Integrali Singolari e Questioni Connesse

(1\1. Riesz);

0) If / is L, then than 1 (Kolmogorov);

Lip

IX

1 is

then /

locally integrable in any power less

d) If / is in both in L and Lip (Privalov).

2) If n

=

2, z

=

x

+i Y= r K (z) =

Developping Q into a

~Fourier

lifllp <

is in LP and

Oirp,

1'-2

IX ,

0

< < 1, IX

then

1 is

in

we have

Q (rp) .

series we have

+=

Q(rp) =~'

k~-=

e

ikrp IXk-9 , 'r~

where the prime indicates that lXo = () (see (6)). vVe may therefore consider the special kernels Kk = r-2 eikrp , and for k = 1 , 2 we find the classical kernels r- 2 cos cp ,

The case n = 2 is already sufficiently typical for the general properties of the Hilbert transform, though the proofs in this case are somewhat easier than in the general case due to the fact that Q becomes a function of a single variable cp. :Moreover we can apply here the th-eory of functions of a complex variable, which is impossible for n > 3 . 3) n = 3. If we introduce polar coordinates 1', cp , (), we have K(x) = r- 3 Q ((), cp).

We may develop Q into a series of spherical harmonics, Q (9 ,cp) 00

roo.)

~

Y k (9 , cp), and we come across special kernels r- 3 Yle (9 , cp) •

Ie-I

§ 2. Before we pass to the proof of Theorem 1 of the preceding section, which is one of our main objects, we consider a number of possible generalizations. Let K (x) satisfy, as before, the conditions

r

Q (x') d .re' = 0 ,

(1 ) I

75

Q E Lip

IX , IX>

0

On singular integrals

[473]

73

and let K. (x) be the trnncated K, that is to say K. (x) 1 xl;;;:::: e and K (x) = 0 otherwise. We write

1. (x) = f ,.. K .

J

=

K (x) if

J

f (y) K, (x - y) d Y

f (y) K (x - y) d y ,

Ix-y l:?'

and we set ;: (x) = Sup

•>0

11. (x) I.

In view of Theorem 1 of § 1, the function ;: is finite almost everywhere for each f E LP, 1 <- p 00. The following res nIts which complement Theorem 2 arc important for applications.

<

THEOREl\'I

3. If f is in LP, 1
<

00,

so is f*, ctnd we hewe

where Ap is a constant independent of f. THEOREM 4.

If f is in LP, 1

<

111-1.llp =

00,

then

0.

In some problems we encounter what may be called convolntions with a variable kernel. Let z = (C j , '2' ... ,Cn) and suppose that (2)

K (x , z)

=

f2 ,x z') 1

z

in ,

where for each fixed x the kernel K satisfies the comlition

J

Q (x , z') d z' = 0 ,

s

and satisfies a Lipschitz condition of positive order with respect to z, and with respect to x, uniformly in the remaining variable. vVe may consider the function (3)

1 (x)

Jf

(y) K (x , x -

76

y) d Y .

74

[474J

A. ZYGMUND

The integral on the right is not a convolution since the kernel ]{ (x , z) depends on x, but it can be shown that the theorems stated above remain valid in this more general case. There are other types of singular integrals which are of interest in applications. Consider, for example, in the two dimensional case the integral '"':' (x 1) = ~ j , Y:n~

(4)

J+00 J+00f

(s , t) d s d t . (x - s) (y - t)

-00 -00

The operation which transforms f(x, y) into l(x, y) may be called a double Hilbert transform, since, formally at least, it consists of successive applications of ordinary Hilbert transforms to each variable x and y. The kernel ]{ (x , y) = 1/:n2 x y has singularities along the x

1

and y axes, and to define we must first eliminate in (4) a neighbourhood of these axes and then pass to the limit making the eliminated neighbourhood shrink indcfinitely. This could be done in various ways, but the following procedure seems to be the most natural: we define l(x, y) as · 21 /' 11m :n.

f

,,11~O

f(s, t)dsdt , (x - s) (y - t)

Ix-'I>. Iy-tl>'l

where e and t] tend to 0 independently of each other. The neighbourhood we remove therefore has the form of a cross symmetric with respect to the x and y axes, but the two beams of the cross have different widths. The above definition can be generalized. Suppose that x = = (t 1 , ~2 , ••• , ~m) and y = (t]l , t]2 , ••• , t]n) are points of Em and En respectively, and let ]{ (x , y) = ]{' (:0) ]{" (y) ,

where ]{' (x) and ](" (y) are of the types discussed at the beginning of the section. vVe consider the double Hilbert transform

(6) l(x,y)

fJf(8,t)]{(;r-S,y-t)dsdt=lim E

In

S'1J-~O

En

77

f

f,

IX- 8 1>.ly-tl>'l

On singular integrals

[475]

which is a transformation on functions t]t, ••• ,t],,) of 1n n independent real transforms we know the following results.

+

75

f (x,

y)

=f

(~1 , ... , ~1n

,

variables. About snch

'l'HEOREM 5. If both f (x ,y) and If (x , y) I log +1 f (x , y) I are integrable, then the limit in (6) exists (md is finite la almost all points of Em+n. THEOREM 6. If If (x , y) Ip is integrable over Em+", and p is greater than 1, then f(x, y) exists almost everywhere and

whe1'e Apis a eon stant which is independent of f. The proof of these results depends strongly on Theorem 3 and its suitable extension to the case p = 1. The curious aspect of the situation, not yet completely clarified, is the requirement that f log+ If I be integrable. It is not known whether the conclusion of Theorem 5 holds for functions which are merely integrable. There is a conjecture that perhaps the limit (6) exists almost everywhere, even for f merely integrable, provided that e and t] tend to 0 in such a way that both ratios e/t] and'r//e remain bounded; the problem remains open and seems rather difficult. vVe add that we can, of course, introduce kernels which are products of p kernels of standard type, and consider corresponding extensions of (6). Theorems 5 and 6 then still hold provided in the former we replace the integrability of Ifllog+lfl by that of If I (log+lfl)p-l. Let ... , X- 1 , Xo , Xi , X 2 ' •• , be a two-way infinite sequence of numbers. We denote by lP the class of all such sequences X for which the number (~I Xm Ipj1lp is finite, and we denote the number

by IIXllp. Consider the transformation

(7)

": _ +00, ;1,,, -

X

rn~-oo

Xm

---.,

n-

(n = ... , -

where the prime indicates that the index mation. vVe write

1 , 0, 1, 2, ... )

1n

X = !;;;,). 'l'he

1n

= n is omitted in sum

transformation is (except for the-

78

76

A.

[476]

ZYGMUND

factor lin) a discrete analogue of the Hilbert transformation

+00

~;' 1 Jf(Y)d Y J(x) = -

(8)

n

-00

x-Y'

I XI12 -:;;;: n I X 112' The result was showed that IIXllp ~ Ap IIXllp for each

and a classical result asserts that

extended by M. Riesz, who p>l. We may consider a generalization of the transform (7) to spaces of higher dimensions. IJet e1 , e2 , ••• ,en be a system of n mutually orthogonal non zero veetors in En. Consider the set of all lattice points h generated by this system. Hence the h are of the form fl1 e1 fl2 e2 fln en, where fl1' ,fln are arbitrary integers. Since the set of all lattice points is denumerable, we may arrange them into a single infinite sequence I hq I = (h o : hi , .•. ) . Given any infinite sequence of numbers I Xr 1= (XO , Xi , ••• ) we may consider the transformation

+

"'2 , ...

+ ... +

~

(9)

X'l

=

+00

2' K(h q

hr)xr ,

-

(q

=

0 , 1 , 2 , ... )

r~O

which obviously generalizes (7), and for which we have the following analogue of l\L Riesz"s theorem: 8. If the kernel K (x) = Q (x')1 Ix In satisfies the conditions (1), then the transformation (9) is from lP into lP, for p strictly THEOiml\I

greater than 1 . .iJ1ore precisely, if X = l;;q 1, then I ~Yllp < Ap II X lip· We shall now consider singular integrals for periodic functions. We recall that parallel to the theory of Hilbert transforms (8) we have a theory of conjugate functions

(10) -n;

The function f here is initially defined in (- n ,n) and then extended to all X by the. condition of periodicity. The properties of conjugate functions are close to those of Hilbert transforms (8),

79

On singular integrals

[477]

77

and this is not surprising since using the formula

(11)

1 (1 -cot -1 x = 1- + +00 ~ - -1-) , 2 2 x n~-oo X 2 11: n 2 :n n

+

we can put (10) in the form l(x) =

~

r

+00

f

11:.

-00

(y) d y •

x-y

We can apply the same procedure to the general kernel K (x) • Suppose for simplicity that the orthogonal vectors e 1 , e2 , ••• ,en are all of length 2 11: and set (12)

K* (.-r)

=

K (x)

+ +00 ~' jK (x + h

q) -

K (h q )\

q~l

assuming, for simplicity of notation, that ho = 0 . Using the fact that Q E Lip IX, it is not difficult to see that the series (12) converges uniformly over any finite portion of Ii] provided we drop the first few terms of the series which have singularities there; the function K* (x) is of period 2 11: in each of the components ~ j of x . Let Qn be the hypercube in En with center at the orig'in and half-dimensions 11:. Consider the function

n,

(13)

f(x) = ff(y) K* (x -

y) d y.

Qn

Since K - K* is continuous in Qn, Theorem 1 implies that the integral (13) exists almost everywhere in Q", and from Theorem 2 we can deduce without difficulty the following resuit: THEOREM

9. If f is in LP on Qn, where p

~ inLP, and llfllp<Apllfllp, where

Ilflip

> 1,

stands for

then

f

is also

(f If(x) Ip dx )l /P. Q"

We conclude this section with a few words about singnlar int.egrals on curves.

80

'is

[478]

A. ZYGMUND

C be a rectifiable curve in the complex plane, and let f (C) be and integrable function on C (that is, f an integrable function of the arc length). Consider the integral I~et

j

(14)

f(C)dC

z-C '

a

where z is also on C. By the pt'incipal value of this integral we mean the limit, as e -. 0, of

j z-C1 ' f(C)d

as (0)

where C. (z) is the part of the curve C which is outside the circle with center z and radius e. One would expect that under these conditions the integral (14) would exist at almost all points of C; whether this is so we do not know, and the best result so far obtained is the following THEOREM 10. If the curve C has bounded curVlttUt'e and. x and f is integrable on C, then the integral (14) exists, in the sense of principal value, at almost all points of C. D~finitions of singular integrals can be extended from Euclidean spaces to curved varieties, but very little is known in the general case, and we do not discuss the topic here.

§ 3. Given a function j (x) =

Fourier transform of

f

(I; 1

, •••

,I;n), we denote by

/'0-

f the

f,

f(x) = (2 n) -

+ff n

(y)

(}-i(xy)

dy ,

+ ... +

where (x y) stands for the scalar product /;1 'f/l /;n 'f/n of the vectors x = (1;1"'" I;n) and y = ('f/t , .•• ,.'f/n)' We take for granted /'0elementary properties of the Fourier transform f, in particular the

f

facts that if fE L2, then exists (in the metric LZ), that = II f 112, and that we have the inversion formula

(1)

f(y)

=

(2

n)-+nj.i(X) ei(xy) r7.1:, 81

IIill2 =

On singular integrals

[479]

79

Moreover we shall need the fact that if we define the convolution h of functions ! and g by the formula

h (x) = (2 n)-

+f! n

(y) g (x -

y) d y ,

and if one of the functions!, g is in L and the other in L2, then

h=!

(2)

g.

/'..

Since a singular integral! is a convolution of two functions ! and g, we may expect (2) to be valid in this case in some sense, and we are led to study the Fourier transform of the kernel K (x) = = Q (x') / I x In. The kernel being not integrable near the origin, we must first define the Fourier transform of K. Let K e ,'} (x) be the function coinciding with K for IJ < I x I < 'YJ and equal to 0 otherwise. "Ve define the Fourier transform of K by the formula K (xl

(3)

=

lim lim K e,') (x) , e---+O

't}-HXJ

/'..

and we first study the behavior of K e ,'}. We snppose temporarily that Q is merely bounded and satisfies the usual condition

r

Q (x') d x' = 0 .

I

I 1=

I I=

We introduce polar coordinates and set x r, y (2 , (x y) = r e cos cp. We have d y = e"-l d e d a, where d a stands for the element of area of ~, and (2 n)n/2

(4)

Jr.,') (x) =

J

da

I

'}

e- 1 Q (y') e-irecoAip (7 e =

:E

=

r

. :E

Q (y') il e

/7~-iecos'l'

. er

e

de.

Let g (e) be the function eq nal to 1 for e < 1 and to 0 for e

>1.

Then, in view of the condition

I

:E

82

Q (y') d y' = (),

the right

80

[480]

A. ZYGMUND

hand side of (4) is

J J 1Jr

de·

e

er

:E

g (e)

e-iecosrp -

Q (y')

Denote the inner integral by I. vVe will show that (.5)

1II

< log 1cos1 ffJ 1+ 0,

where 0 is an absolute constant. Consider first the case 13 r < 1

R~l

1

1=

e-iecosrp -

1

e

er

say. The inequality

1 eit -

d

----;::- d

'Jr

( e-iecosrp

.

e+

-e- d e = + 1 II

.

1

1

1

1

(! / •

"

1

We have

J

J

R e-ie

I

sup

01 =

r; 1", and set

<

2,


On the other hand,

and, consequently, 1

1121 <

Je-

1

Ieosrp I

de = log 1 1 I' cos ffJ

1121

1

< 0 1 + log 1cos ffJ I' ,

according as r; r 1 cos ffJ 1 is < 1 or > 1, so that in any case we have (.5) with 0 = 0 1 1. In the cases r; r < 1 and 13 r > 1 the situation is similar. In the former, using the same estimate as for It above, we obtain 1 II < 1; and in the latter

+

111=/ J

J

I --glle e- ie ~e-(lel<

e-iecosrp

£1'

1

I

£1' ('0':'.(1'

83

I

+, J-erle.I 1Jrleosrpl

1

'Jr

I

I' l

e- ie

I

1

[481]

81

On singular integrals

The contribution, if any, of the interval (1, 17 r I cos cp exceed 1 , and the rest is numerically majorized by

°

1

f

de -=log

erlcos'!'l

Q

1

r I cos cp

8


I-

1

I cos cp I

Il

does not

.

Thus (5) is established. Return to (4). Since the integral I converges as first 1] 00 and then 8 - 0, provided cos cp =F 0, and since the right-hand side of (.5) is integrable over ~, we deduce that

+

/'..

(i) K.,') (x) is bounded, uniformly in (ii) if

1] -

00

8

and

1];

/'..

and then

0, K.,,} (x) tends pointwise to a

8 -

/'..

bounded function which we may denote by K (x) and call the Fourier transforrn of K. Suppose now that fE L2, and set

Since K.,'1 EL, we have, by (2), /'..

1.,'} = f K.,'1 .

(6) ./"-. /'..

/'.. /"...

Since the difference f K - f I(.,,} tends pointwise to 0 and is majorized by a function in L2, we deduce that /""-...............

(7)

Ilf K

/".../"...

-

f

K.,'} 112

-+

o.

J

Let be the function whose Fourier transform is (7) and (6) we have (8)

In other words, the integral

1.,'} (x) =

rf

(y) K (x -

y) d Y

·.slx-yl.:S:'}

converges in L2 to limit

1 as

1]

~

00

84

and then

8

~0.

/'..

""-

f K. From

82

[482]

A. ZYGMUND

In the argument just completed we assumed that Q was bounded over ~. Since the right-hand side of (5) is integrable over ~ in any power 1, an application of HOlder's inequality shows that (8) holds if Q is in some LP, p 1. But we can go one step further. The right side R of (5) is integrable exponentially over ~, that is exp,1. R is integrable for some ,1. O. Using therefore, instead of HOlder's, Young's inequality

>

>

>

u v :::;;: u log (tt

(8a)

+ 1) + e (u , v;;::::: 0), V

we can deduce (8) under the hypothesis that Q log+ I Q grable over ~ : TEO REM

11. If Q log+ I Q

I i.q integmble over 2', and

I

Q

I is

inte-

(x') dx' =0,

:E

we have (8) for each f EL2. The result is not a special case of Theorems 1 and 2 for p = 2: we assume now much less about Q, but we assert only convergence in L2, not pointwise convergence. From the preceding argument we can easily obtain an explicit /'-.

formula for K. Suppose again, for simplicity, that Q is bounded. We have - n ..........

(9)

Let w

(2 n)21

=

K (x)

=

J lj't. e

.~+o 'l~+'"

.2

1

e-teeoscp -~-

Q (y')

declo =

Er

sign (cos cp). The inner integral can be written

I·- . e 00

(10)

lim lim

e-iwi!

e= J- 00

d

'rlco"'!'1

cos 12

Qd

J 00

.

e-~w

.rlc08'!'1

sin e

J

--012'

.rlco"'!'1

e

The real part on the right is

=

Je

cos e d e

.r

~

+ 1- cos 12 .

'''1,"08'!'1

85

Q

+J' e ~

1 if e

de.

.rlc08'!'1

On singular integrals

[483]

83

Of the three integrals, the first is independent of cp and so can be dropped in (9), in view of the fact that the integral of Q over ~ is zero; the second integral tends to as e ~ 0, r remaining fixed; the third integral is log (1/lcos cp I). The imaginary part on the right

~

of of (10) differs from -

17, i w by a quantity tending to 0 with

e. Collecting the results we see that

(11)

/'-

K (x) = (2 17,)

-~nf 2

It

Q (y') log

I cos1 cp I -

1 ( 217, i sign cos cp do .

I

2:

This bounded, over ~. The of degree

formula was proved under the hypothesis that Q was but, of course, remains valid if Q log+ I Q I is integrable formula (11) shows that K (x) is a homogeneous function O.

§ 4. Our next topic of discussion is Theorems 1, 2 and 3 stated previously. Complete proofs of these theorems would take us too much time and we cannot go into them, but there are some aspects of the proofs, and some special cases, which are of independedent interest an which can be diijcussed easily. We call the kernel K (x) even, if K (- x) = K (x), and odd if K (- x) = - K (x). Every kernel K (x) is a sum of its even com-

1 . ) and odd component 21 iIK(x)-K(-x) ) . ponent 2!K(x)+K(-x) It turns out that some of results are comparatively easy to prove

for odd kernels; more precisely, in the case of odd kernels results for n-dimensional Hilbert transforms are deducible from corresponding result for the classical, one-dimensional, Hilbert transform. 'l'his is of interest since a number of important kernels are odd. 'l'his is true in particular of the Riesz kernel K(x) =

x/I x In+1;

it is a vector function, and so transforms every scaJar function

f

into a vector function = f * K. I,et K be an odd kernel, K (- x) = - K (x) , and suppose first that f E L2. Consider the truncated transform

7

7 (x)

•

£

=If I In (x -

y

y) Q (1f') r7lJ =

..

lyl2:£

_If I In

(x _-tkl) Q (1f') r7 Y . JJ

,yl2:£

86

.

,

84

[484]

A. ZYGMUND

(the equality of the last two integrals uses the odd character of the kernel K). It follows that

1. (x) =

-

~J/(X 2

lyl2:·

+ y)Iy- In/(x -

y) Q (y/) dy'

If we set I y I = 'rj, we have d y = 'rjn-l d 'rj do, and if t deuotes a unit vector we can write y = t'rj, and we obtain (1)

l.(x)=-

~

j!J(t) {J/(X-t't])-;;/(x+t't])d't]}dt. 1}2:.

2,'

If we set g.(.1), t) = _ J/(X +t't])--;;/(X-t'Y)) d'Y), 'I'J?!:.e

we have (2)

It is clear that (except for the numerical factor l/n) g. (x ,t) is a trun-

cated one-dimensional Hilbert transform at the point x of the function / restricted to the straight line L t passing through the point x and parallel to the vector t. It is therefore natural to apply here results for one,dimensional Hilbert transform. Suppose that h (x) is a function of the single variable x and of the class I}. A classical result asserts that

II 'h. (x) 112 <

(3)

A

II h (x) 112 .

The inequality is a corollary of Theorem 11, but can be proved directly and immediately by observing that the Fourier transform of the truncated kernel l/x is

jSin 00

(2n) -~J 2 e- ixt t-1 dt=-(2/n) ~2 signx. Itl2:.

t -t-dt,

'Ixl

and so is uniformly bounded in x and 8, and tends pointwise to a limit as 8 ~O. If we apply (3) to the function go (x ,t) we

87

On singular integrals

[485]

85

have

Jig" (x , t) I d L Z

t

< A

I

If (x)

12 d L t ,

Lt

Lt

and, integrating this over all straight lines parallel to the direction

t, we abtain 1

(4)

1

(JI g" (x, t) 12 dxt < A (JI1(X) 12 d X)2 =

A

11111z'

Return to (2).Applying Minkowki's inequality (which asserts that the norm of an integral never exceeds the integral of the norm), we deduce that (5)

III

112< ~

J

I Q (t) III g" (x, t)

112 d t:::;;: ~

A II Q

111 111112,

:4

where II Q 111 designates the integral of I Q lover X . The inequality (3) which we used in the preceding argument is a special case of the more general inequality (6)

of M. Riesz, valid for all p strictly greater than 1. If we assume that 1 (x) is in LP (E n), and use (6) instead of (3), the preceding argument gives the following generalization of (5) :

(7)

Iii (x) lip

<

Ap II Q

111 111(x) lip,

>

for any 1(x) in LP,p 1. The inequality (6) admits of considerable generalization. In connection with Theorem 3 we introduced the function ;: (x)

=

sup ,,>0

11.(x) I

(thus /; is always non-negative, possibly infinite). It is known that the one-dimensional inequality (6) can be strengthened to (p> 1)

(8)

88

86

[486]

A. ZYGMUND

an inequality which is equivalent to saying that for any measurable step function e (x) we have II h e(x ) (x) lip < Ap II h (x) lip,

(p> 1)

where Ap is a constant depending on p, but not the choice of e (x) • Suppose now that the e in (2) is a function of x, say a step function (by a step function in En we mean a function constant in a finite number of non-overlapping n-dimensional rectangles and 0 elsewhere). We have Ih(X) (x) I::::;:

~

J

IQ

(t) ge(x) (x , t)

Id t <

:?:

if we set

J

I Q (t) Ig* (x , t) d t ,

:?:

g* (x , t) = sup I g e (x , t) I , e>O

and so also

~ (x) :::;;: 2, 1 j~

I' I Q

(t)

I g* (x , t) d t ,

:?:

from which, by the previous argument, we deduce

I IJ~ (x) lip <

(9)

Ap II Q Iii Ilf(x) lip

(p> 1).

1.

This inequality implies that < 00 almost everywhere, and in particular that at almost all points x the integral h (x) remains bounded as e ~ O. But at also implies easily that for almost aU x the limit l(x) = lim h (x) necessarily exists. To see this we observe that exists everywhere if f is continuously differentiable and, say vanishes outside a sufficiently large sphere. The class of such functions f is dense in every LP . Now, if we set

1

() (x)

= () (x, f) =

lim suph (x) -

._+0

lim infh (x), e~+O

then (9) implies that (10)

II () (x ,f) lip < 2 Ap II Q 111 IIflip·

Now () (x ,f) = a(x, f- g) for any g such that g (xl = lim g. (x) exists and, selecting g such that Ilf - gl lp is arbitrarily small, we deduce

89

On singular integrals

[487]

that

1

87

110 (x , f) lip =

0 , (} (x , f) = 0 almost everywhere, so that = lim]; exists almost everywhere. Applying this to (7) we see that

(11)

Let us summarize the results obtained so far in this section. Taking for granted results for the one-dimensional Hilbert transform, and assuming that the function Q is odd and integrable over ~ (this, of course, implies that the integral of Q over ~ is 0) we deduced that 1(x) = lim]; (x) eX'ists almost everywhere, and that l zed by a function in LP; in particular

is rnajori-

111-];llp = o.

lim

The method we used, which we may call the method of rotation, may be applied to more general kernels, already considered in § 2, provided the kernels are odd. Oonsider the integral (12)

-ff (- ) I I

f~• (x ) -

y Q (xyn, y') d y,

x

1!l1;;;;':·

where f is in L p , p > 1, and Q is a function of the variable y' E ~ and of the parameter x E En. We assume that Q is odd in y', and that there exists a function Q* (y') integrable on ~ and such that

IQ (x , y') I ~ Q* (y')

(13)

for all x. If g. (x , t) has the same meaning as above, then arguing as before we have

~ (x) f.

IJ

="2

Q (x , t) g. (x , t) d t ,

I

11. (x) I< ~ JIQ* (t) II g. (x , t) I d t , I

from which it follows that

II]; (x) lip <

Ap

II Q* 111 IIf

90

(x)

lip

(p> 1).

88

[488]

A. ZYGMUND

The last inequality can be extended to

h, (x) =

where

=

sup

_>0

I]; (x) I,

which, as before, implies that l(x)

=

lim.f. (x) exists almost everywhere.

§ 5. In this section we cousider the limiting case p = 1, and we show that if Theorem 1 is valid for p = 2, then it is also valid for p = 1. Thus the result is of conditional nature, but is of interest since in the preceding section we showed the validity of Theorem 1 for odd kernels and any p > 1. The method we use is itself of interest and has wider application. We begin with the proof of the following lemma. LEMMA I. Let P be a bounded perfect set in En, and A a sphere containing P. Let b (x) be the distance of x from P. Then for any A. 0 and almost all points x in P we have

>

(1)

I (x) =

f

0" (y) y In+l d y

!x _

<=

.

LI

We shall call the integral in (1) the integral of ltlarcin7ciewicz. It is enough to prove the lemma in the case n = 1 which is entirely typical. We will show that ./I(X) dx

(2)

< =.

p

Clearly

r

. I (x) d x

=

P

ff

f f1 dx

P

b.! (y) x _ y 1J.+1 d Y = d x

LI

oJ. (y) __ Y

1J.+1

dY .

Q

P

where Q = A- P. We have

1x

00

./ I (x) d x P

faA (y) d ~r 1x ~; 1J.+1
Q

P

fbJ. (y) d y.

~

O-J. (y) d Y <

Q

This proves (2), and so also the lemma.

91

~~

~

1Q

I.

On singular integrals

[489J

----

89

LEMMA II. Let f (x) be a functionintegralfle in a cube IC.E, and let y be a positive nttmber sufficiently lctrge. Then there exists a sequence of cubes It , I2 , . " contained in I, without interior points in common with each other, and stwh that

(3)

y

< T;k I JIf I d x ::;: 2" y

(k

=

1 , 2, ... ).

Ik

MOre01)er, if Q =

~

h ,P

=

I -

Q, we have

Ifl
We decompose I into 2" equal cubes, and set aside those cubes over which the average value of If I exceeds y. Each of the remaining cubes we subdivide into 2 n equal parts and set aside those of the cubes over which the average value of f exceeds y. Each of the remaining cubes we again subdivide into 2 n equal parts and proceed as before, and so on. Let Ij' I2 , . .. be the sequence of all the cubes we set aside in this process. The Ik are all contained in I and have no interior points in common with each other. Clearly we have (3). In this process of subdivision each point of P = I - Q = I - ~ Ik is contained in a sequence of cubes converging to it and over which the average value of If I is < y. It follows that If I < y at almost all points of P. This completes the proof of the lemma. We now pass to the proof of the result stated at the beginning of the section, and we assume that

J

Q d y'

=

0 and that Q E Lip A ,

1:

A> O.

Since the existence of f is a local property, we may assume that the function f is 0 outside a cube 1. We may also suppose that f> O. Let y be a number positive and sufficiently large.

92

90

[490]

A. ZYGMUND

Let Ii , I2 , ... , P, Q have the same meaning as in Lemma II. We set \ fin P g =) , Cf7Lf in each I k , where Cf7Lf designates the average value of f in I outside I. If we set f= g h, then

+

k ,

and g = 0

\' 0 in P,

h='

if - Cf7Lf in each I k

and we have

,

Jh(ZX=O, Jl hldX<2jf dX .

(4)

Ik

Ik

rVe now observe that

Ik

g exists almost everywhere, since g is bounded

and is 0 outside I, and we assume the existence of

g if g

is in L2.

It is therefore enough to prove the existence of h . We denote by It the cube concentric with and homothetic to I k , with sides three times those of I k , and we set

Q"'=

~

It,

P"'= I - Q"'.

h at almost all points of p.r., for the proof of Lemma II implies that, if y is large enough then IQ I, and so also IQ* I, is small and, correspondingly, Ip* I is arbitrarily close to II I . We have, formally,

It is enough to prove the existence of

h (x) Jh (y) K

(x -

y)

dY = Jh (y) K (x -

I

y)

dY =

I

= ;

J

h (y) K (x -

y) d Y ,

Ik

since Q =

~ Ik •

Using the fact that p~(y) dy = 0, we can also write Ik

(5)

h (x) =

f Jh (y) [K (x Ik

93

y) -

K (x -

Yk)]

dy "

On singular integrals

[491]

91

where Yk is the center of the cube I k , It will be enough to show that if we replace the integrands by their absolute values, the resulting series will converge almost everywhere in p*, The difference in square brackets is ]( (x _ y) _ ]( (x _ Yk)

=

Q [(x _ y)']

lI

1

_

x - Y I"

=

Ix -

1

Q [(x -

y)'] _ Q [(x - Yk)'j = Ix - Ykl n

Ix - yin

Yk In

+ Q [(x -

]

y)'] -

IX -

Q [(x -

Yk In

Yk)'J ,

Denoting by 0 suitable constants we can write

where Yk is on the segment joining Y and Yk' Since for x in P* and Y in lk we have OJ <: Ix - Y I/ Ix - Yk I<: 0z, it follows that

(6)

IQ [Ix. <: 0

-

y)'] [_ _ Ix - 1 Y_ In _

IY - Yk I <: Ykl n+1 -

Ix -

0

I,r -1 Yk In-J I::;;: Ix -

dk Ykl n +1

,

where dk designates the diameter of Ik ' Let rp be the angle between the vectors x - Y and x - Yk ' Since Q satisfies a Lipschitz condition of' order A, 0 A <: 1 , we have

<

due to the fact that rpl. <: 0 IY - Yk I/ Ix - Yk I, It follows that

IQ I(x -

I

y)'] - Q [(x - y)!c] <: O. d~ , Yk In -- Ix - Yk In+1.

Ix -

Comparing this with (6) and observing that A <: 1, we see that (y

94

Eh, x EP*)

02

[492]

A. ZYGMUND

and (see (5))

(7)

~Jlh(Y)IIK(X-Y)-K(X-Yk)ldY
n

k

~

~

Since, by (4) and (3)

JI hid Y < 2J f d Y < 2,,+1 IY II Ik I , lie

Ik

it follows that the right side of (7) is majorized by

and so also, if /J (x) is the distance of x from P*, by

almost everywhere in P*, by Lemma 1. This completes the proof of the existence of

f

§ 6. Consider a function transform

/'-.

Tf=f (x)

(x) E L~ (- = ,

1 almost everywhere.

+ =)

and its Fourier

1 f+OO

= -=

f(y) e- ixy d y.

V2.n

-00

It is a very well known, and immediately verifiable, fact that if Tg

--

= T f,

then g (x)

----

= f (-

x) •

Let

J (x) = ~ n

+00

Jf

(y) d Y ,

x-y

-00

that is

J = f.,

K, where K = (2/n)11 2 X-I. In this case, /'-.

K (x)

= -

i sign x ,

95

On singular integrals

[493]

93

and

T

(1)

1=

. T K = T f . (- i sign xl ,

Tf

or T f = T (i sign x), from which it immediately follows the inversion formula

1-

r

+oo~

. j(x)

(2)

= -

-f(y) -dy.

-1

:rr. x-y -00

This formula was established for f E L2, but is immediately extensible to f in any LP, p > 1, since it is valid for functions f which are simultaneously in L2 and LP, and such functions form a dense subset in LP. The Fourier analysis of Hilbert transforms in E2 has already a richer content. Write z = r ei

=

eik
(k=+1,+2, ... ).

-2-

r

/'..

We want to determine K k • Let J k (e) denote the lc-th Bessel function. We recall the basic properties

We shall also need the formula

Ie 00

Jk(e) d e

(3)

this~

we have

II 2",

"'. 1 Kk (z) = 2n

o 00

0

~

(lc> 0).

k

o

Using

=

00

e-irecos(tp-fi) eikfi

e

d

e d () =

2",

= eik
94

A.

.f

e'k
o

f

2n

00

=

[494]

ZYGMUND

d

-

(2

(2

1 2n

e

il!'ill'l'-ibl'-i!!.-2 Ie

d

ljl

=

0

The last integral being 11k for k> 0 and (- 1)kj I k I for 7c we we arrive at the formula /'-.

(4)

Kk(z)

=

i)llel

eile

(7c

17c I

= +

1,

+

<0

2 , ... ) .

Consider now the M. Riesz transform R which is defined by the kernel KI (z) = zl Iz 13• We have /'-.

(5)

KI (z)

Hence, if k

=

ei

i),

/'-.

K_I (z) =

e-i'l' ( -

i)

= -

/'-.

1/KI (z).

>0,

(6)

Let Hie be the transformation corresponding to the kernel K le From (6) we deduce that

•

(7) where

1

ric = k

for k

=

1 , 2, ... , and

ric =

(_ 1)1e

-17c- I- for 7c

=-

1,

- 2 , . .. Thus, except for numerical factors, the transformations Hie are 7c-th powers of the Riesz transformation. If we write K (z) = Sf Ie

eile

(where the prime signifies that the term k = 0 is absent in summation) and denote by II the transformation corresponding to the

97

On singular integrals

[495]

95

kernel K, we obtain, formally, (8)

(9)

where the Yk have the same meaning as before. The formula (8) is valid if Q log+ 1 Q 1 is integrable over (O,2n). To see this we observe that the formula is certainly valid if Q (e) is a trigonometric polynomial in 9. Let an be the (C , 1) means of the Fourier series of Q. From the theory of Fourier series it is well known that if Q log+ 1 Q 1 is integrable, then both integrals

J

J 2"

2"

1

Q (ip) -

an (cp) 1 (1 cp ,

o

I

Q (cp) -

an (ip) Ilog+ 1 Q (cp) -

an (cp)

1

d cp

o

converge to 0 as n ~ 00. Consider now the formula (11) of § 3, which in our case can be written

K(r

J 2"

eifi )

=

(2 n)-1

1

Q (cp) llOg cos (: _

ip)

1- ~

ni sign cos (9 -ip)!dCP.

o

Using Young's inequality (8a) of § 3, we con easily prove that the integral 2,.

J

1Q (ip) -

an (cp) 1log

o

cos

191

- cp

I d cp

tends to 0, uniformly in 9. Hence, if in the formula defining

/'..

J( /'-

we substitute an for Q, the re,sulting expression will tend to K, uniformly in e, as n ~ 00. This means that the series in (8) is /'-

summable (C, 1) to K, and since the terms of the series are o (ljn) , the series converges. This proves (8). As to (9), we observe that if Q (e) "" ~ r:x.k eikfi is such that Q log+ 1 Q I is integrable then, as is known in the theory of Fourier series, ~'I r:x.k I k 1-1 is finite. Hence, if K (z, N) is the N-th sym -

98

96

[496]

A. ZYGMUND

/'-.

metric partial sum of ~r-2 (Xk eikf3 , then K /'-.

to K (z), and so, for any

(z

,N) converges uniformly

f EL2,

II (K(z) -

K(z, Nlf(z) 112 - 0,

which means that the operators which represent the N-th partial sums on the right of (9) converge uniformly to the operator H . It is easy to obtain inversion formulas for the operators defined by the kernels K k • Suppose that fE L2 and let ()k = (-i)lkl 17c 1-1 (see (4)). If T f denotes the Fourier transform of f, then T = Tf. eikrp ()k, whence

1=

and, taking conjugates,

Passing from Fourier transforms to functions we have

():12

f(-X)=I

J

K k (x-y)l(-

~)(ly,

and, finally, we get the inversion formula (10)

f

(x)

=

7c 2

J

K'f (y -

We discuss briefly the case n series of spherical harmonics have

K (x)

00

=

x).1(y) d y .

> 2.

Developping Q into a

r- n Y k (x') ,

~ k- l

where Yk (x') is a linear combination of fundamental spherical functions of order k. A remarkable fact about the kernels Y k (x') r- n is that their Fourier transforms are numerical multiples of Y k (cf. (4)); more precisely (11)

n (2 1l) - 2.. 2

f

Y k (y') rn

. _ - e-i(xy)

99

cl y

=

()k

Y k (x) ,

[497]

On singular integrals

97

where (12)

00

Arguing formally, we see that if K(x) = r- n X Y k (x'), then 1 "'-

K (x) = X ~k Y k (x') .

(13)

1

This is an analogue of the formula (8),and it is valid if Q log+ IQ I is integrable over X and the sum is taken, say, in the Abel sense. The proof is essentially the same as before: the formula is valid if ~J is a spherical harmonic; hence it is valid when the development of Q into spherical harmonics converges uniformly; finally if Q log+ I Q I is integrable over X, and Q (e ,x') is the Abel mean of the Fourier series of Q (or, what is the same thing, the Poisson integral of Q) then

fI J

IQ (x') -

Q (e , x') -

I

Q (x') d x' - 0

1

:E

Q (e , x') Ilog+ I Q (x') -

Q (e , x') I d x' - 0

:E

which implies that if on the right of the formula 11 of § 3 we replace Q (y') by Q (e ,y'), the resulting expression tends uni"'formly to K (x) as e - 1. This show that the series in (13) is uniformly " Abel summable to K. REMARK. Return to the formula (11) of § 3. It indicates that K is a spherical convolution of Q with a kernel Q (cos cp) depending only on the angle cp: /'..

(14)

11. (.:v') =

where (15)

J

Q (y')

Q (cos cp) d y',

:E

Q(cos cp) = (2n)

_2..n{ log I 2

I

I } I- I sign cos

cos cp

100

2

ni

cp •

98

[498]

A. ZYGMUND

It is of interest to find the development of this kernel into a 1 series of ultraspherical polynomials pt (cos cp), where a = 2 (n -2)

and the P ku are defined by the equation (1- 2 w cos cp

+ w )-u = 2

00

~

P ku (cos (p)

Wk •

k- O

The elassical formula

+

r(a) (k a)I U Y k (x') =. 2 n U +1 P k (cos cp) Y k (y') d y' I

<

implies that if 0 < e 1 and ~k is given by (12), then for any, say, continuous Q!'.) ~ Y n we have 00

(16)

~ 15."

k~l

e2k Yk (x') =

Normalizing the Pi: over the sphere ~ and using the Riesz-Fisher theorem we easily verify that the series ~ (k a) 15k P ku (cos cp) is the Fourier series of a function S (cos cp) in L2 over ~. Since the

+

/'-.

left side of (16) as well as ~ (l Y k (x') tend uniformly to limits K (x') and Q (y') respectively, we have

K(x') =

r (a+) 2nu 1

I

S (cos cp) Q (y') d y.

I

Comparing this with (14), and observing that the integrals over ~ of both Q (cos cp) and S (cos cp) are 0, we see that Q = (T(a)j2n u +1 j S, or, after simplifications, 1og

I cos1 cp I - -12

n

101

..

~ SIgn COS

cp ("

[499]

On singular integrals

99

Taking real and imaginary parts we obtain formulas for log(l/l cos cP and sign cos cp •

Il

>

§ 7. We now prove Theorem 9. Let p 1 and let the kernel K(x) = Q (x)/1 x In satisfy the hypotheses of Theorem 8. Given any

sequence X = (xzJ = (xo,

we consider the transformation

Xi' •• ,) 00

(1)

';;"" =

X' K (It", -

hi)

XI

l- o

of X into

X=

~1nl. vVe have to show that

where II X II p stands for (X I Xl IP) l/P, and Ap depends on p and the kernel K, but not on X, 1'0 simplify the argument we suppose that the vectors e1 , e2 , ••• ,en which generate the lattice points hm are of length 1. Let Rq be the hypercube with center hq, edges parallel to thecoordinate axes and of length 2- 1• We set if X E R q , q = 0 , 1 , 2 , ' , , ; elsewhere,

\Xq

f(x)

= 10

Since Ilf lip = 2- n /p II X lip, the function

f is in

Lp, By Theorem, 2, 111111'

11111p <

(2)

If

X

E R""

l(tc)

=

<

Ap lif lip. Hence

Ap II Xllp·

then

X [,em

J

K (x -

y).l (y) el y

111

= X

+ JK (x -

y) f (y) c1 Y =

11m

J' K

Xl l,em , Rl

(X -

y) ely

+

Xm

J' K

(X -

y) d y.

, Em

The hypothesis that Q is in Lip A. implies that if y E Rl (l =f: 1n), then C I K (x - y) - K (11m - 11.1) I s:: I' _ 1 In+).· "~m

102

II

X

E Rm and

100

[500]

A. ZYGMUND

It follows that, if x E Rm ,

f~(x) =

2

-n

~ Xl Z¢m

K (h m

hi)

-

+

~ (x)

Xm gJm

+0

{

I IxIII In+J.\ ,

~ Z -h

l¢m

~m

where by gJm we denote the characteristic function of Rm. It follows that

and it is therefore enough to show that

~J Il(x) Ip·d x + Xlxm Ipfl ;Pm (.x) Ip d + m m X

Rm

Rn

Now, we have

:II.f(.X) Ip d x

<J If(x) Ip

R",

JITm

(X)

J

Em

~ I Xm Ipf ITm (X) Ip d X <

m

n II X II ~ .

2- At

Rm

If we set OGm-1

= 2-

E"

and therefore

IX

II~ ,

< At'11 X

Ipd x < f ITm (X) Ipd X :::::;: At' I gJm (X) Ipd X

R",

with

d.T

En

= I hm -

independent of

X (~ 111

l¢m

hi 1-"-\ we have Xl

CXm-l

l¢m

1n,

< < CX

and we can write

I Xl I am-i) p < X (X ml¢m

=

~ (X I Xl I 1X:!'~1 1X~!'~I) P ml¢m

I Xl I CXm-l) I¢m (X CX

1l1 -I)P/P'

This eompletes the proof of Theorem 8.

103

00 ,

nAt' ,

On singular integrals

[501]

101

8. Historically, singular integrals and Hilbert transforms appeared first in the theory of the potential. It is well known that the second derivatives of the potential of an absolutely continuous mass distribution are represented by such integrals, and some of the problems still unsolved in the theory of the potential are essentially problems about Hilbert transforms. We shall now apply some of the results discussed previously to the theory of the potential. We are primarily interested in local problems, and without loss of generality we may assume that all the functions / (x) we consider ,vanish outside a sufficiently large sphere. The class of functions / (x) such that 1/ Ilog+ 1/ I is integrable we shall denote by L*. With the hypothesis just made about the functions / we have the obvious inclusions: (1

Ooncerning

l