1 (Aa u(x))2]dx Ijl
Ei

1

TQT

1.

exp[Dju(x) - C 1 (Aa u (x))2]dx

E,

taj-~J C~] L. exp]2D7u (x) - G1(Aou(x))']dx) l

.C~I Li

1

eXP[-C1(A a u(x))2]dX) 2]

5. Good-A Inequalities for the Density of the Area Integral

162

::;

C~(2i + 4)e-! (\~\ lei exp[N~3u(x) - C (A 1

o u(x))2]dX)

~

::;C where for the last inequality we have used Lemma 5.2.9. For the next to last inequality we have used Lemma 5.2.11. To see this, take the C1 in this last computation to be four times as large as the C 1 in Lemma 5.2.11. So it suffices to show Lemma 5.2.11. To do this, fix an integer j. Then Lemma 5.2.4 with I = [j,j + 1), w = W' = lR+.+l, a = 2/3,p = 5, and the notation Bj(x) = BJ(x) then yields: Under the assumption

\~\

k

\iJhu(x; a) - iJhu(x; b)\5dx < Cia -

Dju(x) ::; 8(4!)Bj(x)

b\~,

then

+ CN~3u(x) + \iJhu(x;j)\.

(5.2.9)

We first claim that the assumption needed here holds. To see this compute as in Lemma 5.2.7 and note that the hypothesis there is only used at the last step ofthe proof. Without such a hypothesis we can use those computations to nevertheless estimate

r-

1 lQf i Q \Dhu(x, a) where

r

E

-

5 1 Dhu(x, b)\5dx ::; Cia - b\2-

Ib iJRnr a

Dnu(x; r) Q2dxdr,

n is as in the proof of Lemma 5.2.7 (now with W = lR+.+ 1 ).

[a, b],

But for each

r Dnu(x;r)5/2dx::; iZ(D hu(x;r))5/2dx ::; C iZexp (Dhu(x; r))dx,

ilR

n

Q

Q

where Q is a cube slightly larger than Q. Because we have assumed (see the comments after the statement of Proposition 5.2.8) that Aou(x) ::; C' for all x E Q (and we may do the same for x E Q), then Lemma 5.2.3, Lemma 5.2.9, and Lemma 5.2.10 combine to show that this last integral is bounded independently of r. This combined with previous computation shows the claim, that is, that the hypothesis of Lemma 5.2.4 holds so that we may deduce (5.2.9). Note that the constant C in (5.2.9) does not depend on C'. We will show: Lemma 5.2.12 There are positive constants C 1 and C 2 independent of j so that

Assume Lemma 5.2.12. Then (5.2.9), Holder's inequality, and Lemmas 5.2.9, 5.2.10 and 5.2.12 show that if C 1 is chosen large enough then the conclusion of Lemma 5.2.11 holds. Thus, we have reduced matters to proving Lemma 5.2.12.

5.2 Sharp control of D by A and N

163

To prove Lemma 5.2.12, it will be convenient to set

so that with this notation

The function exp[(y + 45 )!] is convex for y > 0 and thus:

So Lemma 5.2.12 follows from: (5.2.10) with C 1 ,C2 independent of a,b. To see (5.2.10), let 'T](x) be a positive radially symmetric function supported on B(O, 64vn(3) such that 'T] == 1 on B(O, 32vn(3) and define

iJhu(x;a)

=

r

jffi."j.+l n{ t
t'T]t(x-s)~(u(s,t)-a)+(dsdt).

Note that by definition,

k(x,a,b) =

1

r

jffi."j.+l n {t
1>t(x_s).(\7(u(s,t)-a)+-~(u(s,t)-b)+)dSdtl. la -

bl 2

Set

S(x)

= (

r

jr

h = 32vn/3

1\7(u(s, t) - a)+ (x)

la -

~(u(s, t) -

bl

2

1

b)+ 12 t 1 - n dSdt)

2

164

5. Good-A Inequalities for the Density of the Area Integral

Reasoning as in the proof of Lemma 5.2.10, we obtain: There exists constants C l and C 2 such that

r

1 TQI JQ exp[k(x, a, b) -

Cl(S(x)) 2 jdx:S; C2.

But

where the last equality is just the Gundy-Silverstein formula (5.0.4). Thus,

1 Jr exp [k(x, a, b) - Clla _ 1 bl TQI Q

Ib a

1

D h u(x; r)dr dx:S; C2. A

(5.2.11)

Then (5.2.11) together with the Cauchy-Schwarz inequality and Jensen's inequality show that (5.2.10) follows from:

I~I 10 exp[.z)hu(x; r) -

C 1 (Ac.u(x))2]dx :s; C 2

(5.2.12)

for every r E JR., where C l and C 2 are independent of r. So we only need now to show (5.2.12). To see (5.2.12), let Q be a cube in JR.~+1 with the same center as Q but with £(Q) = 64yin£(Q). Then (5.2.12) follows if we show a similar estimate with Q

i~___place of Q. Then h =

2

vnJCQ ) = 2t:1n~).

Corresponding to .z)hu(x; a) we have

.z)hu( x; a) as given by Lemma 5.2.3. Then (b) of this implies that

But since .z)hu(x; a) = 0 whenever

lal > Ni;8vnf3u(x),

then for all a E JR.,

Then, reason as before, using Cauchy-Schwarz and Lemmas 5.2.9 andJi..2.10 to complete the proof of (5.2.12). Note that we apply Lemma 5.2.10 to .z)hu(x; a); this is defined with a kernel having support in B(O, 64yinf3) and thus, we demand (32yin)(64yinf3) < L¥.

5.3 Application I. A Kesten-type LIL and sharp LV-constants

165

Problem 5.2.13 Prove Theorem 5.0.3(b) under the assumption (3 < a. In the proof we just gave, we didn't always choose the optimal values of constants, but even if we had proceeded with great care, we could not have followed the same line of reasoning to produce a proof valid under this assumption. It seems a different strategy is needed.

5.3 Application I. A Kesten-type LIL and sharp V-constants In [KelJ, Kesten proved the following LI1's for the maximal local time up to time t, L; = sup{Ls : s :::; t}. Theorem 5.3.1

L*

lim sup t t-+oo V2S;(X)loglogSt(X)

=1

(5.3.1)

and . f (log log St(X)) 1/2 L* _ :~~ S;(X) t - 'Y

1.

almost surely on {S (X) = oo}. Here 'Y is a constant such that ~ :::; 'Y :::; qo is the smallest positive zero of the Bessel function Jo(x).

(5.3.2)

-Y2, where 2

The exact value of the constant 'Y was subsequently found by E. Csliki and A. Foldes [CsF] to be V2qo. The exponential square good-A inequality on Lipschitz domains, Theorem 5.0.2(b), together with the proof of Chung-type 1IL in Section 4.3, Theorem 4.3.2, and its corollary, Corollary 4.3.4, immediately imply a lower half version of (5.3.2) (without the precise constant, of course). Theorem 5.3.2 (Lower-Half Kesten-type 1IL for harmonic functions) Let u be harmonic on lR.~+1. Fix 0 < (3 < a and suppose these are points (xo, to) and (xl,h) such that A,B'(u)(xo, to) < 00 and D",'(U)(Xl,h) < 00 for some (3' > (3 and a' > a. Then .. hmmf tlO

(

loglog(A1u(x, t)) ) (AI ())2

for almost every x E {x E lR.n only on a, (3, a', (3' and n.

fJu x, t

: A~u(x)

1/2

1

Dau(x, t) ~ C

= oo}. Here C

is a constant which depends

This result is motivated by (5.3.2). As in the case of Theorem 4.3.2, the upper bound is an open problem. Also both, the upper and lower half analogues of (5.3.1) for harmonic functions, are open. An upper bound would follow, as in Chapter 3, from a Lipschitz domain version of Theorem 5.0.3(b) which we do not have.

5. Good-A Inequalities for the Density of the Area Integral

166

The inequality S;(X) :::; L;X; shows that the LIL's in (5.3.1) and (5.3.2) are closely related to the Chung LIL and the Kolmogorov LIL, at least in as much as we do not ask for best constants on the right hand side. For example, we showed in Chapter 3, (3.0.5), that lim sup t--->oo

X* t

<1

V2S;(X)loglogSt(X) -

almost surely on {S(X) = oo}. This upper bound together with S;(X) :::; L;X; immediately implies a lower bound in (5.3.2). For harmonic functions we analogously have (A~u(x, t))2 :::; 2D~u(x, t)N~u(x, t). (This is just a version of (5.0.5) using truncated cones.) The difficulty in passing from one LIL to another, as for example in passing from the upper half LIL, Theorem 3.0.4, to Theorem 5.3.2, is the technical requirement of the different sizes in the aperture of the cones. As in Section 4.4, we have the following result whose proof follows from Theorems 5.0.2 and 5.0.3 exactly as the ratio inequalities of Theorem 4.4.2 follow from Theorems 4.1.1 and 4.2.1. It is also possible to show a version of Theorem 4.4.3. involving the maximal density, but we will not state this here. Theorem 5.3.3 Suppose u is harmonic on ~~+l, not identically zero, 0 < {3 < 0:, 0 < a :::; 1 and 0 < p < 00. Then

(a)

II (D:~~-a lip :::; C~II(D"utllp,

(b)

II (D~~~-a lip:::; C;II(D"u)all p,

(c)

II (N,,~~l-a lip:::; C;II(N"u)all p'

The constant C~ is O(y'P) as p ~ 00 and Cff, and C; are both O(p) as p ~ If in addition, 0: and {3 are as in (b) of Theorem 5.0.3, we have

p <

00.

In Section 4.4, we proved that for any 0 < (3 < 0:, IIA"ull p :::; CpIIN(3ull p, 0 < 00, with Cp = O(y'P) as p ~ 00, and that this order is best possible.

Using this and the fact that IIA"ull p :::; J2IID"ull~/21IN"ull~/2, shows that, with = 1, the order of the constants in (c) and (d) are also best possible. Hence the decay in E of Theorem 5.0.3 is also sharp. By analogy with the corresponding martingale inequalities which are known to be sharp it should also be the case that the asymptotic behavior of the constant in (a) and (b) should also be best possible. Since we know that IIN"ull p :::; Cy'PIIA"ullp, p :2 2, and the order is a

5.4 Application II. The Brossard-Chevalier L log L result

167

best possible, it suffices to show that the order in (b) is best possible. We have been unable to show this and leave it as a conjecture. As mentioned in Section 4.4, for N and A the best order of the constants can also be proved by calculating the square function of lacunary series and applying the central limit theorem. The problem in our present case is that it seems to be very difficult to compute Du directly for any function, including lacunary series. Given the above inequalities and Davis' [Dav1] result, (4.4.5) above, the following problem naturally arises. Problem 5.3.4 Let 1 inequalities

<

p

< 00. Find the best constants

c~,

C;, C;, C: in the

and Again, Theorem 5.0.1 gives the best order of magnitude of this constant but no additional information. Finally, conspicuously absent from this chapter is a discussion of analogues of these theorems for caloric functions. The authors have not attempted such results. However, progress in this direction has been made recently by N. Eisenbaum and 1. Iribarren. In [EI] they define a density of the area integral for caloric functions u on JR~+l and relate it to the local time of the martingale obtained by composition of u with the space-time Brownian process. This work provides an important tool for the continued study of the density of the area integral in this setting.

5.4

Application II. The Brossard-Chevalier L log L result

In [BC2], Brossard and Chevalier use the density of the area integral to characterize L log L within HI (JR~+l). As further evidence of the utility of the techniques of this chapter, we present here a different proof of their results. We recall (Chapter I, Section 7) that the theorems of Burkholder, Gundy and Silverstein [BGS] and Fefferman and Stein [FS] allow us to regard the Hardy space HI (JR~+1) as the space of all harmonic functions u on JR~+1 with IIN{3ulll < 00. (Recall that the choice of f3 is immaterial by Lemma 4.1.2.) We further recall that if u(x, y) E HI (JR~+I), then the nontangentiallimit of u(x, y), call it u(x) exists almost everywhere (Theorem 1.5.12). The function u(x, y) is just the Poisson integral of u(x), and u(x) as well as all its Riesz transforms Rju(x),j = 1, ... ,n belong to Ll(JRn). Conversely, if u(x) as well as all the Rju(x) belong to Ll (JRn) then the harmonic extension u(x, y) of u(x) to JR~+l satisfies IIN{3uI1 1 < 00. Furthermore, in this case, we have n

IIN{3uI1 1 ~ IIul1 1 + L IIRj uI1 1, j=1

168

5. Good-A Inequalities for the Density of the Area Integral

with the constants in the equivalence depending only on (3 and n. (For this see Stein [St4], Fefferman and Stein [FS], or Torchinsky [To], and the discussion in Chapter 1, Section 7.) We define L log L to be the space of all harmonic U on ]R~+1 such that sup (

y>O JlR n

lu(x,y)l(l +log+ lu(x,y)l)dx <

00.

(5.4.1)

The interest in LlogL, at least in this context, dates to Zygmund's [Zy1] improvement of M. Riesz's [Ri] earlier work on the integrability of conjugate functions. Riesz had shown that if f E LP([O, 211"]), 1 < p < 00, then the conjugate function is also in LP([0,211"]). This is false for p = 1. Zygmund showed that if merely f E L log Lon [0,211"], then the conjugate function Tis in L1 ([0,211"]). Consequently, f + E HI j thus, Zygmund's result is often succinctly written as L log L ~ HI. Conversely, if f ~ 0, f E L1([0,211"]) and E L1([0,211"]), then f E LlogLj this can be found in Zygmund [Zy2] who attributes it to M. Riesz. These results combined yield: If f ~ 0 on [0,211"], then f E HI if and only if f E L log L. In 1952, Calderon and Zygmund [CZ1] extended this to higher dimensions. They showed that for certain singular integral operators T, T fELl (B 1) if f E LlogL(B2) whenever B 1, B2 are balls in]Rn such that B1 ~ B 2. We remark that if T is, say, a Riesz transform, R j , then it is known that if f E L1(]Rn), f ~ 0 then R j fELl (]Rn) only if f = 0 on ]Rn. This is the reason the conclusion ofthis theorem seems awkward in comparison to the conclusion in the case of the disk. Thus, we no longer have the containment of L log L in HI, at least not globally, but we do have this "local" substitute. Stein [St3] showed a converse: If B1 and B2 are balls in]Rn such that B1 ~ B2 and if f ~ 0, f E L1(B2) and all the Riesz transforms Rd are in L1(B2) then f E LlogL(B1). Later, Calderon and Zygmund [CZ2] also generalized Zygmund's earlier work to the case of periodic functions of several variables. They considered functions f on]Rn which are periodic: f(x + m) = f(x) whenever m = (m1, ... , m n ), mi E Z. Then setting Q = {x = (Xl, ... , x n ) E ]Rn : - ~ ::; Xi < ~ for every i} they showed that for certain singular integral operators T, Tf E L1(Q) whenever f E LlogL(Q). Stein [St3] showed a converse for Riesz transforms: If f is periodic, f ~ 0, f E L1(Q) and if all Rd are in L1(Q) then f E LlogL(Q). In all these cases, f E L log L implies that f is in some sense in HI-in an exact sense for periodic functions and in a "local" sense on ]Rn. But in all these cases a converse is possible only under the assumption f ~ o. Consider this converse direction in the case of functions on [0,211"): If f E HI, f ~ 0 on [0,211") then f E L log L. In other words, within HI, f ~ 0 implies f E L log L. It is easy to produce examples to show this containment is strict. Now consider a version of Dau(x; 0) adapted to the disk:

T

iT

T

Dau((); 0) = (

Jr",(IJ)

~Iul

dxdy.

5.4 Application II. The Brossard-Chevalier L log L result

169

If u never takes the value 0 or is constant on r a(B), Dau(B; 0) = 0; in particular if f ~ 0 on [0,27r), the harmonic extension u of f to the disk satisfies this. In fact, the integration here takes place over {(x,y) E ra(B) : u(x,y) = O}. Thus, borrowing language from probability, we could say that Dau(x; 0) measures "the amount oftime" the function u(x, y) is at 0 in r a(B) - a way of thinking which is in accordance with our view of Dau(x; r) as an analogue of local time. By continuity, a measure of the "amount of time" the function is at 0, is, in some sense, a measure of how often it crosses 0, and consequently a measure of how far it is from being positive. Hence Dau(x; 0), in the words of Brossard [Br2J, provides a measure of the "positivity default" of the function u in r a(B). (See Brossard's [Br2] article for an extensive discussion of this point of view. See also Brossard and Chevalier [BC3] for several Fatou theorem type results which further emphasize this point of view.) The following, due to Brossard and Chevalier [BC2] exactly characterizes L log L within HI by means of this "positivity default" .

Theorem 5.4.1 Let u E HI (]R~+1 ). The following are equivalent:

(a)

u

(b)

In~n D,eu(x; 0) log+ D,eu(x; O)dx

E

LlogL.

<

00.

To show that (a) implies (b), Brossard and Chevalier use both probabilistic and analytic techniques. In particular, they rely on a probabilistic analogue of this theorem which they proved in [BCl]. They prove the other implication using purely analytic techniques. We will give a different proof that (a) implies (b) using the techniques developed in this chapter. This proof is from Moore [M03]. Similar ideas can be used to give a proof that (b) implies (a) which is somewhat different than the proof given by Brossard and Chevalier. However this proof is neither shorter nor better nor sufficiently different than theirs. Hence, it is not worth the effort, and we will not show it here. To begin the proof that (a) implies (b) we fix an 0: > (3. Consider a point x E]Rn and a cube Q ~]Rn centered at x. We create an auxiliary domain, O(Q) = Us EQ r3(8), where we choose h so that for every 8 E Q, r,e(8)W3(8) ~ r'"Y(x), where '"Y = a~,e. Such an h can be chosen as h = CC( Q) where C is a constant depending only on 0:, (3 and n. Let Q denote the cube with the same center as Q but with sidelength C( Q) = kC( Q) where k is a constant to be chosen momentarily. We then set O(Q) = UsEQ r~h(s); this is simply a k-fold enlargement of O(Q). We choose k large enough so that k ~ 3 and for P E O(Q), d(p,aO(Q)) is realized by a point (8,0) E ]Rn. Set Po = (x, k2h). We may also assume that k is chosen large enough so that _ Po E r ~ 2 (8) for every 8 E Q. For each cube Q ~ ]Rn we consider O(Q) and O(Q) constructed in this way; since hand k can be chosen to depend only on 0:, (3 and n, each O( Q) and O( Q) can be considered as a translation and dilation of a canonical domain. Consequently, all constants arising in what follows

170

5. Good-A Inequalities for the Density of the Area Integral

will depend only on Q, /3 and n. The proof of next lemma uses techniques similar to those used in the proofs of theorems in this Chapter and in Chapter 4. Lemma 5.4.2 There are a constant C

= C(Q, /3, n) such that for every x

E lR n ,

Here MD{3u(x; 0) is the Hardy-Littlewood maximal function of D{3u(x; 0). Proof: Fix an x E lRn and a cube centered at x and consider O(Q) and O(Q) as constructed above. Let G(P) denote the Green's function for O(Q) with pole at Po. We then apply Green's theorem to G(P) and lui on O(Q). As in other applications of Green's theorem in this chapter, this requires a smoothing procedure. Here this can be accomplished using the same procedure as in the proof of Lemma 5.1.1. We obtain:

jf r

_ G(P)~lu(P)ldP = _ f

J !1(Q)

_ ~G (P)lu(P)lda(P) - lu(Po)1

Ja(!1(Q»

= f

_

Ja(!1(Q»

un

lu(P)ldw(P) -lu(Po)l,

where, as before, w represents the harmonic measure on 8(O(Q)) taken with respect to the base point Po. On lR+.+1, lu(P) I is dominated by the Poisson integral of lui, call it v. Therefore,

f _ lu(P)ldw(P)::; f

Ja!1(Q)

_

Ja(!1(Q»

v(P)dw(P)

=

v(Po)

::; NOi.v(x) ::; COi.Mu(x). Thus,

jf r

_ G(P)~lu(P)ldP::; CMu(x).

J !1(Q)

(5.4.2)

For P E O(Q), let d(P) denote the distance from P to 8(O(Q)). Note that O(Q) is starlike with respect to the point Po. For P E O(Q)\{Po} we let P denote the point of intersection of 8(O(Q)) with the ray from Po through P and for P E 8(O(Q)) we set A(P,r) = 8(O(Q)) n {PI: IF - pII < r}. We recall a result of Dahlberg we used previously, (Lemma 4.2.4): There exists a k > 0 such that for P E O(Q)\{Po}, k-1d(p)n-1G(p) ::; w(A(P, d(P))) ::; kd(p)n-1G(p) where k depends only on n and the Lipschitz constant of O(Q), hence only on n, /3 and Q.

5.4 Application II. The Brossard-Chevalier L log L result

171

For P E O(iJ), z E IRn set X(P, z) = 1 if P E r~(z) and zero otherwise. Note that our choices of h and k insure that there exists a constant ( depending only on a, (3, n such that if P E O(Q) then {z E Q : P E r~(z)} ~ A(P, (d(P)). This, together with a theorem of Hunt and Wheeden [HW, p. 311] on the doubling properties of harmonic measure implies:

w( {z E

Q: P

E

:s w(A(P, (d(P)) :S Cw(A(P, d(P)) ,

r3(z)})

where C depends on (and the Lipschitz constant of O(Q) - hence on a, (3, n. Thus, by Dahlberg's Green's function estimates, Lemma 4.2.4,

fJr r

_G(P)~lu(P)ldP 2:

O(Q)

cfrJr

2: C

II

_d(p)l-nw(A(p, d(P)))~lu(P)ldP

O(Q)

~lu(P)ld(p)l-n O(Q)

1

x(P, z)dw(z)dP

Q

r r ~lu(P)ld(p)l-ndPdw(z) J Jr~(z) 2: C r r ~Iu(s, t)ltl-ndsdtdw(z). J Jr~(z)

(5.4.3)

=C

Q

Q

Note that for the last inequality we have used the fact that d(P) ~ t for P = (s,t) E O(Q). Elementary estimates on the Poisson kernel and Harnack's inequality show that there exists a constant C, depending only on a,{3,n such that w(E) 2: CM whenever

E~

Q. Thus, :; 2:

C#dr

on Q. This, combined with the fact that

~Iu(s, t)1 = 2~u(s, t)+, and (5.4.3) gives

II

O(Q)

G(P)~lu(P)ldP 2: I~I

This and (5.4.2) give:

I~I

k

D3 u (z; O)dz.

k

D3 u (z; O)dz :S CMu(x).

(5.4.4)

As before set D~ u(z; 0) = D,au(z; 0) - D3u(z; 0). Recall that by our construction r,a(z)\r3(z) ~ r -y(x) b = at.6) so that Lemma 2.3.1 and Lemma 5.2.1 combine to show that ID~u(Zl) - D~U(Z2)1 :S CNau(x) for every Zl, Z2 E Q. This and (5.4.4) then yield;

I~I

k

D,au(z; O)dz :S CMu(x)

+ D~u(x; 0) + CN,au(x)

:S C(Mu(x)

+ D,au(x)).

Taking the supremum over all such Q finishes the proof of Lemma 5.4.2.

172

5. Good-A Inequalities for the Density of the Area Integral

We now proceed with the proof of (a)

r

Df3u(x; 0) log+ Df3u(x; O)dx

=

JR."

=}

(b) in Theorem 5.4.1. We compute:

r

Df3u(x; 0) log+ Df3u(x; O)dx

J{x:Dfju(x;O»I}

=

1=(11

Df3u(x; O)dx

-

A

1

~ C1

1=

{x:Dfju(x;O»>'}

)

d)"

I{x: MDf3u(x;O) > C2 )..}ld)".

This last estimate is just equation (6) in Stein [St3]; here C1 , C2 depend only on n. Set b = C 2 /2C where C is the constant appearing on the right hand side of Lemma 5.4.2. Then I{x : MDf3u(x; 0) > C2 )..} I ~ I{x : Df3u(x) > b)..}1

+ I{x : Mu(x) > b)..}I·

Using the fact that I\Df3UI\1 ~ CI\NaUI\1 (which follows from Theorem 5.3.3(c)) and the weak type 1-1 estimate for the Hardy-Littlewood maximal function we obtain:

r

Df3u(x; 0) log+ Df3u(x; O)dx

JR."

~ C11= I{x: Df3u(x) > b)..}ld)" + C1 ~ CI\Df3UI\1

~ CI\NaUI\1

C1=11 +"\ b

C +b

~ CI\Naulh + C

1 b

/\

{x:lu(x)I>H

I{x: Mu(x)

> b)..}ld)"

lu(x)ldxd)"

1

{x:lu(x)I>H

lu(x)1

1=

2IU (X)'

b

1 >.d)"dx

r lu(x)l(l + log+ lu(x)l)dx.

JR"

By Fatou's lemma, the last integral is dominated by

r

sup lu(x, y)l(l y>O JR"

+ log+ lu(x, y)\)dx.

This completes the proof. For various other applications and uses of the D-functional, we refer the reader to [BC2], [BC3], [BC4], [BC5], [Che4], and the survey article [Br2].

Chapter 6 The Classical LIL's in Analysis In this chapter we will describe how the LIL's in Chapter 3 are related to the classical LIL's for lacunary series and to the more recent LIL's for Bloch functions.

6.1

LIL's for lacunary series

The unit disc in the complex plane will be denoted, as in Chapter 3, by D and the unit circle by T. Throughout this section, a real trigonometric series with partial sums m

8 m (8)

=L

(ak cos nk8 + bk sin nk8)

k=l

which has nk+1/nk set

> q > 1 will be called a q-lacunary series. For such a series we

We remark that the definition of Bm used here differs from that in Chapter 3 by the factor in the parenthesis. With this present definition, B! is the variance of 8 m (using the probability measure d8/27r). This will make the following LIL's resemble the Kolomogorov LIL, (3.0.1). It has been shown over the years that lacunary series exhibit many of the properties of partial sums of independent random variables. (In the modern language of probability, lacunary series are examples of "weakly dependent" random variables.) The following, to the best of our knowledge, is the first LIL in analysis.

!

Theorem 6.1.1 (Salem-Zygmund [SZ2]) Suppose that, with the notation above, 8 m is q-lacunary and the nk are positive integers. Suppose also that Bm -+ 00 as m -+ 00 and 8 m satisfies the Kolmogorov-type condition:

173 R. Bañelos et al., Probabilistic Behavior of Harmonic Functions © Birkhäuser Verlag 1999

6. The Classical LlL's in Analysis

174

for some sequence of numbers Km ! o. Then . Sm((}) hmsup <1 m--->oo y'2B;" log log Bm for almost every () E T.

(6.1.1)

As we have clearly demonstrated by now (hopefully!), lower bounds in Kolmogorov-type LIL's are much more difficult to obtain, and this is the case here. Erdos and Gat [EG] were the first to make progress in this direction. They showed that if Sm((}) = 2::=1 exp(ink(}) , and if the nk are integers, then lim sup m--oo

Sm((}) = 1 ymloglogm

(6.1.2)

for almost every () E T. Later, M. Weiss gave a complete analogue of Kolmogorov's LIL in this setting. Theorem 6.1.2 (M. Weiss [We]) Suppose Sm is a lacunary series with Bm as m ----+ 00 satisfying the Kolmogorov-type condition (Ko). Then lim sup m--oo

Sm((}) y'2B;" log log Bm

=1

----+ 00

(6.1.3)

for almost every () E T. The analogous theorem for Abel means of real lacunary series is a consequence ofthis theorem [We]. To state this, let S(p, (}) = 2:~=1 pnk (ak cosnk() + bk sin nk(}) , where the nk are lacunary and set Bp = (~2:~l(a~ + bDp2nk)L Suppose that the ak and bk satisfy the conditions of Theorem 6.1.1. Then · 11m sup

.J

S(p, ())

=1

pj1 2B~ log log B p for almost every () E T. The modern probabilistic approach to the above theorems is via an invariance principle which gives stronger results. Set Bo = So = O. For () E T and t ::::: 0 define

(6.1.4) Theorem 6.1.3 (Philipp-Stout [PSI]) Let Sm be a q-lacunary series with Bm as m ----+ 00. Suppose

----+

00

Mm = O(B;,-O), for some 0 < 8 ::; 1. Then, without changing the distribution of {Ut : t::::: O}, we can redefine this process on a richer probability space together with standard Brownian motion {Wt : t::::: O} such that

(6.1.5) almost surely as t

----+ 00

for each rt

< 8/32.

175

6.1 LIL's for lacunary series

The invariance principle for martingales is the following: Theorem 6.1.4 (Philipp-Stout [PS2j) Let {d n } be a martingale difference sequence satisfying condition (Kd of Chapter 3:

Id 12 < K m

-

a 2 (fm) mloglog(ee + a 2 (fm)) '

almost surely on {a(f) = oo} for some sequence of constants Km a 2(fm) = 2:;;'=1 E(d~IFk-1). For t > 0, define

1 0,

where

m

it =

Lik,

k=1

Then the process it can be redefined on a richer probability space together with standard Brownian motion {Wt : t ~ O} such that ft - W t almost surely as t

-----> 00

= o(tloglogt)1/2,

(6.1.6)

on {a (f) = oo}.

Theorem 6.1.3 and 6.1.4 are the invariance principles discussed in the preface and mentioned several times before. These results, together with the easy LIL for Brownian motion, (equation (**) in the Preface), imply the corresponding results for {8m } and {fm}. Notice, however, that (K1 ) is stronger than (Ko). S. Takahashi [Ta] has obtained results similar to Theorem 6.1.3 with a condition very close to (Ko), and furthermore, with weaker conditions on the lacunarity of the nk. We leave this direction to the interested reader. Besides the Kolmogorov LIL, the above invariance principles have various other remarkable consequences. For example, Theorem 6.1.3 immediately implies the Kac-Salem-Zygmund [Zy2, vol. II, p. 269], [SZI] central limit theorem: 8 :::; r } m { () E T : ~ Bn

----->

1 I
V21f

Jr

2 exp( - -x )dx,

2

-00

(6.1.7)

as n

-----> 00 (here m is the probability measure d()/21f on T), and the Chung-type LIL discussed in Section 4.3:

~~~ COg~~Bn ) 1/2 8~(()) =

:s

(6.1.8)

for almost every () E T, where 8;;"(()) = max1~k~m 18k (())I. In fact, there are even functional versions of these results as Theorems C-E in Philipp and Stout [PSI] show. The above LIL's extend to the case of lacunary power series in the disc. Consider F(z) = 2:%:1 Ckznk with the nk lacunary and for 0 < p < 1 define

176

6. The Classical LlL's in Analysis

Bp = (E%"=llckI2p2nk )1/2. Note that Bp is the variance of F(pe i ()). M. Weiss [We] proves that under the assumptions of Theorem 6.1.2,

IF(pe i ()) I = 1, lim sup pi! J B~ log log Bp

(6.1.9)

for almost every () E T. There are also corresponding statements as in (6.1.7) and (6.1.8). In particular, if we set F;(()) = sUPO
1/2

F* (()) = ~ p 10' y8

(6.1.10)

for almost every () E T. The argument of Step 3 in the proof of Theorem 3.0.6 shows that for lacunary series of the form F(z) = 2:%"=1 akzqk, q> 1, and satisfying the hypothesis of Theorem 6.1.2, Aa (F) ((), p) is comparable with B p for p near 1. In addition, (K3) of Theorem 3.0.4 is satisfied by these functions. Thus, for such functions, Theorems 3.0.4 and 3.0.5 give the existence of two positive constants C l and C2 such that

IF(pe i ()) I C < lim sup
(6.1.11)

for almost every () E T. This, however, is weaker than the M. Weiss result since we do not know the constants. In the same way, Corollary 4.3.4 gives a one-sided version of the Chung-type LIL (6.1.10). In addition to the LIL of Theorem 6.1.1, Salem and Zygmund also considered another type of LIL for lacunary series. They considered a lacunary series 2:;::1 (ak cos nk() + bk sin nk()) where c~ = a~ + b~ satisfies 2:%"=1 C~ < 00. However, here instead of partial sums, they considered tail sums: 00

SN(()) = 2)akcosnk() + bksinnk())' k=N Then, setting

they showed: Theorem 6.1.5 (Salem-Zygmund [SZ2]) Suppose

131 < 00

and that

177

6.1 LIL's for lacunary series

for some sequence of numbers KN

10 as N

----* 00.

Then, (6.1.12)

for almost every e E T. Inspired by this and the fact that except for sets of Lebesgue measure zero,

{x E IR n :N;u(x) < oo} = {x E IR n : A~u(x) < oo}

= {x

E

IR n

:

lim

(s,t)->(x,O)

u(s,t), (s,t) E r~(x),exists and is finite},

(see Chapter 1, Section 8) we are led to the Question 6.1.6 Does there exist a constant C, depending only on n and a, such that if u is harmonic on IR~+ 1 then . 1~rnp hO

u(x, t) - u(x, 0) VA):,u(x;t)210gI0g A~u\X;t)

~C

(6.1.13)

for almost every x E {A;u(x) < oo}? In the absence of any other hypotheses, this seems unlikely to be true. Ideally, it would be true under hypotheses analogous to those of Theorem 6.1.5. In the case of independent random variable such tail 1IL's have been considered by Chow and Teicher [CT] and Barbour [Bar]. Possibly even more useful in the study of (6.1.13) would be the martingale tail 1IL's of Heyde [He] and Kesten [Ke2]. Other authors have shown results relating to these tail 1I1's; Huggins [Hu] approaches Heyde's result in another way and Clark [CI] gives an estimate of the speed of convergence in Heyde's tail 1IL. Our strategy for proving the upper-half 1IL, Theorem 3.0.4, could be simply explained by saying we reduced to the case of dyadic martingales and then applied Stout's upper-half LIL for martingales. It is possible that (6.1.13) could be approached in a similar way: reduce to the case of dyadic martingales and then apply Heyde's or Kesten's martingale tail 1IL. However, there seems to be some technical difficulties in following this plan and we have not done it. One difficulty is that both the tail LIL of Heyde and that of Kesten require stringent hypotheses comparing the local behavior of a martingale to its global behavior, so that in essence, the martingale must behave almost like a sum of independent random variables. It seems likely then, that even if we use the methods of Chapter 3 to reduce to dyadic martingales, we cannot be certain in general that the martingales we produce will satisfy the hypotheses of either Kesten's or Heyde's theorem. Naturally, we could extrapolate backwards to determine further hypotheses on the harmonic function so that its reduction to martingales yields a martingale to which either Heyde's or Kesten's theorems can be applied, but this seems technically complicated and we have not explored it. This approach may not give a good answer to

178

6. The Classical LIL's in Analysis

the question for the reason that all of our estimates for harmonic functions have paralleled estimates for dyadic martingales and both Heyde's and Kesten's results are for general discrete parameter martingales. As we have noted above, estimates for dyadic martingales can sometimes be different than estimates for general discrete parameter martingales. Consequently, an application of Heyde's or Kesten's results may be too restrictive. Perhaps then, a study of (6.1.13) should begin with a study of tail LIL's for dyadic martingales.

6.2

LIL's for Bloch functions

In the particular case of F(z) = .E~1 z2 k , an easy exercise (hint: estimate using an integral) verifies that 2

BP

~ 2.2 k = ~ P '"

1 (1)

log 2 log

1_ P ,

k=1

as

pi 1.

Thus, in this case (6.1.9) and (6.1.10) become, respectively,

!F(pe iO ) I

lim sup PTl

Jlog

l~p log log log l~p

1 v'log2

(6.2.1)

and liminf ( pjl

log log log

1

G

log l~p

) 1/2

F*(O) = p

71'

.j8log 2

(6.2.2)

for almost every 0 E T. The function F(z) = .E~l z2 k is a Bloch function, which, we recall from Chapter 3, means that F is an analytic function on the unit disc D which satisfies

I!FIIB = IF(O)I + sup(l -lzD!F'(z)1 < 00. zED

(6.2.3)

More generally, consider F(z) = .E~1 akz qk with sUPk lakl ~ 1 and q ~ 2. We have

179

6.2 LIL's for Bloch functions 00

+ pqk+l + ... + pqk+l_ l ]

:::; q ~ 1 Eqk [pqk k=l 00

< _q_P" kpk-l = -q-

P q - 1 (1 - p)2 '

- q- 1 ~ k=l

from which it follows that IIFIIB < 00. In [Mal], N.G. Makarov proved that the upper half in (6.2.1) is true for any Bloch function. More precisely, we have Theorem 6.2.1 (N.G. Makarov [Mal]) There is a constant CM such that for any Bloch function F, lim sup pil

IF(pe ili ) I Vlog l~p log log log

l~p

:::; CMIIFIIB

(6.2.4)

for almost every () E T. Proof: If F is a Bloch function, then

A~(F)((}; p) = :::;

1

r",(Ii;p)

IF'(z)1 2dxdy =

r -( 1-r

o:llFlI~ h

r )dr:::;

l

0

p lli+.9.(l-r) 2

IF'(rei1/)12rdrdry

Ii-%(l-r)

o:llFll~log

(_1_). 1-p

It follows from this and Theorem 3.0.4 that lim sup pj1

IF(pe ili ) I V log

l~P log log log l~p

~

C",IIFIIB

(6.2.5)

for almost every () E {() E T: A",(F)((}) = oo}. For almost all those (}'s for which A",(F)(O) < 00, the function F(pe ili ) has finite limit as p i 1 and so (6.2.4) trivially holds. This proves Makarov's Theorem. A computation similar to that above shows the estimate g~(Fp)((}) ~ 211F11~ log l~ ,which together with the 1IL for g*, Theorem 3.4.1, implies Theorem 6.2.1 with CM = 2. Other proofs which give this same constant are given in T. Lyons [Ly] and Ch. Pommerenke [Po]. The question of whether the constant 2 is best possible is an open problem, as far as we know. Suppose that F is a Bloch function which also satisfies the assumptions of Jones [Jo], (3.2.28). Then for all () E T,

A;(F)((};p)

~ C",log

(_1_). 1-p

(6.2.6)

6. The Classical LIL's in Analysis

180

Since Bloch functions clearly satisfy the condition (K3 ), the lower bound of our LIL, Theorem 3.0.5, gives Jones' result:

IF(pe iO ) I

lim sup

Vlog

pj1

l~P log log log l~P

>G -

(6.2.7)

E:'

for almost every () E T. In the same way it follows from Corollary 4.3.4 that under (6.2.6), .. hmmf

(

log log log log

pj1

1 ) 1/2 1-p

1

Na (F) ((); p) ;::: Gc;,a

(6.2.8)

1-p

for almost every () E T. Here,

Na(F)((); p) = sup{IF(z)1 : z E r a((), p)} is the truncated nontangential maximal function. For Bloch functions, Na(F)((); p) is pointwise comparable to F;(()). This, and the result for lacunary series, (6.2.2), raises the following Question 6.2.2 Is there an upper bound in (6.2.8) for Bloch functions?

6.3

LIL's for subclasses of the Bloch space

In this section we will give some applications of our results in Chapter 3 to the study of various classes of Bloch functions for which a lower bound does not exist in Makarov's LIL. A Bloch function F is said to be in the space Bo if (1 - IzI}IF'(z)1 ----t 0 as Izl ----t 1 and it is said to be in B1 if for every sequence {Zn} C D with IF(zn)1 ----t 00 we have (1 - IZnl)IF'(zn)1 ----t O. Plainly, Bo <;;; B 1 . Examples of functions in B1 are given by lacunary power series with coefficients tending to zero; see J.L. Fernandez [Fer] for more on this. We start with the following simple result. To simplify notation we set for the rest of this section

'\(p)

=

log

C~

p)'

0 < p < 1.

Theorem 6.3.1 For F E Bo

lim sup pj1

for almost every () E T.

IF(peiO)1

J '\(p) log log '\(p)

= 0,

(6.3.1)

181

6.3 LIL's for subclasses of the Bloch space

Proof: For any e > 0 there is a Po, depending on e, such that (1 - Izl)lF'(z)1 < e for Izl > Po. Let 1 > P > Po. Then, computing as in the proof of Theorem 6.2.1, we find that

A~(F)(O, p) ::; allFll~ log (1 ~ po) + ae 2 log (1 ~ p) ::; 2ae2log (_1_) I-p

= 2ae 2 )..(p),

if P > PI > Po, with PI depending on Po, e and the Bloch norm of F. Since e > 0 was arbitrary, Theorem 6.3.1 follows from the upper bound of the 1IL, Theorem 3.0.4. By Theorem 4.4.2, for any 0 < p < 00, (6.3.2) A straightforward computation similar to that in the proof of Theorem 6.2.1 shows that for 0 < p < 1 (6.3.3) In fact, if FE B o, we can argue as in the proof of Theorem 6.3.1 to improve (6.3.3) to: A~(Fp)(O) ::; 2ae2)..(p) , whenever e > 0 and P is chosen sufficiently close to 1. This, plus (6.3.2) shows that for all 0 < p < 00, limsup pll

~(

V)..(p)

(27r IF(pei0)IPdO) lip = 0,

(6.3.4)

io

whenever F E Bo. In [Gil, D. Girela proves that (6.3.1) and (6.3.4) also hold for FEBI . Theorem 6.3.2 (Girela [Gil) Let FE B I 1 limsup /\7:\ pTl v)..(p)

(1

27r

a

.

Then for any 0 < p < . ) IF(pe'°)IPdO

lip

= 0

00,

(6.3.5)

and lim sup pll

J

IF(peiO)1 = 0 )..(p) log log )..(p)

(6.3.6)

for almost every 0 E T. The following lemma proves (6.3.5) with some additional information. This lemma, as we will momentarily see, also implies (6.3.6). Notice that in both (6.3.5) and (6.3.6) it suffices to assume F(O) = 0 which we do for the rest of this section.

6. The Classical LIL's in Analysis

182

Lemma 6.3.3 Let F E Bl with F(O) = 0 and c numbers, 0 < Pm < 1, depending on c and with Pm for each m = 1,2, ... ,

>

O. There is a sequence of 1 as m ----4 00 such that

----4

(6.3.7)

I!FIIE'

In fact, we may

I!FIIE ::;

1. By Theorem

for all P > Pm, where eo is a constant depending only on take Pm to be of the form

where Me is a constant which depends only on c. Proof: Fix 4.4.2,

a > 0,

and without loss of generality assume

121' (F;(O)) 2m dO ::; em

(v'2mfm 121' A~m(Fp)(O)dO.

Thus, it is enough to show that for P > Pm, (6.3.8) As in Chapter 5, we let Da(Fp)(O) denote the D-functional of Fp. By Theorem 5.3.3 there is a constant e l , depending only on a, such that whenever 0 < P < 1, (6.3.9) Since F E B l

,

we may choose Me so large so that if IF(z)1 > Me, then (5.0.4), if 0 < P < 1,

(1-lzI)IF'(z)1 < vii. Then, by the change of variables formula,

A~(Fp)(O) =

r

!F'(pZ)12p 2dxdy

J{zaa(&):IF(pz)I<M,,} .

+ =

1

r

IF'(pz)12p 2dxdy

1

J{za a (&):IF(pz)I>M,,}

M"

-M"

Da(Fp)(O; a)da +

::; 2MeDa(Fp)(O)

rare)

cp2

(1 - plzl)

2

dxdy

(6.3.10)

+ CtcA(p).

Now suppose P > PI = 1 - exp(( -Me/c)2). Then integrating (6.3.10) and using (6.3.9) and (6.3.3) we obtain

121' A~(Fp)(O)dO ::; 2Me

121' Da(Fp)(O)dO + acA(p)

6.3 LIL's for subclasses of the Bloch space

::; 2cV),(Pl)Cl

183

127r A2a(Fp)(e)de + 2nac),(p)

::; 2cV),(p)Cl2n~v),(p)

+ 2nac),(p)

= (4n~Cl + 2na)c),(p) which proves (6.3.8) for m = 1 with the constant C continue by induction. Suppose

= 4nV2QCl + 2na.

We now

for p > Pm-l. Then for P > Pm, (6.3.10), (6.3.3), the induction hypothesis, and (6.3.9) give

127r A~m(Fp)(e)de ::; 2Mc 127r A~(m-l)(Fp)(e)Da(Fp)(e)de + ac),(p) 127r A~(m-l)(Fp)(e)de ::; 2Mcam-l(),(p))m-l 127r Da(Fp)(e)de + ac),(p)cm-lcm-l(),(p))m-l ::; 2cmV),(Pm)am-l(),(p))m-lCl

127r A2a(Fp)(e)de

+ aCm-lcm(),(p))m ::; 2cm a m - l (),(p))m-! C l 2nv2a),(p)

=

(4nV2a m-!Cl

+ aCm-lcm(),(p))m

+ aCm-l)cm(),(p))m

::; cmcm(),(p))m. Here the last inequality follows by noting that a m -! ::; cm-lva and so 4nV2a m-!Cl + aC m- l ::; 4nV2QCm - l C l + 2naC m - l = cm. With a fixed, C is independent of m, c and F (as long as IIFIIE ::; 1). This completes the proof of (6.3.8) and thus completes the proof of the lemma. Before proceeding with the proof of (6.3.6), we make some observations about (6.3.5). If we define for 0 < p < 00,

then for 0 < p ::; 2, Jensen's inequality gives (6.3.11)

184

6. The Classical LIL's in Analysis

or (6.3.12) Thus, to prove (6.3.5) it is enough to do it for p = 2 and the requisite estimate for this is provided by Lemma 6.3.3. Therefore, estimate (6.3.5) does not require information on the behavior of the constant Cp in the inequality IIF* lip::; CpIIA"llp. However, the fact that Cp ::; Cyfp for p 2: 2 is crucial for the proof of (6.3.6) which we now come to.

Proof of {6.3.6}: Let c > O. Let rm = 1 - e-e m and Am = ve m logm. We claim that 00

(6.3.13) m=l

for some constant C. For this, let [logm] denote the integer part oflogm. Observe that if m is large enough, rm > P[logmj, where {Pj} is the sequence of Lemma 6.3.3. By Chebychev's inequality and Lemma 6.3.3,

I{O E T: F* (0) > CyeA }I < m

rm

1 - C2[logm]c[logm]A m2[logm]

10r

27r

F* (0)2[logmjdO rm

< cgOg m] c[log mj ([log m]) [log m] (A(r m)) [log m] =

C2[logm]dlogm]A~logm]

(Co) [logm] C2

([log m]) [log m] < (logm)[logm] -

(Co) [logm] C2

If we take C = e.JCo, we have (6.3.13). The Borel-Cantelli Lemma now gives that F:m (0) ::; CyeAm eventually for almost every 0 E T. That is, for almost every 0 E T, there is an mo = mo(O) such that F:m (0) ::; CyeAm for all m 2: mo. Let P > rmo and choose m 2: mo such that rm::; P < rm+1· Since F;(O) is increasing in P and Am = VA(rm)loglogA(rm ),

F;(O) < F;",+l (0) ::; CyeAm+l ::; CyeAm ::; C yeV A(p) log log A(p) , which proves (6.3.6) since c was arbitrary.

6.4 On a question of Makarov and Przytycki

6.4

185

On a question of Makarov and Przytycki

In [Gi], Girela also proved that if the L 2 -norms of F(pe iIJ ) grow slowly enough as p i 1 then, with no other assumption on F, (6.3.6) still holds. The following generalizes his result.

Theorem 6.4.1 Let F be a Bloch function. Suppose that

, h(27r IF(peiIJWdB ~ ",(p) log (_1_) I-p

where '" satisfies 00

" ~

",(1 _ e-e"')

(logm)/3

<00

for some (3 2: O. Then (6.3.6) holds. Girela's result, [Gi, Theorem 3], is the case

",(p) = (

1

log log l~p

)0<

for some 0: > 1. The proof of Theorem 6.4.1 is very simple. We may assume (32: 2. With rm and Am as in the proof (6.3.6), Chebychev's inequality and (6.3.12) (with p = 2(3) gives that * () r::. }I 0/3 ",(rm ) I{BET.. FrTn B > yeAm ~ e/3 (logm)/3·

Thus the series in (6.3.13) is again finite. We can then continue as before. Notice that if F is a lacunary series then since

B;

=

2~ 1027r IF(peiIJWdB,

the M. Weiss LIL shows that if

· sup ~ 11m A( ) = 0 pi!

then, lim sup pll

P

(6.4.1)

IF(pe;8)1 = VA(p)loglogA(p)

0

for almost every BET

and if lim sup pp

IF(pe;8)1 VA(p)loglogA(p)

then,

rl~\\n . f~ A(p)

=

0.

=0

on a set of positive measure, (6.4.2)

186

6. The Classical LIL's in Analysis

We have shown above that for Bloch functions F in Bo or B 1 , both limsups in (6.4.1) are zero (Theorem 6.3.2) and that with some additional information on B~/ >..(p) we have (6.4.1) in general (Theorem 6.4.1). This represents our knowledge of (6.4.1) for general Bloch functions. Of course, under the stronger hypothesis that IIA;(F)(8j p)lloo = o(>..(p)) as p iI, we do have the conclusion in (6.4.1), as shown above. Concerning (6.4.2) we have the following result proved in Banuelos and Moore [BM5] which answered in the negative a question raised by N. Makarov [Ma2, p. 42] and F. Przytycki [Prz, p. 154]. Theorem 6.4.2 There is a Bloch function F with lim sup Ptl

I

F(pei(J) I

V)..(p)loglog)..(p)

=0

(6.4.3)

for almost every 8 E T and with B2 liminf \(p) > Ptl

A

P

o.

(6.4.4)

Proof: We shall construct a function F of the form

L 00

F(z) where

bk ( Z ) =

a4kZ

4k

=

m=l

L bk(z) 00

amz m =

k=l

+ a4k+lz4k+l + ... + a4k+1_1Z 4 k +1 _1

and which satisfy Ilbkll oo ::::: 1 for all k,

(6.4.5)

>0

(6.4.6)

I bk(ei(J) I lim sup k=l = 0 m---+oo vmloglogm

(6.4.7)

~ (fa;)

liminf m---+oo og m

j=4

and

f

for almost every 8 E T. It is known that (6.4.5) implies that F is a Bloch function (see [Ma2], [Prz]). Also, as in M. Weiss [We], the results (6.4.6) and (6.4.7) for partial sums imply the corresponding results for the Abel means (see also Chapter 3, Section 3) and so we only need to construct F with (6.4.5)-(6.4.7). To do this, we claim it is enough to construct bj's such that

6.4 On a question of Makarov and Przytycki

(i)

Ilbj 1100

(ii)

If 4£ ~ j < 4£+1, then

187

~ 1 for all j.

\t

i=4l

bi \

~ V4£+1.

To show this claim we first note (i) and (6.4.5) are the same. Now, fix m and choose r so that 4r ~ m < 4r+1, and then choose s so that 48 ~ r - 1 < 4S+ 1. Then making repeated use of (iii) we arrive at

So, (iii) implies (6.4.6). For (6.4.7), fix m and again pick r so that 4r ~ m < 4r+1. Then

and so (6.4.7) follows from (ii). So we have reduced matters to the construction of bk which satisfy (i)-(iii). Once again, the inspiration for this comes from the construction of a dyadic martingale with similar properties. We inductively define the bk's (which play the role of the martingale difference sequence here) in such a way that the partial sums of bk (which play the role of the martingale sequence) are "stopped" at the places where they begin to grow too large and thus we will get an estimate like (ii). However, we must continue to add enough so that (iii) will be valid. Fix an integer f. For j = 4£ we set bj = z4j, Z E T, and for j = 4£ + 1, 4£ + 2, ... ,4£H - 1 and z E T we recursively define

We now check that these bj's work. First note that (i)-(iii) are true for bj for j = 4£. Now fix a k such that 4£ < k ~ 4£H - 1 and assume (i)-(iii) are true for j = 4£, 4£ + 1, ... , k - 1. We also assume that for each such j, bj(m) = 0 if

188

6. The Classical LIL's in Analysis

m ¢:. {4 j , 4j + 1, ... ,4j +1 - I} where bj (m) is the coefficient of zm in the series for bj(z). Note that if we set

then gk(m) = 0 if m ¢:. {_4k, -4 k + 1, ... ,0, ... ,4k -1, 4k}. Thus, if m < 2· 4k 4k = 4k, then bk(m) is also zero. In the same way, ifm > 4 k +2·4k = 3·4 k , bk(m) is zero. But 3·4 k < 4k+ 1 -1 and we conclude that bk(m) = 0 ifm ¢:. {4 k , ... ,4k+1-1}. Thus bk (z) has the desired form. By the induction hypothesis,

and so we trivially have

Ilbklloo :-: ; 1. In the same way,

where we have used the fact that if x is a real number with 0 < x < v'4H1 then x + 1 - 4-(£+1)x 2 :-::; v'4H1. (An easy calculus exercise.) This shows(ii). Finally,

2

i=4t

~ ~(k =

~ (k -

+1-

2

2 . 4-(H1)

L2(T)

i=4 t

4£)(1 - 2· 4-(H1») 4£ - 2· 4-(H1)k

+1

+~)

peT)

+

~ ~ (k - 4£ - ~) + 1 = ~(k + 1 -

1 4£).

Thus (iii) also holds for j = k. This completes the proof of Theorem 6.4.2.

6.4 On a question of Makarov and Przytycki

Remark 6.4.3 In [Prz]' Przytycki conjectures a variant of (6.4.2): If

lim sup pll

IF(pei ())I

.JA(p) log log A(p) =

0

on a set of positive measure, then B2 limsup A(P) = O. pj1 P

This is false even for lacunary series. An example is given by

F(z) =

L L z2K 00

K",

m=lj=O

where Km = 22'" (see also Makarov [Ma2]).

",+j ,

189

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Subject Index Cauchy-Riemann equations, 2, 40 Cauchy-Riemann equations, generalized, see Generalized Cauchy-Riemann equations, Stein-Weiss system of conjugate functions Central limit theorem for lacunary series, 175 Change of variables formulas, 136, 137 Chung-type LIL - and Kesten's LIL, 165 - for continuous martingales, 119 - for harmonic functions, 119 Coarea formula, 136 Conditioned square function, 45, 46 Cone, vii, 16, 43 - doubly truncated, 60, 65 - truncated, 43, 60

A

Area function, vii, 37 - doubly truncated, 65, 67 - truncated, 44 Atoms, 49 Auxiliary domains, 105, 106, 110, 140, 144

B Bagemihl and Seidel construction, 116 Best constants - in the LIL for - Bloch functions, 179 - g*, 87 - harmonic functions, 66, 67 - in LP inequalities for - discrete martingales, 128 - g*(f) and f, 133 - L* and S(X), 167 - L* and X, 167 - S(X) and X, 128 - A and N, 129, 132 - in ratio inequalities for - A and N, 129, 132 - A and D, 166, 167 - Nand D, 166, 167 Bidisc,49 Bloch function, 81, 87,178-180 - subclasses Bo and Bl, 180 BMO,104 - boundedness of A on, 110, 119 - estimates for A on Lipschitz domain, 110 - estimates for D, 154, 157 - lemmas for, 104 Burkholder, Gundy and Silverstein theorem, 41, 42 Burkholder-Gundy inequalities, viii, 42, 43, 93, 95, 97, 103, 128

D

Density of the area integral, 136, 137 - for caloric functions, 167 - for harmonic functions, x, 136-138 Dirichlet problem for the ball, 5 Dirichlet problem for the half-space, 10 Doob h-conditional Brownian motion, 88 Double dyadic martingale, 49 Dyadic cube, 45 Dyadic martingale, 45 E

Elliptic operator, 49, 113 Erdos-Gal LIL, 174 Exponential martingale, 47, 64

F Fatou's Theorem, 21 Fourier transform, definition of, 28

c

G

Calderon reproducing formula, 49, 55 Caloric functions, vii, 58, 74, 102, 112, 133, 167

g*-function, 86, 133, 134, 179 g-function, 31, 36,114-117 200

Subject Index Garsia, Rodemich and Rumsey lemma, 153, 154 Generalized Cauchy-Riemann equations, 25, 40, 73 Girela's LIL, 181 Good->' inequalities - for A and D, 138, 139 - for A and N, 97, 98, 104 - for D and N, 138, 139 - for rand 8(f), 96, 97 - for L* and 8(X), 136 - for L* and X*, 136 - for PA and PN, 102, 112, 133 - for X· and 8(X), 93, 95, 96 - integration of, 93 - of Fefferman, Gundy, Silverstein and Stein, 43, 97 - of Murai and Uchiyama, 43, 97 Gradient estimates - using A or N, 60 - using D, 140 Green's function, estimates for, 106 Green's theorem, 2 H

Hardy space - for Hl(IR~+l) and LlogL, 167 - for the disk, 22 - for the half space, 25 Hardy-Littlewood maximal function, 15 Harmonic functions, 1 - converse of the mean value property, 9 - maximum principle for, 4 - mean value property for, 2 Harmonic measure, 105, 106 - and estimates involving surface measure, 106 - and estimates involving Green's function, 106 Heat equation, 58 Hilbert space valued functions, 29 Hilbert transform, 40 I

Invariance principle, 174, 175 - of Philipp and Stout, viii, 175 - of Takahashi, 175

201 J

Jensen's formula, 23, 24 Jones' lower LIL for Bloch functions, 81, 179 K

Kesten's LIL, 177 Kolmogorov's condition, 67, 81 Kolmogorov's LIL, 63, 67, 81 L

LlogL, 167, 168 LP inequalities, 127, 128 - for A and N, 128, 129 - for f and g* (f), 133 - for caloric A and N, 133 - for X and 8(X), 128 - for X· and 8(X), 95 Lacunary series, 67, 173 - and sharpness of Kolmogorov condition, 82 - Erdos-G6J LIL for, 174 - Salem-Zygmund LIL for, 173, 174, 176 - Salem-Zygmund tail LIL for, 176, 177 - Weiss LIL for, 82, 174 Laws of the iterated logarithm (LIL's), 63 - Bloch functions and L2 growth, 185 - Chung-type for harmonic functions, 119 - for B o, 180 - for g., 86, 87 - for N and A, 65 - for caloric functions, 74, 81 - for continuous martingales, 65 - for Stein-Weiss systems, 73 - Kolmogorov-type for harmonic functions, 65, 66 - of Chung, 119 - of Erdos-G6J for lacunary series, 174 - of Girela for B l , 181 - of Jones for Bloch functions, 81, 180 - of Kesten, 165 - of Kesten type for A and D, 165 - of Kolomogorov, 63, 173 - of Makarov for Bloch functions, 179

Subject Index

202

- of Salem-Zygmund for lacunary series, 173 - of Salem-Zygmund for tail sums, 176 - of Stout for discrete martingales, 63, 64 - of Weiss for lacunary series, 82, 92, 174, 176, 185 Liouville's Theorem, 12 Lipschitz constant, 103, 105 Lipschitz domain, ix, 98, 103, 105 120, 122, 123 - starlike, 105 Lipschitz function, 103 Littlewood-Paley g-function, see g-function Littlewood-Paley g.-function, 86, 133 179 ' Littlewood-Paley theorem, 36, 117 Local time, ix, 135 - maximal, ix, 135 M

Makarov's LIL, 179 Martin kernel, 92 Martingales - conditioned square function of,45 - difference sequence of, 45 - dyadic, 45 - maximal function of, 45 - square function, 46 N

Nontangentiallimit, 18, 20 Nontangential maximal function, vii-ix, 16, 103 - comparisons of distribution functions for two apertures, 99 - doubly truncated, 65, 74 - truncated, 43, 99, 119 Nontangentially bounded, 43 p

Parabolic cone, 74 Parabolic Lusin area function 74 Parabolic nontangential maxi~al function, 74 Poisson kernel for the ball, 5

Poisson kernel for the half space, 10 Positivity default, 169 Przytycki, conjectures of, 186, 189

Q q-lacunary series, 173

R Radial maximal function, 15 Ratio inequalities - for A and D, 166 - for A and N, 129 - for Nand D, 166 - for caloric (parabolic) A and N, 133 Reflection principle, 12 Riesz theorem, 22, 40, 41 Riesz transform, 41, 132, 134, 167, 168

S Salem-Zygmund LIL, 173, 174, 176 - for tail sums, 176 Sawtooth regions, 98, 102-104, 110 Schrodinger operator, 46, 92 Sharp constants, see Best constants Singular integrals, 28, 40, 41 - vector valued singular integrals, 30 Stein-Weiss systems of conjugate functions, 25, 73 Stout's LIL, 63, 64 Subharmonic functions, 4, 23 Subharmonicity - of IIIP, if I analytic, 23, 24 - of IFlq for Stein-Weiss systems, 25 Super harmonic functions, 5

T Tail LIL's, 176, 177

v VMO (space of vanishing mean oscillation), 119

w Weiss' LIL, 82, 92, 174, 176, 185

Notation Index A~)u(x) ........................ 44' ~~

50 gQ ................. .................54

gg) ............................. ~3

Ahu(x) ........................ , 65 A hu(x· t) ......................... '103 0: , •••••••••• A"u(P) ................ . ..... vii, 37 A u(x) .................. 120 A "u(x t) ......................... 37 (A 1 )"u(x) ................. : ....... 37 (A 2 )"u(x) ................ . Q

,

'60

•••••••

Bo ...............................

~~~

BI ............................ Bp ......... ................... . ...... , Br(x) ..................... 162 B·(x) ....................... 82'

176 153 163 173

'82

B:' .. ......................... ,

22 HP ................... . .............25

HP(R~+I) ....................... .

k(x a b) ................... ...... .163 66

K,,(u)(x) ................. ::::::::. 66 K,,(u)(x; t) .............. .

163 iJhu(x; a) ....................... : 149 Dhu(x) ......................... 149 Dhu(x;r) .................... ::::. 67 d,,(u)((J; p) ........................... 66 d,,(u)(x;t) ........................ 138 D u(p· r) ................. 136 a: , •.....•. X, D"u(x) .) ...................... x, 136 D"u(x; r ............... 149 D~u(x) ......................... : 149 Dku(x;r) ....................... 149 Dr.} u(x; r) ........................ 149 D~u(x) .......................... 161 Dlu(x) ............................ 45 dk ........................ . .... 151 i5h u(x' a) .................. 87 d: ... : ............................ 138 Du(x) ............................ 138 Du(x;r) ......................... .

............ 45 f(Q)) ...................................... ~x L(a ............ . ........ IX L* ........................ 165 L; ............................... 54 A (j) (x) ............................ 180 m ),(p) ................ . . . . . . . . . . . . .. ... 50 Am(x) ................... .

Mt

.................. .

............. 68

Nhu(x) ..................... ...... 65 N hu(x· t) ......................... '103 a , . N"u(P) ................... . . .". . .vii 16 N u(x) ...................... '120 N:u(x, t) ......................... 151 2h IU(X) ...................... . N ",W 'PN u(x) ....................... ... 74 -n A "u(x) .......................... 74 r a 'P,,(x) ..................... ....... .. 74 -nh(X' c) .......................... 9 ro. , P[f](x) ................. . .......... .

46 f* .................... ............. 176 P* ((J) ................................... 45 p .............. ••••••••••• Pm . .. 45

1m

60

rh(x) .......................... '119 r :(x, t) ........................ 65 r~(x;t) ........................ 66' 67 r h ((J. p) ........................ vI'i' 16 r a (X), ................... ...... , 16 r:((J) ............................ .

........ . .........

Qf((J) ................... . ........ 118

35

G(P) ........................... 105 36 g(u)(x) .................................... .' 87 g.(up)((J) ............. . ........ 36 gl (u) (x) ............ ::::::......... 36 g2(U)(X) ........... .

R~+ 1 ................ . . . . . . . . . . . . . . vii

(S(f))2 ......................... . .. 46 203

204

Notation Index

Sri?() (x) ............................ 55

Sm(x) ............................. 50 S(f) .............................. 46 S(fm) ............................. 46 S(f~))(x) ......................... 54 SeX) ............................. viii S2(f) ............................. 46 63 St(X) ............................ viii u(fm) (x) .......................... 46

8m

Ut

........................

viii, ix, 174

v(x,t) ............................. 58

vex, t) Vm(x) Wt

.•..••.•..•••••.••••..••.•..

56 70

ix, 174

.......•..•.......•.........•...

X* ............................... viii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Xt

Progress in Mathematics Edited by Hyman Bass, Columbia University New York, NY, USA Joseph Oesterle, Institut Henri Poincare, Universite Paris VI, France Alan Weinstein, University of California, Berkeley, CA. USA Progress in Mathematics is a series of books intended for professional mathematicians and scientists, encompassing all areas of pure mathematics. This distinguished series, which began in 1979, includes research level monographs, polished notes arising from seminars or lecture series, graduate level textbooks. and proceedings of focused and refereed conferences. It is designed as a vehicle for reporting ongoing research as well as expositions of particular subject areas.

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Kashiwara, M. et al. (Ed.), Topological Field Theory, Primitive Forms and Related Topics_ 1998.494 pages. Hardcover. ISBN 3-7643-3975-6

PM 161

Sagan, B.E.I Stanley, R.P.(Ed.), Mathematical Essays in Honor of Gian-Carlo Rota_ 1998.480 pages. Hardcover. ISBN 3-7643-3872-5

PM 162

Arnold, V.l.1 Greuel, G.M.I Steen brink, lH.M. (Ed.), Singularities_ The Brieskorn Anniversary Volume 1998.495 pages. Hardcover. ISBN 3-7643-5913-7

PM 163

Berndt, R. 1Schmidt, R., Elements of the Representation Theory of the Jacobi Group. 1998.235 pages. Hardcover. ISBN 3-7643-5922-6

PM 164

Roussarie, R., Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem_ 1998. 224 pages. Hardcover. ISBN 3-7643-5900-5

PM 165

Migliore, lC, Univ. of Notre Dame, India, Introduction to Liaison Theory and Deficiency Modules. 1998. 232 pages. Hardcover. ISBN 3-7643-4027-4

PM 166

Elias, 1 et al. (Ed.), Six Lectures on Commutative Algebra_ 1998.416 pages. Hardcover. ISBN 3-7643-5951-X

PM 167

Facchini, A., Module Theory_ Endomorphism Rings and Direct Sum Decompositions in some Classes of Modules 1998. 290 pages. Hardcover. ISBN 3-7643-5908-0

PM 168/169

Balog, A et al. (Ed.), European Congress of Mathematics. Budapest, July 22-26, 1996_ Set: Vols.1 + II 1998. 736 pages. Hardcover. ISBN 3-7643-5496-8

PM 170

Paterson, A, Groupoids. Inverse Semigroups and their Operator Algebras. 1998. 288 pages. Hardcover. ISBN 3-7643-4051-7

PM 171

Reznikov, A 1 Schappacher, N. (Ed.), Regulators in Analysis. Geometry, and Number Theory. 1999. Approx. 400 pages. Hardcover. ISBN 3-7643-4115-7

PM 172

Brylinski, J-L. et aI., Advances in Geometry. 1998. 404 pages. Hardcover. ISBN 3-7643-4044-4

PM 173

Draxler, P.I Michler, G.I Ringel, CM. (Ed.), Computational Methods for Representations of Groups and Algebras_ Euroconference in Essen (Germany), 1997. 1999. Approx. 372 pages. Hardcover. ISBN 3-7643-6063-1

PM 174

Goerss, P.G.I Jardine, lE, Simplicial Homotopy Theory_ 1999. Approx. 500 pages. Hardcover. ISBN 3-7643-6064-X

PM 176

Bass, H.I Lubotsky, A., Tree Lattices_ 1999. Approx. 232 pages. Hardcover. ISBN 3-7643-4120-3

PM 177

Birkenhage, CI Lange, H., Complex Tori_ 1999. 272 pages. Hardcover. ISBN 3-7643-4103-3

PM 178

Puig, L, On the Local Structure of Morita and Rickard Equivalences between Brauer Blocks. 1999. Approx. 268 pages. Hardcover. ISBN 3-7643-6156-5

PM 9

Springer, lA, Linear Algebraic Groups. 1998.350 pages. Hardcover. ISBN 3-7643-4021-5

PM 5

Helgason, S., The Radon Transform. Second Edition 1999. Approx. 208 pages Hardcover. ISBN 3-7643-4109-2

Probability and its Applications Edited by T.M. Liggett. University of California, Los Angeles, CA. USA C. Newman. New York University. NY, USA L. Pitt, University of Virginia, Charlottesville. VA, USA

F+r:ibability and its AppflCi.ltions publishes research-level monographs and advan· ced graduate texts dealing with all aspects of probability theory and stochastic processes, as well as their connections with and applications to other areas such as mathematical statistics and statistical physics.

Aebi, R., Schriidinger Diffusion Processes. 1996.194 pages. Hardcover. ISBN 3-7643·5386·4

Borodin, A.N.! Salminen, P ., Handbook of Brownian Motion - Facts and Formulae. 1996. 476 pages. Hardcover. ISBN 3-7643-5463·1

Boyarsky, A. ! G6ra, P., laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension 1997. 415 pages. Hardcover. ISBN 3-7643·4003·7

Chung, K.L. I Williams, R., Introduction to Stochastic Integration. Second Edition 1990 (3rd printing 1997).292 pages. Hardcover. ISBN 3-7643-3386·3

Fristedt, B. I Gray, L., A Modern Approach to Probability Theory. 1997.776 pages. Hardcover. ISBN 3·7643-3807·5 Holden, H. et al. (Ed.), Stochastic Partial Differential Equations: AModeling, White Noise Functional AnalYSis Approach. 1996.244 pages. Hardcover. ISBN 3·7643·3928·4

Kwapien, S. ! Woyczynski, W.A., Random Series and Stochastic Integrals: Single and Multiple. 1992. 360 pages. Hardcover. ISBN 3·7643-3572-6

Lawler, GJ., Intersections of Random Walks. 1996.230 pages. Softcover. ISBN 3-7643-3892-X 1991. 230 pages. Hardcover. ISBN 3-7643-3557·2

Madras, N. ! Slade, G., The Self-Avoiding Walk. 1993 (2nd printing 1995). 426 pages. Hardcover. ISBN 3-7643-3589-0 1996.442 pages. Softcover. ISBN 3-7643-3891-1

Progress in Probability (formerly Progr~s In Probability and Statistics I PPS) Edited by 1M. Liggett, Universrty of California, Los Angeles. CA, USA C. Newman, New York University, NY, USA l. Pitt, University of Virginia. Charlott~ville. VA, USA

Ftogress in Prcbabi7ity is designed for the publication of worlcshops, seminars and conference proceedings on all aspectS of probability theory and stochastic processes, as well as their connections with and applications to other areas such as mathematical statistics and statistical physics.

PP 32

Nualart, D.I Sanz Sole, M. et at. (Ed.), Barcelona Seminar on Stochastic Analysis. 51. Feliu de Guixols. 1991 1993. 234 pages. Hardcover. ISBN 3-7643-2833-9

PP 33

~inlar,

PP 34

Freidlin, M. (Ed .), The Dynkin Festschrift. In Celebration of Eugene B. Dynkin's 70th Birthday 1994.446 pages. Hardcover. ISBN 3-7643-3696-X

PP 35

Hoffmann-Jorgensen, J. I Kuelbs, J.I Marcus, M.B_(Ed.), Probability in Banach Spaces - 9. 1994. 440 pages. Hardcover. ISBN 3-7643-3744-3

PP 36

Bolthausen, E. I Dozzi, M. I Russo, F. Ed.), ( Seminar on Stochastic Analysis, Random Fields and Applications. Centro Stefano Franscini, Ascona, 1993. 1995.404 pages. Hardcover. ISBN 3-7643-5241-8

PP 37

Bandt, C. I Graf. S. I Zahle, M. (Ed.). Fractal Geometry and Stochastics. 1995.258 pages. Hardcover. ISBN 3-7643-5263-9

PP 38

Kbrezlioglu, H. I UstUnel, A.S.I 0ksendal, B. (Ed.), Stochastic Analysis and Related Topics V. The Silivri Workshop 1996. 296 pages. Hardcover. ISBN 3-7643-3887-3

PP 39

Adler, R.J. I Muller, P. I Rozovskii, B.L. (Ed.), Stochastic Modelling in Physical Oceanography. 1996. 480 pages. Hardcover. ISBN 3-7643-3798-2

PP 40

Chauvin, B.I Cohen, 5.1 Rouault, A. (Ed.), Trees. Workshop in Versailles, June 14- 16, 1995 1996. 168 pages. Hardcover. ISBN 3-7643-5453-4

PP 41

Bovier, A .I Picco, P_ (Ed.), Mathematical Aspects of Spin Glasses and Neural Networks. 1998.392 pages. Hardcover. ISBN 3-7643-3863-6

PP 42

Decreusefond, L. et at. (Ed.), Stochastic Analysis and Related Topics VI. The Geilo Workshop, 1996 1998. 424 pages. Hardcover. ISBN 3-7643-4018-5

PP 43

Eberlein, E. I Hahn, M. I Talagrand, M. (Ed.), High Dimensional Probability 1998. 344 pages. Hardcover. ISBN 3-7643-5867-X

PP 44

Bramson, M.I Durrett, R.T., Perplexing Problems in Probability. Festschrift in Honor of Harry Kesten 1999. Approx. 432 pages. Hardcover. ISBN 3-7643-4093 -2

PP 45

Dalang, R. I Dozzi, M. I Russo, F. (Ed .), Seminar on Stochastic Analysis, Random Fields and Applications. Centro Stefano Franscini, Ascona, September 1996. 1999. 306 pages. Hardcover. ISBN 3-7643-6106-9

E. I Chung, K.L. I Sharpe, M.J. (Ed.), Seminar on Stochastic Processes, 1992. 1993. 276 pages. Hardcover. ISBN 3-7643-3649-8