Singular integrals and rectifiable sets in Rn: Au-delà des graphes lipschitziens (Astérisque)

193

,

ASTERISQUE 1991

SINGULAR INTEGRALS AND RECTIFIABLE SETS IN Rn Au-dela des graphes lipschitziens Guy David, Stephen Semmes

Subjects Classification : 49F20, 42840.

lntrod uction

There are a number of natural ways to look at the goals and results of this monograph. The first can be stated broadly as the problem of relating the geometry of a set E in R n to the analysis of functions and linear operators on E. A specific question of this type that we shall be concerned with here is the followin.e;. Let E be a subset of R n that has Hausdorff dimension d, 0 < d < n. We equip E with d-dimensional Hausdorff measure restricted to it, and we assume that this measure is locally finite. Under what conditions on E is it true that plenty of singular integral operators are bounded on L 2 (E)? Examples of the sort of singular integral operators that we have in mind are the Cauchy integral when d = 1 and n = 2 and the double-layer potential when d = n - 1. It is known from the work of Coifman, Mcintosh, and Meyer [CMM] that this is true when E is a Lipschitz graph. There are several more general conditions on E which are known to be sufficient to ensure the boundedness of lots of singular integral operators, but there has not been much progress on finding necessary conditions. Our main result provides geometrical characterizations of the sets E for which a fairly large class of singular integral operators are bounded on L 2 (E), at least if we make an auxiliary technical assumption onE (Ahlfors regularity). See Section 1 for the precise statement. Unfortunately we do not know at this time how to work with smaller classes of operators; for example, when d = 1 and n = 2 we would like to use only the boundedness of the Cauchy integral. The geometrical conditions that arise in the aforementioned theorem can be thought of as quantitative analogues of the classical notion of rectifiability. Recall that E is said to be rectifiable if it is contained in the union of a countable family of Lipschitz images of Rd, except for a set of d-dimcnsional Hausdorff measure zero. Rectifiability is a qualitative con-

INTRODUCI10N

clition, and it is not strong enough to imply the boundedness of singular integral operators. Using real-variable methods as in [Dl, 3] it can be shown that various quantitative versions of rcctifiability are strong enough to imply the boundedness of plenty of singular integral operators, and our theorem provides a converse to this. There is a great deal of information available about rectifiable sets. See [Fe], [Fl], and [Ma], for instance. Not so much seems to be known about quantitative analogues of rectifiability. In particular there are many characterizations of rectifiability, and these give rise to many candidates for the notion of quantitative rectifiability, but the complete relationship between these various candidates is not at all clear. Our theorem provides some nontrivial equivalences between some of these conditions. Although this is a purely geometrical issue, it turns out that singular integral operators provide a useful tool for passing between some of these conditions. Our main result also gives a higher-dimensional version of Peter Jones' travelling salesman theorem ([J3]). That is, we give two other conditions on E that are equivalent to the others, and which are roughly as follows. One of these conditions says that E is contained in a set that admits a nice parameterization by Rd. The other condition is a bound on certain quantities that measure the extent to which E can be approximated by d-planes. Again, Section 1 should be consulted for the precise statement. Although there are several ways of looking at what we are doing and what it means, there is an uoderlying common theme. To a large degree we are trying to produce methods for analyzing the geometry of sets, in much the same way that more traditional harmonic analysis (as in [St]) is concerned with the analysis of functions and operators. Some of the ideas of harmonic analysis make sense in this context, but mostly the techniques don't work so well, because of the absence of a linear structure. The methods that have grown out of Carleson's corona construction seem to be more cooperative in this geometrical setting. In connection with the analogy with traditional harmonic analysis it is interesting to look at the theorem in Section 1 from the perspective of Littlewood-Paley theory. In some sense this theorem gives a LittlewoodPaley characterization of a class of good sets that is analogous to well-knmvn results for Sobolev spaces. It turns out that this analogy is somewhat misleading, in that there are some other results in our geometrical context that do not have a natural counterpart for Sobolev spaces. Such a result is discussed just before Lemma 5.13, but its details will appear elsewhere.

INTRODUCDON

The precise statements of our main results are given in Section 1, along with some background information and organizational details. It is perhaps worth mentioning now that there is a discussion of open problems in Section 21. In that section there is also some limited description of other work in this general area. More information of that nature can be found in

[D5]. The authors would like to thank R.R. Coifman and P.W. Jones for many helpful discussions. Portions of this work were completed while the second author was visiting the first author at the Ecole Polytechnique and UCLA. The second author is partially supported by the U.S. National Science Foundation and the Alfred P. Sloan Foundation.

Table of Contents

Page . Intrcxluctlon ... ..................................................... ........... .........•...... ............... ........... ......

1

.'Iableofcontents .......................................................................................................... 5 1. Statelnentofthetnabl:results ................................................................................... 7 2. Dyadic cubes and the corona decomposition ...................................................... 17 3. From (Cl) to (C2) ........................................................................................................... 21 4. (C2)1Inplles a local synuneby condition .............................................................. 23 5. The local synunetry condition (LS)implies the weak geometric lemma ... 27 6. Approximation of E lnxneasure ............................................................................... 33 7. Building the stopping-time reglons,and some of their properties ............. 37 8. The construction of the approximating Upschitz graph ............................... 43 9. Pushing the square function estimates from E to the graph of A ............ 51 10. Controlling a square function of A in terms of J (S. ) ............................... 57 11. The end of the proof that (C2)1Inplies (C4) ...................................................... 73 12. The proof that (C3)1Inplies (C4): prelllninary discussion .......................... 81 13. Pushing estimates on

/3 1(x,

t) from E down to the approximating

lipschitz graph .............................................................................................................. 83

14. The end of the proof that (C3) llnplies (C4) ...................................................... 91 15. (C4)1mplies (C3) ........................................................................................................... 93 16. The main step in the proof that (C4)1Inplies (C5) ...................................... 101 17.Anex:tenslon theorem ............................................................................................. 107

18. Tile proof that (C4) ilnplles (C7) ........................................................................ 111 19. A variant of(C2) and (C3) ........................................................................................ 125 20. Acounterexainple ....................................................................................................... 135 21. Sonle open problenls ................................................................................................ 139 .~hy

................................................................................................................ 143

. Abstract (In French) ................................................................................................ 145

1. Statement of the main results

Let E be a subset of Rn with Hausdorff dimension d, 0 < d < n. Unless explicitly stated otherwise, d will always be an integer. Assume that d-dimensional Hausdorff measure Hd is locally finite when restricted to E. Consider singular integral operators on E of the form

Tf(x)

(1.1)

= p.v.

j K(x- y) f(y) dy, E

where dy denotes Hd satisfies

IE' and where K(x) is smooth on Rn \ {0}, odd, and

I vJK(x) I:S C(j) I X 1-d-;,

(1.2)

j = 0, 1,2, ...

We would like to know what conditions onE are needed in order for (1.1) to define a bounded operator on L 2 (E), say. For technical reasons it is better not to look for an L 2 estimate for ( 1.1) but rather an estimate like (1.3)

~~~

JI J E

En{!,-y!>f}

K(x- y) f(y) dy

2

1

dx

~ C(K)

J

If

12

dx,

E

for all f E L 2 (E). This formulation avoids the problem of the existence of a principal value. An important example of such a function K( x) is the Cauchy kernel, 1 + for x E R 2 , d = 1, n = 2. This case is of course relevant i.e., K(x) = %1 1X2 for complex analysis; for instance, the L 2 boundedness for the associated operator is closely related to the analytic capacity of E and its subsets. (See

[G2], [Mu], [Cl, 2].) A higher-dimensional analogue of the Cauchy kernel is K(x) = d = n- 1. One way that this kernel arises is in connection with the double-layer potential, which can be expressed in terms of K( x ). It is easy to see that an operator as in (~.1) is bounded on L 2 (E) if E is a smooth submanifold (e.g., C 1 ,o, a > 0) which is nice at oo, or a subset of such a submanifold. This is also true if E is a Lipschitz graph over some d-plane, but it is much harder to prove. For the Cauchy kernel, this is a theorem of 9oifman, Mcintosh, and Meyer [CMM], improving an earlier result of Calderon (which covered the case of graphs of Lipschitz functions with small norm). The case of general kernels was derived from this result in [CDM], at least when d = 1, n = 2. The higher-dimensional case can easily be obtained from this using the method of rotation, just like the argument in Section 13 of [CMM]. The fact that Lipschitz graphs are O.K. for these operators shows that the smoothness of E is not the issue, but it is not obvious how wild E can be. The following two results help to clarify the situation. The first says that if d = 1, E is a curve, and, say, n = 2 and K is the Cauchy kernel, then Tis bounded on L 2 (E) if and only if there is a C > 0 such that

fxln'

(1.4)

H 1 (E

n B(x, R)):::; CR

for all x E R 2 , R > 0, where B( x, R) denotes the ball with center x and radius R. Such curves are often called regular curves. This was proved in

[Dl]. The second result goes as follows. Let G be the Cantor set in [0, I] obtained from the usual construction, except that you remove the middle half of the interval at each stage. Then E = G X G satisfies (1.4), and also H 1 (E

n B(x, R));::::

c- 1 R

for all x E E, 0 < R:::; 1, but the Cauchy integral operator is not bounded on L 2 (E). This follows from [Gl], see also [G2J, [D2], [Jl], and [Ma3]. These two results suggest that rectifiability plays a role here. To make this precise it is helpful to recall a couple of definitions and facts from geometric measure theory. Let A be a subset of R n with Hausdorff dimension d. We say that A is (count ably) rectifiable if there is a countable family JJ of Lipschitz maps of R d into R n such that

Hd(A \ ( U!j(Rd))) = 0, J

so that A is almost covered by the union of the images of the fi 's. It turns out that this is equivalent to requiring that A be almost covered by a countable family of d-dimensional Lipschitz graphs, or even C 1 submanifolds. A is said to be unrectifiable if Hd(A n B) = 0 for all rectifiable sets B. A basic fact is that any set A with Hd(A) < oo can be written as the union of a rectifiable set and an unrectifiable one. For example, any subset of a curve of finite length is rectifiable, but it can be shown that G x G is unrectifiable (with d = 1 in both cases). References for these topics include [Ma], [Fl], and [Fe]. You might hope that if E is rectifiable, and if you have some control on Hd (like (1.4), when d = 1), then singular integrals have to be bounded on L 2 (E). This is not true, but for a good reason; rectifiability is a qualitative condition, while estimates on singular integrals are quantitative. It is not hard to build sets that are rectifiable, but which have pieces that approximate unrectifiable sets (like G x G) on which singular integrals do not define bounded operators, in such a way that singular integral operators are not bounded on the set you've constructed. Thus we need to look for quantitative notions of rectifiability. We give an example of such a notion after the following definition.

IE

DEFINITION.

A set E s; R n is regular (with dimension d) if it is closed and

if

(1.5) for all x E E, R > 0, where C does not depend on x, R. We shall assume throughout the rest of this paper that E is regular, and we shall often write IAI for Hd(A) when As; E. It is not hard to show that if singular integrals like (1) define bounded operators on L 2 (E), then the right hand inequality in (1.5) must hold. (See [D5], [Sl].) The left hand inequality should be thought of as a nondegeneracy condition. Note that it is translation and dilation invariant. Notice also that if E is closed and if there is a measure a supported on E for which the analogue of (1.5) holds, then a must be equivalent in and E must be regular. Thus we lose nothing by restricting size to Hd ourselves to Hausdorff measure here.

IE'

DEFINITION. E bas BPLG (big pieces of Lipschitz graphs) if it is regular and if there exist C, f > 0 so that for every x E E, R > 0 there is a

d-dimensional Lipschitz graph

r

with constant ::; C such tbat

(1.6) When we say that r is ad-dimensional Lipschitz graph with constant ::; C we mean that there is a d-plane P, an (n- d)-plane p.L orthogonal to P, and a Lipschitz function A : P -+ p.L with norm ::; C such that

r = {p + A(p): pEP}. This condition is a good example of what we mean by a quantitative version of rectifiability. It is not hard to see that E is rectifiable if it has BPLG, but the converse is not true; rectifiability only allows you to co'uclude (1.6) with an e that depends on x, R. Notice that if you fix C, e and look at the class of sets having BPLG with constants C, e, and which also satisfy (1.5) with this same C, then this class is invariant under translations, rotations, and dilations, and it is also elosed in the Hausdorff topology on closed subsets on R n. (In this topology E; - E if for every e, R > 0 and all j sufficiently large we have that each point in E; n B(O,R) is within e of an element of E, and vice-versa.) We do not know whether it is true that any class of rectifiable subsets of R n with these same invariance and closure properties has to be contained in the class of sets that have BPLG. It follows from [Dl, 3] that if the regular set E has BPLG, then the estimate (1.3) holds for all K as before. We do not know if the converse is true. However, there are some other "quantitative rectifiability" conditions similar to (and a priori weaker than) BPLG which also imply (1.3), and for which we are able to obtain a converse. We also obtain other analytical and geometrical characterizations of these sets. THEOREM. Let E ~ R n be a regular d-dimensional set in R n. The following conditions ( Cl )-( C7) are equivalent.

(Cl) If K(x) is any smooth odd function on Rn \ {0} that satisfies (1.2), then (1.3) holds. Using a standard fact from Calder6n-Zygmund theory (Cotlar's inequality - see [JL]) we have that (1.3) is equivalent to

(1.7)

/sup •>O

E

1/

Jsnttx-yi>•D

K(x-y)f(y)dyi 2 dx::;C(K)jlfl 2 E

for all f E L 2 (E). (To be honest we should admit that in order to apply the techniques of Calder6n-Zygmund theory we should first observe that E is a

space of homogeneous type, in the sense of [CW], when E is equipped with the measure Hd and the Euclidean distance.] Calder6n-Zygmund theory also implies that these L 2 estimates are equivalent to their LP counterparts, 1 < p < oo, etc. (See [JL] again.)

le

(C2) For each smooth odd function '1/; on Rn with compact support we have that 00

2::

(1.8)

k=-oo

is a Carleson measure on E

X

R+.

Here '1/;k(x) = 2-kd!f(2-kx) and dhs(t) denotes the Dirac mass at sin t. A Carleson measure on Ex R+ is a measure p for which there is a C > 0 such that for every x E E and R > 0 we have

1R1 o

d11 < CRd.

B(x,R)

Thus Carleson measures are measures on E X R+ that behave as though they are d-dimensional near Ex {0} ~E. This condition is quite natural despite its technical appearance. Because '1/; is odd, fe 1/Jk(x- y)dy is zero if E is ad-plane; thus this quantity measures in some way how close E is to being a d-plane. One can think of (C2) as a geometrical analogue of classical characterizations of various function spaces in terms of the size of expressions like '1/;k(x- y)f(y)dy This geometric Littlewood-Paley point of view is discussed somewhat more thoroughly in [DSl].

I fR•

(C3)

I·

/31 (x, t) 2 d~dt

is a Carleson measure onE

For x E E, t > 0 we define f3 1 ( x, t )

p

R+.

/31 (x, t) by

• f 1

=Ill

X

d t

1

dist(y, P) t dy,

EnB(x,t)

where the infimum is taken over all d-planes P. Thus {31 (x, t) measures how well E can be approximated by a d-plane.

Peter Jones was the first person (to our knowledge) to look at this kind of condition. Actually, he worked with an L 00 version, i.e., with f3oo(x, t), where

(1.9)

/3q(x, t)

.

= mf

( 1

p

td

1

(dist(y,P))q t

EnB(:z:,t)

dy )

~

In [Jl] he showed how to use the fact that Lipschitz graphs satisfy (C3) (with f3 00 (x, t)) to give a new approach to the estimates for Cauchy integrals on Lipschitz graphs. He later found a characterization of subsets of rectifiable curves in terms of a (related) quadratic condition on the f3oo's. In particular he showed that subsets of regular curves cart be characterized by a quadratic Carleson measure condition on the {300 's. Our results give analogues of this characterization for d > 1. If we replace /31 by /3q in (C3), then we still get an equivalent condition as long as q < d2_!2 (q ::S oo if d = 1). However, Jones and Fang have produced 3-dimensional Lipschitz graphs so that (C3) does not hold for f3oo·

There is a classical counterpart of (C3) for functions just as there was for (C2). Given a function f on Rd, set

(1.10)

'Yq(x, t)

= inf ( a

~

t

.l.

{

( if(y)- a(y)i) q dy)

Jn(:z:,t)

9 ,

t

where now the infimum is over all affine functions. Notice that for Lipschitz functions the 'Y's for f are essentially equivalent to the (3's for the graph of

f. The 'Y's can be used to characterize smoothness properties of J, e.g., whether f lies in a particular Sobolev space. (See [Do], for instance.) This is closely related to the corresponding results using second differences instead (see [St]), which are perhaps more familiar. In Section 19 we shall give a condition ( C8) that is a geometrical version of a second-difference condition for functions, and we shall show that it is equivalent to the others.

(C4) E admits

a corona decomposition.

The precise explanation of this condition is complicated and will be postponed until the next section. Roughly speaking it means that you can decompose E x R+ into two pieces, the good and the bad parts, with the

following properties. The bad part is not too big, in that it is controlled by a Carleson measure. The good part can be subdivided into stoppingtime regions on each of which E is well-approximated by a Lipschitz graph. There aren't too many of these regions, in that they satisfy a Carleson measure packing condition. This condition plays a central role for us, acting as a bridge between (C1)-(C3) and (C5)-(C7). Although it is awkward to state, it carries a lot of useful information, and is not so hard to work with. (C5) E bas very big pieces of bilipschitz images ofRd inside Rn", n* = max(n, 2d + 1). This means that for every e > 0 there is an M > 0 so that for each x E E and R > 0 there is a mapping p: Rd-+ Rn" which is bilipschitz with constant M, i.e.,

(1.11)

1

Mix- Yi ~ ip(x)- p(y)i ~Mix- Yi

and whose image almost contains B(x, R)

for all x, y E Rd,

n E, that is,

Here we identify R n with a subset of Rn" in the obvious way. (C6) E bas big pieces of Lipschitz images of subsets of Rd. This means that there exist e, M > 0 so that for every x E E, R > 0 there is a Lipschitz mapping p with norm ~ M from the ball Bd(O, R) in R d into R n such that

It follows from the main result in [Jl] that ( C6) is equivalent to the condition you get by replacing B(O, R) with a subset F of B(O, R) and "Lipschitz" with "bilipschitz." [To apply that result it is useful to notice that Hausdorff measure is equivalent in size to Hausdorff content for subsets of a regular set.] Also, if p: F-+ Rn is bilipschitz, F ~ Rd, then you can extend p to a bilipschitz mapping on R d, at least if you replace R n with R n. , as we shall discuss in Section 17.

(C7) There is an A 1 -weight won Rd and an w-regular mapping z : Rd -+ Rn+l whose image contains E. Recall that w( x) is an A 1 -weight on R d if it is a positive locallyintegrable function such that for each ball B, (1.12) It is well-known that this implies that w is an A 00 weight, which can be characterized by the existence of C, 6 > 0 such that

(1.13) for all balls B. (See [JL] for basic facts about weights.) As in (D3] we say that z : Rd -+ R"+ 1 is an w-regular mapping if w is an Aoo weight, z has locally integrable distributional derivc~.tives, jV'zi ~ Cw~ a.e., and {1.14) for ally E R"+I, R > 0, where w(A) =JAw. If z{·) is w-regular, and B is any ball in Rd, then

(1.15)

diam(z(B)) ~ Cw(B)~.

This can be derived from IV'zl ~ Cw~ and (1.13) using standard results. Conversely, it is not hard to show that if {1.15) holds for an A 00 -weight w, then z has locally integrable distributional derivatives and IVzl ~ Cw~ a. e ..

It is also not hard to show that if z(·) is w-regular, then its image z(Rd) is a regular set, and w is equivalent in size to the pull-back of Hausdorff measure. Notice that z(·) is !-regular if it is bilipschitz. It is proven in [D3] that (CI) holds forE = z(Rd) if z(·) is an wregular mapping with wE A 00 • Thus (C7) implies (Cl) even if we weaken the requirement w E A 1 tow E A 00 • When n 2: 2d we can replace R n+l in ( C7) by R" and still have an equivalent condition. This is proved using the methods of [D3], Section 5. The main interest in this observation comes from the case d = 1, n =

E=

=

2. Notice that when d = 1 we can always take w 1, because we can reparameterize z(R) by arclength. We originally derived the equivalence of (C7) with (C1)-(C6) when d == 1 from Peter Jones' [J3] characterization of the subsets of regular curves in terms of his version of (C3). The equivalence of (C3) and (C7) when d > 1 provides a higher-dimensional version of his result, at least for regular sets. This finishes the statement of conditions (C1)-(C7) (except for (C4)), whose equivalence is stated by the theorem. Notice that the ambient dimension n does not play a serious role. Let us now describe the routing of implications that we follow in proving the theorem. It is relatively easy to show that (C1) implies (C2). The proofs that each of (C2) and (C3) imply (C4) are quite similar and they constitute the main step in the proof of the theorem. The proof that (C4) implies (C3) is pretty straightforward but messy. Both of (C5) and (C7) will be obtained from (C4) by direct constructions. Of course (C6) is a trivial consequence of (C5). It follows from [D3] that (C1) holds if any of (C5), (C6), or (C7) do. (In the case of(C6) we also use the result of [J3] as discussed above.) You can also derive (C1) directly from (C4), as in [S4]. We should point out that our methods for deriving (C5) or (C7) from ( C4) are quite constructive, although somewhat messy. The stopping-time argument given in Section 7 for producing a corona decomposition (if one exists) is both constructive and fairly simple, and one could imagine asking a computer to do it. The difficult part- proving that a corona decomposition does exist if (C2) or (C3) holds- is not the computer's problem. A curious feature of our arguments is that we do not know how to pass from ( C2) to ( C1) analytically, without going through the geometry. One can look for analogues of some of the well-known methods for controlling singular integrals on R n using square functions (via reproducing formulas, for example), but we have not been able to make anything like that work here. Similarly, it is not so clear how to pass from (C6) to (C5), or from (C5) to (C7), without going through singular integrals. We also take up a version of the main theorem for fractional dimensional sets. Conditions (C1) and (C2) still make sense in this case, although (C3)-(C7) don't. We shall prove that if Eisa d-dimensional regular set, d noninteger, then neither ( C1) nor ( C2) can hold. The organization of the remaining sections is as follows. In Section 2 we cover some preliminary material and also give the precise defintion of a

corona decomposition. We prove that (C1) implies (C2) in Section 3. We derive in Sections 4, 5, and 6 some geometrical consequences of (C2), and in particular we show in Section 5 that neither ( C1) nor ( C2) can hold if E has fractional dimension. We set up in Sections 7 and 8 the initial machinery common to the proofs that (C4) holds if either (C2) or (C3) do. We also give an outline of the argument used to show that (C2) implies (C4) in Section 7, and the details are carried out in Sections 9, 10, and 11. The proof that (C3) implies (C4) is given in Sections 12, 13, and 14. We show that (C4) implies (C3), (C5), and (C7) in Sections 15, 16 and 17, and 18, respectively. In Section 19 we state condition (C8), a variant of (C2) and (C3), and we indicate why it is equivalent to the other conditions. We give a counterexample in Section 20 to show that the "weak geometric lemma" (see Section 5) is not strong enough to imply rectifiability, even if E is regular. In the last section we discuss some open problems, concerning the theorem and its proof as well as other related topics. In so doing we also give some small and partial indications of other work in this general area. A more substantial overview can be found in [D5]. We should also indicate the interdependence of the sections. Section 2 is essential for most of what we do. Sections 3, 15, (16 + 17), 18, 20, and 21 are all independent of each other and also Sections 4-14, with minor exceptions. The proof that (C2) implies (C4) is given in Sections 4-11, while the proof that (C3) implies (C4) uses parts of Section 5 and also Sections 7, 8, 12, 13, 14, and 11, in that order. The details of Section 19 rely on the proof of the equivalence of (C3) with (C4).

2. Dyadic cubes and the corona decomposition

As in [D4] one can build a family of subsets of E that play much the same role that dyadic cubes do for Rd. More precisely, there is a family of partitions 6.j of E, j E Z, into "cubes" Q with the following properties: (2.1) if j :S k, Q E b.j, and Q' E b.k, then either Q

n Q'

=

0 or Q

~ Q';

The cubes can also be built in such a way that they have relatively small boundary, like ordinary cubes in R d do: (2.3) if Q E b. J and T > 0, then I {x E Qj: dist(x, E \ Q) :S r2i} I:S Cr!2id. Of course it is important that the constant C in (2.2) and (2.3) does not depend on j, Q, or T. The properties of the Q's and b.j's given in [D4] are not quite the same as those above, but the same kind of construction can still be used. For a slightly better proof, see (D5J. An extension of this result can be found in

[C2]. We shall follow the standard practice of referring to the cubes that contain a given cube as its ancestors, referring to its subcubes in the next generation as its children, etc. Let b. = Ub. 1 denote the set of all our cubes. We can think of b. as providing a discrete version of E x R+, by letting (x, t) E E x R.t correspond to the Q E b. J with x E Q and 2i :S t < 2i+I.

We say that E admits a corona decomposition if for each 7J > 0 (think of 7J as being small) there is a C = C(ry) > 0 such that we can partition~ into a good set g and a bad set B with the following features. The bad set is not too large, in that it satisfies the Carleson measure packing condition

LIQI:s;CIRI

(2.4)

for all R E

~-

QEB

9~R

The good set g can be partitioned into a family :F of subsets S of g such that: (2.5)

each Sis coherent, which means that it has a maximal element Q(S), and that if Q E S, Q' E D.,

Q ~ Q' (2.6)

s; Q(S),

then Q' E S;

from the viewpoint of each S, E is well-approximated by a d-dimensional Lipschitz graph r with constant :::; ry, in the sense that for each Q E S we have dist(x,r):::; 7JdiamQ whenever x E E,dist(x,Q):::; diamQ;

(2.7)

there aren't too many of the S's, in that they satisfy the packing condition

L

IQ(S)i:::; CIRI

for all R E

~-

SE:F

Q(S)~R

There are a number of places where something like a corona decomposition has been used before. One is the work of Garnett and Jones [GJ] on the corona theorem for Denjoy domains. Although what they did is somewhat different in details it is quite similar in spirit. Another example is Peter Jones' proof of the L 2 -boundedness of the Cauchy integral on regular curves [Jl], and later in his quadratic estimates on the f3 00 's for rectifiable curves [J3]. A corona decomposition also arose in [54] for a certain class of hypersurfaces in R n, in connection with square function estimates for the Cauchy integral from Clifford analysis. In each of these examples something like a corona decomposition was obtained by applying Carleson's corona construction to a function that

somehow controlled the geometry. In our case we cannot apply the corona construction so directly, but we shall use many of the same ideas. This notion of a corona decomposition is somewhat technical and complicated, but it is very useful. It includes enough control on the geometry of E to imply other things, and it is set up in such a way as to make it amenable to proving it using a stopping-time argument. In fact, there is sort of a. universal stopping-time argument for deciding whether E admits a. corona. decomposition, which is described in Section 7. This argument is universal in the sense that it produces a corona decomposition for E whenever one exists. Let us give an imprecise outline of this procedure. To get started you need to know that E satisfies the weak geometric lemma. This condition is defined in Section 5; roughly speaking, it means that for most cubes Q, E is well-approximated by ad-plane PQ. The cubes for which this is not true are put into the bad set, and then you use a stopping-time argument to partition the good cubes into regions S for which (2.5) holds, the angle between PQ and PQ(S) is small for Q E S, and such that the minimal cubes Q of Seither have a bad son or have angle (PQ, PQ(S)) being not too small. If you choose the parameters correctly (2.6) holds, and the hard part is to verify (2.7). In most of the examples the verification of (2.7) works as follows. Your hypothesis is some sort of square function condition on E, such as (C2) or (C3). The main step is to show that if S has lots of minimal cubes with angle (PQ, PQ(S)) not too small, then there has to be a substantial contribution to the square function condition on E coming from S. To do this you push the contribution from S down to a square function estimate on the Lipschitz graph, and you can usually work with that using classical results. This gives you control on the Lipschitz graph (namely, control on the oscillation of its tangent plane) that permits you to show that S can't have too many minimal cubes of the above type. It is natural to ask after seeing the definition of a corona decomposition whether the graph of a Lipschitz function A necessarily has one. Of course our main theorem says that it does, but it is not so difficult to prove this directly, by applying the corona construction to, say, the Poisson extension of VA. This is similar to the approach taken in [S4], although [S4] applies in more generality, and can certainly be simplified in this case. There are a number of variations that we can make in the definition of a corona decomposition that would still yield an equivalent condition. For

example, we could replace dist(x,r)::;., diamQ

by

dist(x,r)::; CdiamQ.

Because we shall not need this fact we shall content ourselves with merely an outline of the proof. Suppose that E admits this weakened version of the corona decomposition, and let us show that we can find one of the stronger type. We replace each S E F by a subregion S as follows. We require that if Q E S then RES for every cube R in the same generation b.i as Q which con~ains a point x such that dist( x, Q) ::; dia.m Q. We also require that if Q E S then all its subcubes for the next m generations lie in S, where m ~ log~, and that the cubes R as above also have this property. If we take all the cubes inS\ Sand add them to B for each S, then the resulting augmentation of B still satisfies (2.4). (This can be checked using (2.3) and (2. 7).) It is not difficult to then decompose the S's into subregions that satisfy (2.5), (2. 7), and the stronger version of (2.6). (You have to decompose the S's because they do not have maximal elements.) Similarly, in (2.6) we can replace dist(x,Q)::; diamQ by dist(x,Q)::; kdiamQ for any given k > 1. Indeed, if E admits a corona decomposition as above, we can remove the top m layers of each S E F, m ~log k, put them into B, and reorganize what's left of each S into new coherent regions. We also don't really have to require that the Lipschitz graphs r in (2.6) have small constant. Indeed, if E admits a corona decomposition where the f's merely have uniformly bounded constants, then we can build one where they have small constants by applying the corona decomposition to each of these r's to get new ones with small constants, and combining these corona decompositions for the f's into one for E. It is also not hard to show that whether E admits a corona decomposition does not depend on the choice of b., as long as (2.1) and (2.2) are satisfied. (Our second variation of the corona decomposition - the one with the k - is helpful in this regard.) Hopefully these variations give an indication of the flexibility of the notion of a corona decomposition. In practice it is particularly convenient that the only requirement on the bad set is (2.4 ).

3. From {Cl) to (C2)

It is helpful to introduce an intermediate condition.

(C2') Given any smooth odd function 'ljJ on Rn with compact support,

f j j I

k=-oo for all

f

E

'1/Jk(x- y)f(y)dy

1

2

dx

~ C j lfl

E

2 dx.

E

E L 2 (E).

That (C2') implies (C2) is well-known, and is obtained by applying (C2') to characteristic functions of balls. (The converse is not too difficult either; it is essentially a square function version of the T(1) theorem.) To prove that (C1) implies (C2) we use a familiar artiface. Let n denote the space of all sequences w = {wj}, j E Z, of ±1's, with the usual product topology and the product measure that gives each choice of ±1 equal probability. Define €j : n -+ {±1} by €j(w) = Wj· As usual we observe that

(3.1)

JI .f J

€j(w)'l/Jj(x- y)f(y)dy 12

n

&.;

J=-m E

=

.f If J=-m

'1/Jj(x- y)f(y)dy

1 2

•

E

for any m. This follows from the orthonormality of the €j's. We want to apply (C1) to the kernel m

K(x) = Km(x,w) =

L j=-m

€j(w)'l/Jj(x)

(which is certainly odd and satisfies (1.2)) to conclude that

JIJ

j !f! dx.

E

E

Km(X- y,w)f(y)dy \2 dx::; C

(3.2)

E

2

Unfortunately, however, the way we stated (Cl) only gives us (3.2) with a C that depends on m and w. Using a "completeness" argument we shall show that (Cl) actually does imply that (3.2) holds with a C that is independent of m and w. Once we've done that it will follow immediately that (C1) implies (C2), because of (3.1). Let N( m, w) denote the smallest constant for which (3.2) holds for all f. For each m < oo choose w( m) so that

N(m,w(m))

= maxN(m,w) =: N(m). w

We want to show that supN(m) < oo.

(3.3)

m

Suppose not. Choose mj, j = 1, 2, 3, ... , such that

N(mj)?:: 2mi- 1 •

(3.4) Define w(oo) to see that

En by w;(oo) = w;(mj) whenever mj-1


It is easy

N(mj) = N(mj,w(mj))::; 2N(n:tj,w(oo)) + Cmj-1·

(3.5)

This uses only the fact that the number of i's such that -mj ::; i ::; mj and w;(mj) ::f w;(oo) is at most 2mj-l + 1. On the other hand, we have for any w that

j j I

E

Km(x- y,w)f(y)dy

12

j !fl dx + C !~~ j

I

dx

E

::; C

2

E

E

j

Koo(x- y,w)f(y)dy

2

1

dx,

En{lz-yl>a}

where C does not depend on w, m, or f. This is not hard to check. Of course (C1) says that the right side is at most C J lfl 2 dx, where now C E

depends on w but not m or f. Applying this to w = w( oo) and then recombining with (3.5) we find that there is a C > 0 not depending on j such that N(mj).:::; C + Cmj-1· For j large enough this is incompatible with (3.4). The ensuing contradiction establishes (3.3).

4. (C2) implies a local symmetry condition

To simplify notations, we'll use from now on the convention that

( 4.1)

>.Q = {x E E: dist(x, Q)::; (>.- 1) diam Q}

for Q E ~ and >. > 1. Given a small number T' let n( T) denote the set of cubes Q E ~ such that there exists two points x, y E 2Q with dist(2x - y, E) 2: T diam Q. Thus if Q ~ n(r), then for any x, y E 2Q, the point z = 2x- y is near E, and z is of course the point on the line through x andy which is opposite to y about X. In other words, if Q ~ n( T)' then E is approximately symmetric near Q about each point in Q. DEFINITION 4.2. We say that E satisfies the local symmetry condition (LS)

if for each T > 0 the set of cubes n( T) satisfies the packing condition (4.3)

L

IQI::; C(r)IRI

for all R E

~-

QE'R(r) Q~R

In the next section we'll show that if E satisfies (LS), then E is wellapproximated by d-planes around most cubes Q. PROPOSITION 4.4. If E satisfies (C2), then E satisfies (LS).

Fix T > 0. Our strategy for proving (4.3) will be to find a finite family of 'lj;'s so that each cube in n( T) gives a substantial contribution to (1.8) for one of these 'lj;'s. This will allow us to derive ( 4.3) from (1.8).

Let Q E R(r) be given, and let k E Z be such that Q E ~k- Let x, y E 2Q be such that dist(2x-y,E) ~ rdiamQ. Denote by Co the constant in (2.2). Set Yo= 2-k(y- x) and B = B(yo,r/lOCo). Let '1/J = tPQ be an odd c= function such that

(4.5)

supp '1/J ~ B U (-B), '1/J 2: 0 on B, and '1/J

= 1 on B(yo, r /20Co)-

We can even find a-finite family Ill of functions so that for any Q E R( r) we can take tPQ to be an element of Ill. This is because Yo E B(O, lOCo)\ B(O, r /3C0 ) independently of Q. Let x' be any point of B(x,2"r/40C0 ). Because of our choice of '1/J, tPk(x'- u) 2:0, except perhaps when u E B(2x- y,2kr/5Co), and in this case u ~ E because of our assumptions. Also, if u E B(y, 2"r/40Co), then tPk(x'- u) = 2-kd. Using our assumption that E is regular (1.5) we get that

J

tPk(x'- u)du

2"r

~ rd /C

for all x' E B( x, -----;::::;- ), 40vo

E

whence

(4.6)

j I j '1/Jk(x'- u)du 12 dx';::: c-lr3d2kd_ 3Q

E

For k and x' given, there are at most a bounded number of cubes Q such that Q E ~k and x' E 3Q. Hence, for any R E ~.

2:

IQI::; C(r)

QE'R.(r)

2:

j j (tPQ)dx'- u)du I

QE'R.(r) 3Q

Q~R

1

2

dx'

E

Q~R

:S C'(r)

2:

2:

j j tPk(x'- u)du I

1/JE'll 2t$CdiamR 3 R

1

2

dx' :S C"(r)IRI-

E

The last inequality uses (1.8). This proves the proposition. REMARK 4. 7: Our local symmetry condition implies a slightly stronger version of itself. For each r > 0 and each k > 1, let R( r, k) denote the set of cubes Q such that there exist x, y E kQ such that dist(2x- y, E) 2: r diam Q. If E satisfies our local symmetry condition, then

(4.8)

2: QE'R.(r,k) Q~R

IQI::; C(r,k)IRI

for all R E /}.and all r, k. The proof is not difficult. (If Q E R(r, k), then there is a not-too-distant ancestor of Q in R( r'), if r' is chosen suitably.)

5. The local symmetry condition (LS) implies the weak geometric lemma

For Q E

(5.1)

~and

1

~

q

~

f3 (Q) =.1 f q ~

oo, define

(-1IQI J

(dist(x, P)) q d ) diamQ x

t

'

2Q

where the infimum is taken over all d-planes P. This is of course a minor variation of (1.9). We say that E satisfies the weak geometric lemma if for each e > 0 we have

(5.2)

L

IQI ~ C(e)IRI

for all R E

~-

Poo(Q)>l Q~R

The name stems from the practice of saying that E satisfies the geometric lemma (of Peter Jones) if

(5.3)

L f3oo( Q) IQI ~ GIRl 2

for all R E

~-

Q~R

Note that if E satisfies our version of the geometric lemma- i.e., (C3) -then it satisfies the weak geometric lemma. This can be readily derived from the fact that

(5.4)

f3oo(x,t)

~

1

CJ'I(x,2t)7+I.

To prove (5.4), let P be the d-plane that realizes the infimum in the definition of fi 1(x,2t), and let y be the point in B(x, t) n E furthest from P. Set D = dist(y, P). If D ~ t, then

J

dist(x, P) dx ~ cd- 1 IE n B(y, D/2) I D

EnB(y,D/2)

~

c-1(t-1 D)d+1 ~ c-1 fioo(x, t)d+l_

If D ~ t, then fi1(x,2t) ~ C, and there's nothing to prove. PROPOSITION 5.5. Suppose that E satisfies (LS). For each f > 0 let Q(e) denote the set of cubes Q E ~ such that there is a d-plane Pq with the following two properties:

dist(x,Pq)

(5.6)

(5.7)

~

edia.mQ

ifw E Pq and dist(w,Q)

Let 8(E) =

~

~

for all x E 2Q;

diamQ, then dist(w,E)

~

ediamQ.

\ Q(e) denote the complement ofQ(e). Then

L

IQI ~ C( e)IRI

QEB(t) Q~R

for all R E

~

and all

f

> 0.

Thus the conclusion tells us that E is well-approximated by d-planes in a stronger sense than the weak geometric lemma. Notice that the conclusions of Proposition 5.5 imply (LS). To prove the proposition we shall first prove two lemmata. We shall also use these lemmata to prove that neither (Cl) nor (C2) can hold if d is not an integer, and so for their statements and proofs we allow d to be noninteger.

Fix Q E ~' and let (d) denote the smallest integer greater than or equal to d. Then tl1ere exist (d)+ 1 points Yo, ... , Y(d) in Q such that dist(yj,Lj-I) ~ A- 1 diamQ for j = l, ... ,(d), where Lk denotes the

LEMMA 5.8.

k-plane passing through y0 , .•• , Yk, and where A depends only on d and the constant in (1.5). This is obtained using an easy induction argument. Suppose we have found points yo, Yb ... ,Yj as above for some j <(d). Suppose that there does not exist a suitable point Yj+l• so that for ally E Q. Then it is not hard to see that we can cover Q by less than (A+ 1)i balls of radius l~n diam Q, say. From (1.5) we then get that IQI::; C(A

·

10n

+ 1)' ( A

d

diamQ )

This is impossible if A is large enough, because j <(d), and so y1 + 1 exists. This proves the lemma.

5.9. Let M be a large integer, and set k = 4(d + 1)M. Let R( r, k) be as in Remark 4. 7, and assume that Q ¢. R( r, k). If r is small enough (depending on M), then given an integer£, 1 ::; f::; (d)+ 1, £ + 1 points yo, ... , Yl of 2Q, and a 1 , ... , a1 E 21 Z n [-M, MJ, there is a point z E E such that LEMMA

t

I z- {Yo+ l:a;(y;- Yo)} I::;

C(M,f)rdiamQ.

i=l

We prove this by induction on f, beginning with £ = 1. For notational convenience we take y 0 = 0. The point of the proof is of course to use repeatedly the fact that Q fl. n( r, k ). Taking X = Y1 and y = 0 in the definition of R( r, k) we see that there is a point z2 E E such that lz2 - 2y 1 j ::; r diam Q. Since z2 E kQ we can take x = z2 and y = y 1 to get a point z 3 E E such that lza - 2z2 + y1 1 ::; r diam Q, so that lz3 - 3y 1 1 ::; 3r diam Q. Repeating this argument we see that for 2 ::; j ::; M there is a z j E E such that lzi- iYtl::; (2i-l- 1)rdiamQ. Taking x = 0 andy= Zj in the definition of n( r, k) gives a point L j E E with jz_j + iYtl ::; 2i-lr diam Q. That takes care of thee= 1 case, with C(M, 1) =2M-I_ Of course, we could also have done the same thing with y 1 replaced with any of the other yj's.

G. DAVID, S. SEMMES

Assume now that the lemma holds for some £, and let us prove it for £ + 1 (if£+ 1 ~(d)+ 1). Let ai, ... ,at+ 1 E 2t+Iz n (-M, M] be given. By induction hypothesis we can find w 1 , w2 E E so that l

(5.10)

I WI- L

1 2aiYi

I:S C(M,f)r diam Q

j=I

(5.11) If r is small enough, WI and w2 both lie in kQ, we can find a point z E E such that lz- 2wi + w2l-::; rdiamQ, so that from (5.10) and (5.11) we get l+I

I z- LaiYi 1::; C(M,f+ 1)rdiamQ, i=I

where C(M,£+ 1) = 2C(M,£) +2M-I+ 1. Let us now prove Proposition 5.5. For this we require d to be an integer again. Let t > 0 be given. Let M be large, to be chosen soon, depending on f, and put k = 4(d + 1)M again. We shall chooser after M, depending on both M and f. Fix Qo E 6. Choose Q ~ Qo so that diamQ"' fdiamQo, and assume that Q rt 'R.( r, k ). Let Yo, ... , Yd be as in Lemma 5.8, and take P = PQ 0 to be the d-plane that they span. If r is small enough, then (5.7) holds for Q 0 • Indeed, every point on P which is at distance -:::; diam Q 0 from Q 0 is at distance ::; diam Q from a point of the form d

Yo+ L ai(Y;- Yo), with a 1 E 2dz, lail::; cf- 1 • i=I

If M is large enough (M 2: Ce- 1 ), we can apply Lemma 5.9 to conclude that any such point is at distance ::; C ( M, d)r diam Q from E. If r is small enough, we see that every point of P which is at distance ::; diam Q0 from Qo is at distance ::; Cfdiam Qo from E, so that (5.7) holds for Q 0 with t replaced by C f. Let us now show that (5.6) holds for Qo if we assume also that Q 0 ~ 'R.( r, k ). Suppose not; let Yd+I E 2Q0 be such that dist(Yd+I, P) 2:: t diam Q 0 •

5. (I.S) IMPLIES THE WEAK GEOMErRIC LEMMA

Apply Lemma 5.9 to Qo and the points yj; 1 ::::; j ::::; d + 1, to conclude that for each a= (a 1 , ... ,ad+l), a; E 2d+ 1 Zn [-M,M], there is a z,. E E so d+l that if x,. =Yo+ I: a;(y; -Yo), then i=l

We want to use this to contradict the assumption that E is regular (1.5). If a f= a', then lx,. - x,.• I ~ tJE diam Qo. This is easily checked using the fact that for each j ::::; d + 1, Yj is at distance ~ tJE diam Q0 from the (j -1)-plane passing through y0 , ••• , Yj-l· By taking T to be small enough, depending on M and f, we get that

I z,.- z,., I ~ 2~fdiamQo

for a

i= a'.

Thus the balls B,. = B(z,., f. diam Q 0 /4C) are pairwise disjoint. From

(1.5) we get

,. and also lkQol::::; CMd(diamQ 0 )d. This is impossible if we choose M to be much larger than E-d. This contradiction tells us that Yd+l does not exist, and so (5.6) holds for Qo. Thus we have proved that Q0 E Q(CE) if Q0 ¢ 'R.( r, k), if the associated cube Q rt n( T, k), and if T, k are chosen correctly. Proposition 5.5 now follows from Remark 4.7. Let us now indicate why E cannot satisfy (Cl) or (C2) if its dimension d is not an integer. It is enough to show that E cannot satisfy (LS). Suppose it did. Let Q 0 be a cube with Q 0 ¢ 'R.( r, k ). We use an argument very similar to the one we just did, but with f.= 1, Q = Q 0 • Use Lemma 5.8 to select points y 0 , y 1 , ... , Y(d) in Q0 . Given a = (a1, ... , a(d)), a; E 2(dlz n [-M, MJ, let x,., z,., and B,. be as above (but with f. = 1). If T is small enough (depending on M), we obtain once again that the B,. 's are disjoint, and hence

,. while lkQol < CMd(diamQ 0)d. This is impossible if M is large enough and (d) i= d.

5.12: The weak geometric lemma is not strong enough to imply rectifiability. We shall give a counterexample in Section 20. However, a modification of an argument of Peter Jones (J2] shows that E has big pieces of Lipschitz graphs if it is regular, satisfies the weak geometric lemma, and has big projections. This last condition means that there is a (} > 0 so that for each x E E and R > 0 there is ad-plane P so that JII(E n B(x, R))J ~ ()Rd, where II is the orthogonal projection onto P. See [DS3] for more details. Although the weak geometric lemma is not strong enough by itself to imply recti:fiability, it seems to be a very useful intermediary condition. It certainly plays that role in proving that (C2) and (C3) imply (C4). It turns out that (LS) is equivalent to (C1)-(C7). We hope that this is as big a surprise for the reader as it was for the authors. The proof will be given in a separate publication. When d = 1 there is a direct construction that shows that (LS) implies (C7). This combines with the arguments given here to provide a much simpler proof of the fact that (C1) or (C2) imply (C7) when d = 1. The argument for showing that (LS) implies (C1)-(C7) when d > 1 is much less direct and it relies in particular on the fact that (C1) implies the other conditions. Thus it does not enable us to dispense with what we are doing here and in the succeeding sections. We conclude this section with an easy lemma which says that the good d-plane Pq in Proposition 5.5 is almost unique. REMARK

LEMMA 5.13. Let Q E .6. be given, and suppose that P 1 and P 2 are two d-planes such that dist(x, Pi):=:; ediam Q for all x E Q, i = 1, 2. Then

(5.14) (5.15)

for all wE P1

dist(w,P!) $ CediamQ+Cedist(w,Q)

for all wE P2.

In particular, Angle (P1, P2):::; Ce.

Let y0 , ••• , Yd be the points in Q provided by Lemma 5.8, and let P denote the d-plane passing through them. It is not hard to prove (5.14) and (5.15) by comparing each of P 1 and P 2 toP using the fact that dist(yj, Pi):=:; f diam Q for 0 $ j :::; d, j = 1, 2. [An important point is that this last fact not only implies that all the points of P are close to P;, but that the points of P; are also close to P. To see this it is useful to notice that if Zj,i are points in P; close to Yj, then the Zj,i generate P;.]

6. Approximation of E in measure

From Propositions 4.4 and 5.5 we know that if E satisfies ( C2) then most cubes can be well-approximated by d-planes in the sense of (5.6) and (5.7). This result can be strengthened, in that the Hausdorff measure on E will be well-approximated by a constant times Lebesgue measure on the d-plane. This is what we prove now. Let k be a large constant. We denote by 9( t:, k) the set of cubes Q E 6. such that there is a d-plane Pq with the properties

(6.1)

dist( x, Pq)

~

t: diam Q

for all x E kQ

{6.2) if wE Pq and dist(w, Q) ~ kdiamQ, then dist(w, E)~ t:diamQ.

If E satisfies the conclusion of Proposition 5.5, then 6.\ 9( ~::, k) satisfies the usual Carleson measure packing condition for all t:, k. This follows from the observation that if Q rt 9( t:, k) and if Q"' is an ancestor of Q such that kdiamQ ~ diamQ* ~ CkdiamQ, then Q* rt 9(~::/Ck), where 9(15) is as in the statement of Proposition 5.5. Given Q E 6. and ad-plane P, define a measure P,Q,P on P by

(6.3)

t-tq,p(A)

=I II- 1 (A) n (k/2)Q 1

for all (Borel sets) A ~ P, where II denotes the orthogonal projection onto P. In fancier language, 1-tQ,P is obtained by taking the restriction of Hausdorff measure onE to (k/2)Q and pushing it down toP using II. Let us call 11.( t:, k) the subset of 9( t:, k) consisting of those cubes Q such that for each d-plane P with Angle (P,Pq) ~ 110 there is a constant >..Q,P > 0 that satisfies

(6.4)

G. DA laD, S. SEMMES

for every cube A

~

P with

cdiamQ

::=;

k diamA:::; diamQ,dist(A,II(Q)):::; 10 diamQ.

6.5. Suppose that E satisfies (C2). Then for every 10 there is a C = C( c, k) such that

PROPOSITION

and k (6.6)

~

L

IQI :::; c IRI

for all R E

€

> 0

~-

Q¢1i(E,k)

Q<;R

By an easy covering argument one sees that ( 6.4) holds if (6.7) for every cube A ~ P such that dist(A,II(Q)) :::; (k/5)diamQ, A has sidelength exactly rydiamQ, and has sides parallel to a given (fixed) set of axes, provided that we choose 7J small enough with respect to € and then choose T small enough. To verify (6.7) it is certainly enough to show that (6.8)

I f.LQ,P(A)- f.LQ,P(A') I :::;

r( diam Q)d /2

whenever A, A' are two cubes in P that satisfy the properties just listed and also dist(A, A')~ diamQ. Let us prove the proposition. Let r 1 be much smaller than T, to be specified later. We must show that most cubes lie in 'H.( t:, k ), in the sense of (6.6); it is enough to show that most cubes in 9( T1, k) lie in 'H.( t:, k), by the remarks at the beginning of this section. Fix Q E g( T1, k) and ad-plane P with Angle ( P, Pq) :::; 110 . We want to find conditions that imply (6.8) and such that the cubes that don't satisfy these conditions satisfy a Carleson measure packing condition. Of course we want these conditions to be given in terms of something controlled by (C2). The argument we use is similar to the proof of Proposition 4.4. Let A, A' be given. Let p be the center of A and q the point of Pq such that II(q) = p, and define p' and q' similarly. By the version of (6.1) in this context, rr- 1 (A) n (k/2)Q is contained in the n-dimensional cube D centered at q whose projection is A. A similar result is true for A' and the corresponding n-cube D'. From the version of (6.2) in this context we get that there is a point x 0 E E at distance :::; r 1 diam Q from the midpoint between q and q'. Let j E Z be such that Q E ~ i, so that diam Q "' 21. We want to choose a function 1/J so that 2id'I/Jj(x 0 - y) looks a lot like the characteristic function of D minus the characteristic function of D'. More precisely, we ask 1/J to be odd, Coo, compactly supported, and we want it to satisfy

6. APPROXIMAT10N OF E IN MEASURE

(6.9a) tPj(Xo- y) = 2-jd

if dist(y,D) :$Crt diamQ

(6.9b) 0 :$ tPj(xo- y) :$ 2-d (6.9c) tPj(xo- y)

if dist(y,D) :$ 2Cr1 diamQ

= -2-jd if dist(y, D') :$ Cr1 diam Q

(6.9d) -2-jd :$ tPj(Xo- y) ~ 0 (6.9e) t/Jj(x 0 -y)=0

if dist(y, D') :$ 2Cr1 diam Q

if dist(y,DUD')2:2CrtdiamQ.

We can even find a (fixed) finite family cJl of functions so that we can find a 1/J E cJl with these properties no matter what Q, P, A, and A' are, subject to the constraints imposed above. (We allow ct> to depend on r 1 and all the other constants.) An argument like the one used in the proof of Proposition 4.4 can be used to show that if E satisfies (C2), then I 1/Jj(xo- y)dyl is as small as we

J

E

want except for a class C of Q's that satisfies a Carleson measure packing condition. (To be precise, I tPj(Xo- y)dyl should be as small as we want

J

E

for all x 0 that arise from admissible choices of P, A, A'.) Because of (6.9), we get that if Q rf. C, then 1 (6.10) IE n D I:$ IE n G' I +4r(diamQ)d (6.11)

IE n D' I:$ IE n G I +~r(diamQi,

where G is the cube concentric with D but whose sidelength has been increased by 4Cr1 diam Q, and similarly for G'. From chasing definitions we see that (6.8) holds if (6.10), (6.11), and (6.12)

IEn(G\F)I +

IEn(G'\F')I$~r(diamQ)d

do. It is not hard to see that (6.12) is true if Q E Q(r, k) and r1 is small enough: because En (G \F)~ {z: dist(z, Pq n (G \F)):$ 2rt(diam Q}, we can cover En ( G \F) by less than Cr1-(d-I) balls of radius 3rt diam Q, and then use the regularity of E (1.5) to prove (6.12). To summarize, we have shown that if Q E Q(r, k) and Q rf. C, where C satisfies a Carleson measure packing condition, then (6.10), (6.11), and (6.12) hold, and they imply (6.8), and hence that Q E 'H(E, k). This completes the proof of the proposition.

REMARK 6.13: In this proof, as in many places, we do not use the fact that we are dealing with d-dimensional Hausdorff measure onE; the same argument would work if we used any positive Borel measure supported on E and satisfying (1.5). (Of course such a measure must be comparable to Hausdorff measure.)

c-

1 ::;; >.P,Q ::;; 6.14: The constants AP,Q in (6.4) always satisfy C. This is easily seen by applying ( 6.4) to a cube of diameter ::;; C diam Q that contains TI( Q).

REMARK

7. Building the stopping-time regions, and some of their properties

In proving that (C2) or (C3) imply (C4) we shall use stopping-time arguments, and these arguments will be very similar to each other and ha.ve a substantial overlap. In this section we give a construction of the stopping-time regions that will be used in both cases. Let E be given. We assume that E satisfies the weak geometric lemma. Suppose that we are given ~:, h, 0 < f < h, both of which are as small as we want, and with h/t: large. Let k > 0 be large, to be chosen later (but not depending on for h). Suppose also that we are given a decomposition 6. = B U Q, where B satisfies (2.4), and for each cube Q E g there is a d-plane PQ such that (6.1) holds. The reader should keep in mind that for us the only important property tha.t B satisfies is (2.4). Thus we do not mind adding cubes to Bas long as (.2.4) is preserved. LEMMA 7.1. Under the preceeding assumptions we can find a new decomposition 6. = B' U Q', where g' ~ g and B' still satisfies (2.4), and where we can partition g' into a family :F of stopping-time regions S such that each S satisfies (2.5) and also:

(7.2)

ifQ E S, then Angle (PQ,PQ(s)):::; h;

(7.3)

if Q is a minimal cube of S, then at least one of the children of Q lies in B', or else Angle (PQ, PQ(S)) ~ h/2.

Before proving this - which is not difficult - let us make a few remarks about how this fits into the big picture. We want to prove eventually that if E satisfies (C2) or (C3), then it admits a corona decomposition. To

do this it is enough to show that for all sufficiently small ~, 8, the stoppingtime regions S provided by the lemma also satisfy (2.6) and (2.7). We shall see in the next section that (2.6) is always true under the circumstances of the lemma, while (2.7) is harder to prove. We shall give some preliminary reductions in that direction after proving the lemma. Let us prove the lemma. We start with a slightly simpler version of it. Given Ro E ~. let Q(Ro) denote the subset of g of cubes contained in R0 . We first show that we can partition Q(Ro) into a family F(Ro) of stopping-time regions with the above properties. This is easy, because there is pretty much only one way to do it. Let Q0 be an element of Q(Ro) of maximal size (i.e., Q0 E b..j for j as large as possible). It is easy to see that there is a subset S of Q(Ro) that has Qo as its maximal element and which satisfies (2.5), (7.2), and (7.3). (S can be built using the obvious stopping-time argument.) Remove S from Q(Ro) and repeat the process: pick an element of Q(Ro) with maximal size, and then build the associated stopping time region. Repeating this we get our partition F(Ro) of Q(Ro). For many purposes this localized version of the lemma is adequate, but it is not hard to prove the more global version either. To do this we need a sequence {Rj} of cubes which are pairwise disjoint, whose union is all of E, and which have the property that for each £ there are at most C cubes in b..t not contained in any of the Rj 's. Once we have this sequence of cubes we set 9' = UQ(Rj), B' = ~ \ Q', and F = UF(R1 ), and it is not hard to check that these choices satisfy the conclusions of the lemma. Let us indicate how to find such a sequence {Rj}- If E = Rd, it is easy to write down such a sequence of dyadic cubes explicitly, and the general construction is in a similar spirit. Fix a point Po E E. For each k 2: 0 consider the set of cubes in ~k which intersect B(p0 , 2k) or which have a brother that intersects it. (Two cubes in ~k are called brothers if they have the same father.) If we now take the union over k 2: 0 of the cubes so selected, we get a sequence of cubes which have the desired properties except for being pairwise disjoint. The minimal elements (with respect to inclusion) of this sequence gives a new sequence having all the desired features. That completes the proof of Lemma 7.1. We now give a preliminary reduction that is useful for checking (2.7). Give S E F, let m(S) denote the minimal cubes of S. Let m 0 (S) denote the set of minimal cubes of S which have at least one child in B',

and let m1(S) denote the Q E m(S) with Angle (Pq, PQ(S)) 2: 6/2. Thus m(S) = mo(S) U m1(S), by (7.3). Define Fo, F1, and F2 as follows: Fo = {S E F:

I U

Q

12

JQ(S)J/4}

QEmo(S)

fl = {S E F:

u

I

Q I 2 IQ(S)J/2}

QEm1(S)

F2 = {S E F:

I Q(S) \ (

u

Q) I 2 JQ(S)J/4}

QEm(S)

Clearly, then, F LEMMA

(7.5)

= Fo U F1

U F2.

7.4. Fori= 0, 2,

2::

JQ(S)J

for all R E ~-

< CJRJ

SE:F; Q(S)~R

Thus (2. 7) holds if we can prove (7.5) fori

= 1.

The case i = 2 easily follows from the fact that the sets Q( S) \ Q) are pairwise disjoint in E, whic~ is itself a consequence of the

( U

QEm(S)

pairwise disjointness of the S's and (2.5). The case i = 0 can be derived without difficulty from the requirement that 8' satisfy (2.4). Let us say a few words mor~ about how we'll show that (2.7) holds when E satisfies (C2). Similar ideas will be used for (C3). For each S E F we define a function d( x) on R n by (7.6)

d(x)

= QES inf {dist(x, Q) + diarn Q}.

This function encodes a lot of information about S, e.g., where its minimal cubes are very small. Also define an associated "summing region" u = u(S) ~EX Z by (7.7)

u = {(x,f) E EX Z: x E k 0 Q(S) and ')'od(x) ~ 21 ~ diarnQ(S)},

where the constants k0 (large) and 'Yo (small) will be chosen later. (They will not depend on e, 6. Also, k will be chosen after k0 , and will be much

larger.) One should think of a as an enlarged and smeared-up version of {( x, £) : x E Q for some Q E S n ~~}. Given an odd smooth function 1jJ with compact support, set

(7.8)

J(S,tjJ)

=

JL J

tPt(x- y)dy

I

(r,l)E,.

1

2

dx.

E

LEMMA 7.9. Suppose that we are given k, t, 6, ko, and ')'o, and let F be as in Lemma 7.1. To prove that (2.7) holds if E satisfies (C2), it suffices to find a finite family "Ill of 1jJ 's such that for some r > 0

(7.10)

L J(S,'lj;) ~ riQ(S)I

whenever S E F 1 .

1/JE'I!

Here r, "Ill are allowed to depend on all the constants above, but not on S. This follows from Lemma 7.4 once we show that for any

L

J(S,'lj;) :S GIRl

1/J,

for all R E ~­

Q(S)!:;R

This inequality is an immediate consequence of (C2) and the fact that E F, have bounded overlap in E X Z. This last fact comes from the disjointness of the S's and a little definition-chasing: if ( x, £) E a( S), then there is a Q E S such that

a( S), S

For any given (x,f), there are only a bounded number of cubes with these two properties, and so there are only a bounded number of S's with ( x, f) E

a(S). Thus to prove that (C2) implies (2.7) we want to show that if k0 is large enough and -y0 , f, 6, and t/6 are small enough, then we can find \ll such that (7.10) holds. The proof of this is complicated but the basic idea is fairly simple. We first show that there is a Lipschitz function A (on some d-plane) whose graph approximates Every well on the scale of d(x), with errors on the order of f. We use this approximation to push down estimates on J from E to the graph of A, proving eventually that the left side of (7.10) controls a square function applied to A, modulo terms that are small compared to 6, and that this square function controls the £ 2 mean

oscillation of 'VA. In other words, if the left side of (7.10) is very small, we show that '\7 A must be almost constant, in a way that is incompatible with S E :F1. {If S E :F1, then the graph of A cannot be too flat, by definition of :F1 and m1(S).) The proof that {2. 7) holds if ( C3) does will be very similar in structure, the main difference being that we have to work with a different kind of square function. REMARK 7.11: As we mentioned in Section 2, this kind of procedure will always produce a corona decomposition for E if there is one. That is, if E admits a corona decomposition, then it satisfies the weak geometric lemma, a.nd we can apply Lemma 7.1 (with g taken to be exactly·the set of Q's for which there is a PQ such that {6.1) holds) to obtain the good regions S E :F. The results of the next section imply that {2.6) holds, while Lemma 7.4 tells us that we need only check {7.5) for i = 1 to prove {2.7). This one can do using the assumption that E admits a corona decomposition, if the parameters are chosen correctly.

(The reason for this last assertion is that if Q E m 1 (S), and if E has a corona decomposition with 77 chosen small enough (depending on 6), then it is not hard to show that there must be a cube R such that Q ~ R ~ Q(S), and such that R is either a bad cube or a minimal or maximal cube for a. stopping-time region associated to the corona decomposition of E with constant 77-]

8. The construction of the approximating Lipschitz graph

Throughout this section we use the same assumptions and notations as in Section 7. Fix S E F as in Lemma 7.1, and set P = PQ(S)· Let p-L be a.n (n- d)-plane orthogonal toP, and let II and rr-L denote the orthogonal projections onto P and p-L. We shall often identify P with R d, and in particular we equip P with dyadic cubes. We denote by L the diameter of Q(S). In addition to the function d(x) defined in (7.6), we shall also use the function D defined on P by

(8.1)

D(p) =

inf

xEIT- 1 (P)

d(x) = inf {dist(p,II(Q)) + diamQ}. QES

8.2. Tbere is a Lipschitz function A : P C fJ such that

PROPOSITION ~

(8.3)

-+

p.l witb norm

dist(x,(II(x),A(II(x)))) :S Ct: d(x)

for all x E ko Q( S). Thus the points of k0 Q(S) are close to the graph of A. As with Lemma 7.1, we only need to know that E satisfies the weak geometric lemma for this proposition. It will be very important that we have an t: in (8.3) instead of a h. When we later try to push square function estimates from E down to the graph of A, we will need to know that the errors are small compared to 6, and (8.3) is one of the reasons why. Set Z = {z E E: d(z) = 0}, so that D(p) = 0 iff p E ll(Z). We first define A on II(Z). To do this, we have no choice: we must prove that IT is 1-1 on Z, and that its inverse is Lipschitz. We shall even have use for the following more general result.

LEMMA 8.4. Ifx, y E 10k0 Q(S) satisfy jx-yj2: 10- 3 min(d(x),d(y)), then

I IIJ.(x)- IIl.(y) I ::; Assume that

lx- Yl2:

26 I II(x)- II(y)

I.

10- 3 d(x). Let Q E S be such that

dist(x, Q)

+ diamQ::; Cjx- yj.

We can replace Q by one of its ancestors, if necessary, to get diam Q "' jx- yj. By our assumptions in Section 7, there is ad-plane Pq for which (6.1) holds, and so dist(x, Pq)

+ dist(y, Pq)::; CEix- Yl ~ 6jx- yj.

The lemma now follows from Angle (P, Pq) :S 6. From the lemma we see that

(8.5)

A(II(z)) = IIl.(z) for z E Z

defines a 26-Lipschitz function on II(Z). To define A on the rest of P we'll use arguments from the proof of the Whitney extension theorem (see [St ]), and in particular a variation of the Whitney decomposition of P \ II(Z). For each x E P with D( x) > 0 and x not on the boundary of a dyadic cube, let Rx be the largest dyadic cube in P containing x and satisfying

(8.6)

diamRx::; 20- 1 inf D(u.). uER.,

Let R;, i E I be a relabelling of the set of all these cubes Rx without repetition. Thus the R; 's are pairwise disjoint, they cover P \ II( Z), and they do not intersect II(Z). [Here we use the convention that dyadic cubes are closed but are called disjoint if their interiors are disjoint.] LEMMA 8.7. If lORn lORi =f 0, then

c- 1 diamR1 ::; diamR;::; CdiamRi· It is clearly sufficient to check that (8.8)

10 diam R; ::; D(y) ::; 60 diam R;

for all y E lOR;.

Because D is Lipschitz with norm 1, D(y) 2: min D(u)- 10diamR; 2: lOdiamR;. uER;

For the second inequality in (8.8) we use the fact that the father R of R; fails (8.6). Thus there is a z E R such that D( z) < 20 diam R = 40 diam R;, whence D(y) $ D(z) + 20diamR, $ 60diamR;. Let us proceed now to the construction of A on the hall Uo = p n B(II( x 0 ), 2koL), where xo is any fixed point of Q(S), and Lis still diam Q(S). Fbr future use we set for all j EN. We also restrict ourselves to the set Io of i E I for which R; meets U0 . Given i E I 0 , let us choose a cube Q(i) E S such that (8.10)

c- 1 diamR; $

diamQ(i) $ CdiamR;

and

dist(II(Q(i)),R,) $ CdiamR;. The existence of such a Q(i) is not a problem. If pis any point of R;, there is a cube Q E S such that dist(p,IT(Q)) + diamQ $ 2D(p) $ 120diamR;, by definition of D(p). We then take Q(i) to be a suitable ancestor of Q (possibly even Q itself). Note that a single cube Q may correspond in this wa.y to more than one (but not too many) R;. Let B; denote the affine function from P to p.l. whose graph is the d-plane Pq(i)· Because of (7.2), the Lipschitz norm of B; is $ 26. For each i let ~i be a C 2 bump function such that

(8.11)

0$ ~i :$ 1,~;

I "Vi~; I $

= 1 on 2R;,~, = 0 off 3R;,

C(diamR;)-i

and

for f = 1, 2.

Because of Lemma 8.7, there are, for each i, at most C cubes Rj with 3Rt n 3Rj =f. 0; in particular the supports of the ~; have hounded overlap. We can define a partition of unity for V = U 2R; by iElo

(8.12)

l/>;(p)

= J;(p) {

?= Jj(p)} ]Elo

-l

for p E V, i E Io.

Using Lemma 8. 7 again we have that (8.13) We define A on V by

A(p) = 2: tPi(p)B;(p).

(8.14)

iElo

Notice that V n II(Z) = 0 (by 8.8), and of course Uo \ II(Z) ~ V, so that (8.5) and (8.14) combined define A on U0 • Let us prove that A is Lipschitz with norm CfJ on Uo. We first check that the restriction of A to 2Rj, j E Io, is 3fJ-Lipschitz. Given p, q E 2Rj we have (8.15)

I A(p)-A(q) I::;

ll:
I

+ 12:{¢,(q)-¢;(p)}B;(q)

I

i

::; 2fJ I p- q I {2: ¢;(p)}+ 12: {tPi(P)- tPi(q)}{B;(q)- Bj(q)} i

I.

i

(In the last step we used the fact that E(¢;(p)- ¢;(q)) = 0.) If ¢;(p) =/= 0 i

or ¢;(q) =/= 0, Lemma 8.7 gives diamR1 "'diamRj, and then we get from (8.13) that (8.16)

I ¢;(p)-¢;(q) I::;

C(diamRj)- 1 lp-q 1-

To estimate B;(q)- Bj(q) we use the following lemma. LEMMA 8.17. If lORi

(8.18)

n lORj =/= 0, then dist(Q(i), Q(j))::; C diam Rj and

I B;(q)- Bj(q) I::; CEdiamRj

for all q E lOORj

To prove the first part, pick any x E Q(j) and y E Q( i). We may safely assume that lx - y I 2: t diam Q(j), and since d( x) ::; diam Q(j) by definition, we can apply Lemma 8.4 to obtain

I II.l(x)- II.L(y) I::; I II(x)- II(y) I· Because III(x )-II(y)l ::; C diam Rj by (8.10) (and Lemma 8.7), we conclude that dist (Q(i), Q(j))::; lx- Yi::; C diam Rj.

This implies that Q(j) ~ kQ(i) (if k is large enough), and (8.18) now follows at once from Lemma 5.13 (with Q = Q(j), P1 = PQ(j), P2 = PQ(i))· Combining (8.15), (8.16), and (8.18) we get IA(p)- A(q)l :S 2<5lp- ql

(8.19)

+ C(diam Rj)- 1 IP- qlt: diam Ri

:S 3t5jp- qj

for p, q E 2Rj,

if t:/<5 is small enough. We used the fact that there are a bounded number of i's for which c/Ji(p)- cp;(q) f= 0 for fixed p, q. Next, let us show that (8.20)

jA(p)- A(po)! :S Cblp- Pol

if PoE II(Z), p E

U Ri· jElo

Choose j so that p E Rj, and picky E Q{j). Thus we have IA(p)- A(po)l :S

a1

+ a2 + a3 + a4,

where

a1 =

IA(p)- Bj(P)I,

a2

= !Bj(P)- Bj(II(y))l, a3 = !Bj(II(y))- Ilj_(y)l,

a4

= IIIj_(y)- A(po)!.

and

Notice that D(p) :SIP-Pol, since D(po) = 0, and so diamRj :SIP-Pol· From the definition of A and Lemma 8.17 we get a1

:S Ct:diamRi :S Ct:!p- Pol-

Next, a2 :S 2t5lp-II(y)l :S CbdiantRj :S Cblp-pol, because of the 2t5Lipschitzness of Bj and (8.10). The definition of Bj gives a 3 :S 2 dist(y, PQ(j)) :S 2€ diam Q(j) :S Celp- Po ITo estimate a4 we apply Lemma 8.4 to x = (Po, A(po)) E Z and Y to get

(Here we have used (8.10) for the last inequality.) Combining these various estimates gives (8.20). Combining the fact that A is 2<5-Lipschitz on II(Z) with (8.19) and (8.20) it is easy to see that A is Lipschitz with norm :S Ct5 on Uo. We can use the Whitney extension theorem to extend A from U0 to a Lipschitz function on all of P with norm :S Cb. We now turn to the proof of (8.3). If x E Z, then x is on the graph of A, and there is nothing to prove. For the remaining case we'll use the following lemma, which will also be needed later.

LEMMA 8.21. Let p E U0 and r

> 0, D(p) ::; r ::; k0 L, be given, and let

Q E S be such that dist(p, TI(Q)) :::; Cr a!!d c- 1 r :::; diam Q:::; Cr. Then n- 1 (B(p, r)) n 2k0 Q(S) is contained in CoQ, where Co depends only on C and ko. As a consequence,

C0 1 d(x):::; D(ll(x)):::; d(x)

for all x E k 0 Q(S),

where Co depends only on ko.

To prove the lemma, pick X E Q and let y be any point in n- 1 (B(p, r ))n 2k0 Q(S). If lx- Yl:::; diamQ, then y E CoQ, as promised. Otherwise, we can use the fact that d( x) :::; diam Q to apply Lemma 8.4 to get ITI.L(x)- n.L(y)l:::; ITI(x)- TI(y)l:::;

c diamQ.

(The last inequality comes from our hypotheses concerning p, Q, and r.) The second affirmation of the lemma is obvious when ll(x) E TI(Z); apply Lemma 8.4, for instance, to see that n- 1 (TI(x)) n k0 Q(S) contains only one point. Otherwise, if TI(x) rf: TI(Z), take p = TI(x) and r = D(p). By definition of D(p) there is a Q as above. The first part of the lemma tells us that x E C0 Q, whence d(x) :::; C0 D(TI(x)). The other inequality follows from the definitions. Coming back to (8.3), let x E k0 Q(S) be such that d(x) > 0, and set p = ll( x ). The lemma tells us that D(p) > 0, and so p lies in some R,. Applying the lemma with r = D(p) and Q = Q(i), we get that x E C0 Q(i). We'll choose k to be much larger than Co, and so (6.1) gives ITI.L(x)- Bi(TI(x))l:::; 2ediamQ(i):::; CED(p):::; CEd(x), and since IBi(TI(x))- A(TI(x))l:::; CED(p), by Lemma 8.17 and the definition of A, we get ITI.L(x)- A(TI(x))l :::; CEd(x), as desired. This completes the proof of Proposition 8.2. We end this section with one more estimate on A. LEMMA 8.22. IV 2 A(u)l:::; CE(diamRj)- 1 ifu E 2Rj. Indeed, if 00 0p is any second partial derivative,

8a8pA = 8o8p(L ¢iBi) =

2: (8o8p¢i) B, + 2: (8o¢i)(8pBi) + 2: (8p¢i)(8oB;).

(Since B; is affine Oc.f)(JB;

OcrO(JA

=L

= 0.)

Because L, Ocrc/>;

00 0pc/>;(B;- Bj) +

= Oa(L, c/>;) = 0, we have

L Oac/>;(ofJB;- OfJBj)

+ L O(Jc/>;(oaBi- OaBj)· It is not hard to obtain the desired estimate from (8.13) and Lemma 8.17. (We also use the fact that IVB;- VBil ~ CE if u E supp(Vc/>;); this can be derived from Lemma 8.17, or proved using a very similar argument.)

9. Pushing square function estimates from E to the graph of A

In this section, and in the following two sections as well, we assume that E satisfies (C2), and we want to prove that E must then satisfy (C4). We follow the outline given at the end of Section 7, and we make the same assumptions and notations as in Sections 7 and 8. To implement the program described in Section 7 we must first specify the subsets Q and B of ~- We take Q = 'H( f, k ), as defined in Section 6, and B = ~ \ Q. As we said in Section 7, k is large and will be chosen later, but it will not depend on f, 6. Let J = J(S,.,P) be as in (7.8), with tjJ fixed. One of the things we have to do to carry out the program described in Section 7 is to transform an estimate on J into an estimate for a suitable square function applied to A, and it is this issue that we begin to take up now. Let p, denote the measure on P obtained by pushing down Hd lkoQ(S) using IT, i.e.,

p,(F)

= jn- 1 (F) n koQ(S)j.

Define I= I(S, tjJ) by

(9.1)

I=

j L (p,l)Eut

I

J

.,Pt((p,A(p))- (q,A(q)))dp,(q)

2

1

dp,

p

where the "summing region" a 1

~

P

X

Z is defined by

U4 is as in (8.9), L is still diam Q(S), C 1 > 1 will be chosen soon (large, and independent off, li, and -y0 ), and dp denotes Lebesgue measure on P.

PROPOSITION

9.3. There is a constant C

~

0, independent of li and f, such

that

+ Cf 2 IQ(S)I,

I:::; CJ

at least if we assume that supp Set at( x) =

J t/Jt( x -

t/J

~ B(O,

1 10

ko).

y )dy and

E

bt(x) =:

j 7/Jt((II(x), A(II(x)))- (II(y),A(II(y))))dy. E

We first want to replace at by bt in the integral that defines J, with only small errors. Actually, we shall even ask for less than that, namely,

J1 :::; 2J + Ct2 jQ(S)j,

(9.4) where

(9.5) (9.6)

u2

1 = {(x,i) E Ex Z: x E 2koQ(S), lod(x):::; 21

:::;

L}.

To see this, we first observe that for (x, i) E u 2 we have

J

lat(x)- bt(x)l:::; CIIVt/JIIoo

andy is in En B(x, 110 k0 21 ), then d(x) :::; ~~ 1 21 and d(y) :::; /0 ko2 1 :::; C21 . Thus we also have iat(x)- bt(x)l:::; Cf, whence

If (x,i) E

d(x)

+

t:(d(x) + d(y))2-l2-tddy.

u2

lat(x)- bt(x)l 2 Consequently,

J1 :::; 2J + 2

J

:::;

j j

J

L

(z,t)E 172

:::; 2J + Cf2

(d(x) + d(y))2-t2-udydx

EnB(z,fa ka2t)

L2- 1d(x)dx + Ct:2

(.,,t)Ea2

.$ 2J + Cf 2

(d(x) + d(y))2-t2-lddy.

L lbt(x)- at(xWdx

(z,t)Ea2

:::; 2J + Ct: 2

j

Cf2

j lkoQ(S)

dx

+ Cf 2

j l:aQ(S)

j

'2:

yEkoQ(S)

2t ~C-1 d(y)

dy:::; 2J +

C~:2 IQ(S)I,

2- 1d(y)dy

as desired. Next we want to push our integrals down to P. Because the points y that appear in the definition of bt( x) always lie in koQ( S) when ( x, i) E u2 we have that

bt(x)

(9.7)

=

J

t/Jt((p,A(p))- (q,A(q)))dp;(q),

p

where p

= IT( x ).

(9.8)

Let us check that

J2

=

j

L ict(P)I dp;(p):::; Jl, 2

(p,t)E"'s

where ct(P)

= ftflt((p,A(p))- (q,A(q)))dp;(q)

and

p

(9.9) (Thus u3 is a little larger than the region u 1 used to define I.) Because of (9. 7), we only have to check that if (p, i) E u 3 and x E 1 rr- (p) n k0 Q(S) then (x,i) E u 2 • Using (6.1) with Q = Q(S) we see that such an x must lie in ~k 0 Q(S). Also, if ~C1 is larger than the constant Co from Lemma 8.21, then d(x) :s; ~C1 D(p) :s; ~~ 1 2t, and so (x,i) E u2. This establishes (9.8). To finish the proof of Proposition 9.3 we have to be able to replace the measure dp;(p) in J2 by dp. Notice that dp;(p) 2: dp on IT(Z), whence

(9.10)

To control the rest we restrict our attention to each Ri for which R; nU4 and compare dp;(p) to dp on R;. Let I 4 = {i : R; n U4 f. 0}.

f. 0

LEMMA 9.11. Ifi E [ 4 and Tis a cube satisfying T ~lOR; and diam T ~ Mt:diamR; with M large enough (not depending on for 8), then

~

J :s; J dp

T

dp;(p):::;

T

c

J

dp.

T

Here C does not depend one or 6.

To prove the lemma we first apply Lemma 8.21 to any p E Ri with r = 60 diamRi and Q = Q(i). We get that rr- 1 (10Ri)nk 0 Q(S) is contained in CoQ(i) for some C0 • The restriction of f.1. to 10Ri is therefore the same as the restriction to 10Ri of the measure f.J.Q(i),P introduced in Section 6 (see (6.3)), at least if k > C0 • We chose our good set of cubes g to be 'H(e, k), so that (6.4) holds for each Q E Q, for Q = Q(i) in particular. We also pointed out in Remark 6.14 that the constant Ai = AQ(i),P satisfies c- 1 ::; Aj::; C. Lemma 9.11 follows immediately from (6.4) and definition-chasing. We want to use the lemma to control I in terms of J2. Partition Ri into dyadic cubes Ti,j such that

Me diam Ra ::; diam Ti ,j $ 2M e diam Ri, where M is as in the lemma. Let m denote the minimum of ict(P)i on T,,j. Straightforward estimates of the oscillation of ct(P) over Ti,j yield

J

ct(P)

I

12

dp $

T;.;

J[m + J

CediamR;2- 1 ] 2 dp

T;,i

::; Cm 2

df.J.(p)

+ Ce2 (diam Ri) 2 2-2t

nJ

J

< C

I

j

dp

nJ ct(P)

12

d~t(P) + Ce2 ( diam R;) 2 2- 21

T;,;

J

dp.

T;,;

Summing this in J we get

J

I

R;

ct(P)

12

dp::; C

J

I

R;

Ct(P)

12

df.l.(P)

+ Ce2 (diamR;) 2 2-2l

J

dp.

R;

Before summing this in i we need some notation and an observation. Let .C( i) denote the set of R's such that (p, £) E a 1 for some p E R;. If R E .C(i), then 21 2: C1'YoD(p) 2: tcl'YoD(p') for any other p' E Ri, and

(p',f) E a3. Hence

J

I ct(P) 12 dp ~

2:

I ct(P) 12 dp

iEl4 lEL.(•) R;

(p,l)E 111 pEP\II(Z)

~CL

L L j

L j

I

ct(P)

2

1

di-'(P)

+ Cf 2

iEl4 lEL.(i) R;

~c

JL

I

Ct(P)

L L iEl4 lEL.(i)

2

1

(p,t)E.,.3

dj.~(p) + Cf 2 L

2-2l(diamR;) 2

j dp R;

J

dp

iEl4 R;

< CJ2 + Cf2 IQ(S)I. (For the second to last inequality we used (8.8).) '

This last estimate, combined with (9.10), (9.8), and (9.4), finishes the

proof of Proposition 9.3.

10. Controlling a square function of A in terms of J(S, 'ljJ)

According to the program outlined at the end of Section 7, we want ~show that for suitable choices off, 6, etc., we can find a finite family ~·of '1/J's so that (7.10) holds. To do this we want to show that we can ·cxmtrol square functions of A in terms of J(S,'IjJ), modulo certain types Of errors. Proposition 9.3 was a first step in this direction, but we need 'aomething better. The problem is with I in (9.1), in particular its nonlinear dependence on A and the appearance of dp( q) instead of dq. In this section we show that if W is chosen properly we can indeed control a more useful aquare function of A in terms of L J( '1/J, S) . .PEiTI

We continue to use the same assumptions and notations as in the preceeding 3 sections. Let us now choose the class W that we shall work with. Let Po and Pl denote the translates of P and p.l_ that pass through ~he origin. Pick 11 E C;:"(Po) which is radial, not identically zero, supported in B(O, 210 ), zero on a neighborhood of the origin, and satisfies

f

11(p)f(p)dp = 0

Po

for all polynomials f of degree S 2. We shall sometimes commit the following minor abuse of notation. Given a function on P0 , such as 11, and a function on P, such as A, we note by 11 *A the function on P defined by

J

11(p- q)A(q)dq.

p

To simplify notations we allow our '1/J's to be vector-valued. Write the generic element of Rn as (p, w ), p E P0 , w E Pl. For our first '1/J we take any odd, function with compact support and values in and which satisfies

Pi-,

coo

'1/J(p,w) = v(p)w

(10.1)

for all (p, w) such that

lwl S iPi·

This function will give us control on A, but we also need a function to give us control on the measure dp.. For our second ¢ we take any function with values in P which is odd, supported in B(O, 1 ), and satisfies

coo,

'1/J(p, w)

(10.2)

= v(p)p

for all (p, w) with

lwl S IPI·

We can make these choices in such a way that the family of all '1/J's that arise is finite, by making sure that the set of all P 0 's that arise is finite. This we can do, in a way that depends on E. (We could also use a slightly different approach in which the family of '1/J's is much smaller. This would complicate further the notations and presentation of our argument, but it would not present any serious problems.) We take for our family W the set of '1/J's just described (in (10.1) and (10.2)) as well as some of their dilates. That is, we also take the functions sid¢(six), ¢as above, where s = 2~, j = 1, 2, ... ,m 0 -1, where mo is a large integer to be chosen. (It will not depend on E or any of the other constants; just n, d, and the choices of '1/J's above.) We denote these functions by ¢i ( x ), and we take ¢: ( x) = 2-ld¢1 ( ¥) as always. Define vi, similarly. As we said before, we want to control a square function of A in terms J( S, ¢1), ¢1 E \}1. Set

v:

If= If(S) =

(10.3)

j L2(p,l)Etr4

21 \

f

vj(p- q)A(q)dq \2 dp,

p

where u4 (10.4)

Here 8

= {(p, £)

EP

X

Z : dist(p, II( Q( S)) S 100£

1J D(p) S 2t S 8L }.

> 0 is

a small constant to be chosen later (before f).

and

PROPOSITION

10.5. Assume that

(10.6) for all .,Pi E llt. If E, li, 'Yo, 8, and k01 are small enough (how small E has to be will depend on the other constants), then for j = 0, ... , m 0 - 1,

(10.7) where

c,

C' do not depend on

f.,

li, ore, and C' does not depend on 'YO·

We should perhaps point out that this proposition also depends on m 0 being sufficiently large, but this is not important because our choice of m 0 ,pn not depend on the other constants. We shall prove Proposition 10.5 in this section, and use it to show that (? .10) holds (with T = E2 ) in the next section. Suppose that (10.6) holds, so that Proposition 9.3 can be applied to qonclude that

(10.8) for each .,pi E 111. Let us write down explicitly what that means. For .,P as fu (10.1), (10.8) becomes

(10.9)

j L j si2- vj(p- q)(A(p)- A(q))d~(q) 1

I

(p,t)Eul

2 1

dp

~ Ct=.2 !Q(S)!,

p

while for .,P as in (10.2) we get

(10.10)

j

L I v1 * ~(p) 12 dp ~ CE21Q(S)I,

(p,l)Eut

where ii(p) = v(p )p. In both cases j runs from 0 to m 0 - 1. To derive (10.7) from (10.9) we want to replace d~(q) in (10.9) by dq with only acceptable errors incurred. To do this we shall use (10.10) to show that d~ can be approximated by dq. We shall also need the following lemma to help control the errors.

d~'l (1 ::::; r ::::; oo if d = 1), 0. For each p E P and f E Z there is an affine function Ap,t : P --+ p.l.

LEMMA 10.11. Let r, M be given, 1 ::::; r

<

M > such that

2

12_, (A(q)- A,,,(q)JI" dq}. dpd6,.(t)

(10.12)

is a Carleson measure on P course::::; C(M)6 2 ).

X

R+ with nonn:::; C(M)II\7 All~ (which is of

See (Do] for a proof (or [Jl] ford= 1). In (Do] the case of \7 A E L 00 is not discussed, but this lemma follows immediately frotn the results for \7A E £ 2 • Notice that the lemma is still true if we also require that Ap,t be independent of p for p inside a dyadic cube with sidelength 21 . This is not hard to derive from the lemma. [One way to do this is to observe that if M ~ 2..;J, Q is any dyadic cube in P with sidelength 21 , and if a is an affine function so that IQI- 1 JIA- al is as small as possible, then for all p E Q we have Q (10.13)

sup !Ap,lQ

al :::; C2-ld

J

lA- Ap,tl·

PnB(p,M2 1 )

Thus we can replace Ap,l by a without changing much.] From now on we assume that Ap,t has this extra property. There are two easy consequences of this observation that we shall use. The first is that (10.14)

L J2-t [A(p)- Ap,t(P)]\

2

dpd62t(t)

t

is a Carleson measure on P X~- (Taker= 2 in (10.12) and use Fubini.) The second is that Ap,t satisfies (10.15) We shall now use (10.10) to replace A in (10.9) by A-Ap,t· Afterwards we use Lemma 10.11 to control the errors that arise when we replace dp. by something more convenient.

LEMMA 10.16. For j

= 0,

1, ... ,mo -1,

1 L IJ

si2-lvJ(p- q) [Ap,t(P)- Ap,t(q)] dfL(q) j 2 dp:::; Ce:2 jQ(S)j.

(p,t)Eut

p

Indeed, because Ap,l( q) is an affine function,

J

si2- 1vf(p- q)(Ap,t(P)- Ap,t(q)) dfl(q)

p

'it~,

for each p, f, a linear combination of integrals of the form

j si2- vJ(p- q)(p;- q;)dfl(q), 1

p

with coefficients that are dominated by CfJ (because of (10.15)). Here p; oenotes the ith component of p with respect to some basis. This integral is j~t the i'h component of iif * fL(p), and so the lemma follows from (10.10). Combining Lemma 10.16 with (10.9) yields

00.17)

(10.18) The advantage of (10.17) over (10.9) is that we have better control over .ap,t(q) (coming from Lemma 10.11) than A(p)- A(q), and this will be ~ded when we try to replace dfL( q) by something better. To analyze 11 using (10.10) we first need to understand the operator

Here we include a scalar product in the convolution iij * iiJ. (Remember that i1 is vector-valued.) Let Tl(e) be the Fourier transform of il * il. · Because each component of il is real-valued and odd, each component of b is imaginary, and so 11(0 :::; 0 everywhere. Also, 17 is radial, because

v is radial and ii(p) = v(p)p. Since we required that v have two vanishing moments we have that 171( ~)I ~ C 1~1 2 • These properties (and the fact that

71 is rapidly decreasing) imply that

is a finite negative constant. Set

..\(0

.

mo-l

oo

.

L: E

=logs

A

7J(s- 1 21C), so that (Tf)" =..\f. We can

j=O l=-oo

write..\ as

,\(~) =

2: :L 1

mo-l

s-i+ 1 2 1

oo

j=O l=--oo

8 -j

dt 71(s-j2i~)-. t

2l

As m 0 gets large, ..\( ~) tends to J000 77( t~) ~t, and so our previous remarks now make it clear that there is a constant a > 0 so that

1

3

-a<-..\< -a 2 -2

if m 0 is large enough. (Remember that s = 2~ .) We shall assume from now on that mo is sufficiently large for this to happen. Thus T is invert1 /,and y-t can be written as a convolution singular ible, (T- 1 (!))" = integral operator such that the ph derivatives of its kernel are dominated by iPi-d-i. Set ( = T- 1 (ii), so that (is also vector-valued. We have that

.x-

(a) IVi((p)l ~ C(j)(1 (10.20)

(b)

J

((p)a(p)dp

=0

+ ipi)-d- 2-j,

j

= 0, 1, ...

for all affine functions a(p).

These properties of ( follow from the fact that ii is smooth, compactly supported, and satisfies (b), and the estimates on the kernel of y-t noted above. By definitions we have that mo-l

(10.21)

J.t

= logs

oo

2: 2: <: * v1 * J.t, j=-0 t=-oo

where (f(p) = sid2-id((si2- 1p), and where the series converges in the sense of distributions. [Again this includes a scalar product in <1 * vf.J We

are going to use this formula to split p. into three pieces, of which one is smooth and the other two will produce contributions to (10.17) that can be controlled. The remaining part of (10.17) (coming from the smooth piece) will be used to control / 1 and thereby prove (10.7). Let Rj, j E I, be the family of cubes from Section 8, with the corresponding partition of unity {¢>1 } of P \ TI(Z). Let us write fL = f + g + h, where mo-l

f(p)=logs

L

:L<;*v;*p.(p)

j=O 2 1 >L

(10.22)

mo-l

j (Hp-w)vl *P.(w)dw,

L L

+logs

J=O 2'"S;L P\U4

mo-l

~10.23)

L

g(p)=logs

j=O

mo-l

h(p)=logs

{10.24)

L

j=O

j

"L,(J(p-w)vj*p.(w)dw,

(w.t)Eat

j (w.t)

'L,
For the definition of h we have used e to denote the set {( w, £) E P X Z : ·wE u4, 21 < CnoD(w )}, where c] is as in the definition (9.2) of Ut. These sums should be viewed as converging in the sense of distributions. We want to replace the dv(q) in (10.17) by f(q)dq, and so we have to .control the corresponding contributions of g and h. The main point for the g term is that (10.10) allows us to control the £ 2 norm of g. LEMMA 10.25.

J g(p) 2 dp:::; Ce 2 IQ(S)I, with c independent off, 6, and e.

p

in

This is proved using the usual duality argument. Let F be any function P). Then

L2 (

I

J

Fg

p

mo-l

I : :;

clogs

L

j=O

j L I d * F( w) I I vj * JL( w) I dw. (w.t)Eu,

Using (10.10) and Cauchy-Schwarz we can dominate this by 1

2 CE IQ(S)lt (logs mfl 2: J (f * F 2dw) :5 CE IQ(S)It IIFik I

)=0

l

1

p

The last inequality is a standard square function estimate, and in this case it can be derived simply from Pla.ncherel. This proves the lemma. Next we use this to control the contribution of g to (10.17). LEMMA

10.26.

Jf J

o:;,l(q)g(q)dq

I

p

2

1

dp < Cf 2 IQ(S)I,

p

l=-oo

where C does not depend on

f,

6, or 8.

Let r E (2, d~2 ) be arbitrary, let r' denote its conjugate exponent, and set

1

G!(p) = {/lu!(P- q)[[g(q)( dq}.,, (10.27)

Ff(p)

=I T 1 [A(p)- Ap,t(P)] j,

r 1

(10.28)

Hi (p) = {/lui (p- q)[ [2-' [A(q) -

A,,,(q)]l'

dq

By Holder's inequality, the quantity to be estimated is at most

(10.29) We know from (10.12) and (10.14) that

(10.30)

I: (Ff(p) l

2

+ Hf(p) 2 )

dpd6 2 t(t)

is a Carleson measure with norm ::5 CfJ 2. Because r' < 2, we can control the L 2 norm of the maximal function

by the L2 norm of g, and so Carleson's inequality (see p236 of [St]) and Lemma 10.25 imply that (10.29) is ::5 Ce2fJ 2IQ(S)I. Lemma 10.26 follows. Next we want to control the contribution of h to (10.17). Set a 5 =

{(p,£) E P x Z: p E U4, '"YJ D(p) ::5 21 ::5 L}. (Recall that Ui was defined in (8.9).) LEMMA

10.31. There is a C independent of e, fJ, '"'fo, and () sucb tbat

J L I Ja~,t(q)h(q)dq (p,l)Eas

12 dp ::5 C'"'(ofJ21Q(S)I.

p

The notation h(q)dq is convenient but somewhat misleading, since h is a priori only a distribution and at best a measure. The idea behind the lemma is that although h is not very regular, it has a lot of local oscillation to help its integral against a!,t (which is relatively smooth) to be pretty small. Let us record a couple of simple estimates on a~,t· It is readily seen from the definition (10.18) of a~,l (and also (10.15)) that (10.32) and also that supp a~,t ~ B(p, 21 ). Set a = aip, 1 = pJ api ,1 ( q)h( q)dq for (p, P) E as. Thus mo-l

a= logs

?: t=O

J LJ

a!,t(q)(l(q- w)dq

(w,t)

Ee

iil * p(w)dw

p

by (10.24). We need to get estimates for the interior integral AZ

= AZ(j,p,P,i,k,w) = faip, iq)(L(q- w)dq. p

When

k;::: for Jp- wJ;::: 2t+ 1 we have jAZj :5 CHt(p)
(10.33) where

J

Ht(P) = 2-td

2- 1 jA(q)- Ap,t(q)Jdq

+

2-1 JA(p)- Ap,t(P)J

B(p,2l)

and IP~:,t(x) = 22 1:(2" Jp- wj :S 21+1 then

+ 21 + jxl)-d- 2 • This uses (10.20a). If k :S .e and JAZj :5 Ch'2"2-t(d+tl.

(10.34)

This uses the bound (10.32) on \i'a~,t and both parts of (10.20). Let AZ1 (p, £, k, w) denote the right side of (10.33) when k ;::: for jp- wJ ;=:: 2t+ 1 , and let it be zero otherwise, and let AZ2 (p, £, k, w) denote the right side of (10.34) when k :S £, Jp- wj :S 2t+ 1 , and zero otherwise. Set e1 = {(w,f) E e: k;::: for jp- wj;::: 2l+ 1} and e2 = {(w,f) E e: k < f and Jp- wJ < 2l+ 1 }, and split a into a(1) + a(2) accordingly. Thus a(1,2) :5 C

J

L

AZ(I,2 )(p,f,k,w)2-kd

J d~-t(r)

dw.

B(w,21:)

(w,1o)Ee(l,2)

We want to show that

JL

(10.35)

Ja(1,2Wdp :5 Cf'0 62 JQ(S)J.

(p,l)Ea5

We start with a(1). By definitions a(1) :5 CHt(P)

J

L

(w,I:)Eel

J

2"( 2 -d)(2"

+ 21 + Jp- wl)-d- 2 d~-t(r)

dw.

B(w,21:)

Ifr E B(w,2k) and (w,k) E e1 , then 21: < Cn0 D(w) [see the definition of e just after (10.24)] and D(w) :S Jr- wj + D(r), and so D(w) :5 2D(r) if 'Yo is small enough. Thus 21: < 2C1 'YoD(r). Also, (w, k) E e1 implies that 21 :S 21: or 21+1 :5 Jp- w J and this implies that 21: ~ 21 or jp- rJ ;=:: 21 when Jr- wJ :5 21:. Thus (w, k) E e1 and Jw- rJ :5 21: imply that (r, k) E e1 ,

el

=

{(r, k) E p X z : r E u3, 2" < 2CnoD(r), and either 21 2: 21 or

jp- rl 2: 21 }, whence a(1)

~ CHt(P)

j 4= j k

IP- rl)-d- 2 dw

dl-l(r)

B(r,2k)

(r, )Eel

~ CHt(p)

2l:(Z-d)(2" + 21 +

j L 2 "(2" + 2 + jp- rl)-d-zdl-l(r). 2

1

(r,k)Eel

To control this expression it will be convenient to break up the rintegral. Let {Ri} be the cubes chosen in Section 8, which cover the set {q E P: D(q) > 0}. Let /3 denote the set of i such that R; n U3 =/= 0. Then a(1)

j L

~ CHt(P) L iEls

Consider

22 "(2"

+ 21 + IP- rl)-d- 2 dl-l(r).

(r,l:)Eel rER;

J(2" + 2£ + \p- rl)-d-Zdl-l(r).

If p ~ 2R;, then the integrand

R;

'is roughly constant on Ri, and this integral is comparable to the one you .get by replacing dl-l( r) by dr, because of Lemma 9.11. If p E 2R;, then 1

lOdiamR; ~ D(p), by (8.8), and of course 21 2: 'YJ D(p) if (p,f) E a 5 . 'From Lemma 9.11 we get that 1-l(T) ~ ITI for subcubes T of R; of size 2t, at least iff is small enough, depending on 'YO· [N.B.: Although we have to let the choice off depend on 'Yo here, the constant from Lemma 9.11 doesn't depend on anything, not for 'Yo in particular.] Because (2"+2 1 +\p-r\)-d- 2 is roughly constant on the scale of 21 , we get that /(2" + 2l + IP- rl)-d-2dl-l(r) R;

~ /(2" + 2l + IP- rl)-d-2dr R;

in this case as well. Let us use this to control a(1). If (r,k) E C'Yo diam R; by (8.8), and so

a(!) :'0 C Ht(P)

~

~ CHt(P) L

1(,. ~cf.•mR; 2")

e1 ,

r E R;, then 2" <

(2 1 + [p- rl) -d-2 dr

j 'Y~(diamR;? (2t + IP- rl) -d- dr.

iEl3 R;

2

Let Ft(P) denote this last expression without the CHt(P) when (p,£) E a 5 , and Ft(P) = 0 otherwise. To check (10.35) in this case we want to use Carleson's inequality and the fact that E IHt(p)i2dpd6 2 t(t) is a Carleson l

measure with norm :S C6 2 (which comes from (10.12) and (10.14)). Thus we need to control the L 2 norm of

F.,(p) ==

sup

Ft(q).

q,t

lq-pl9 1

The definition of F clearly gives F.(p) :S C sup Ft(p). Because 2t ~ l 1

"Yi D(p) if (p, f) E a

5,

we have

diamR; :S D(r) :S D(p) + IP- rj :S "Y;t(2t

+ IP- rl)

when r E R;,

so that

F.(p) :S C

L j 7~ diam R;('Y~ diam R; + IP- rj)-d- dr. 2

iE/3 R;

To estimate the L 2 norm of F., we integrate it against an arbitrary L 2 function G and observe that

j F.IGI

:S C

L j 'YoG"'(r)dr, iE/3 R;

where G*(r) denotes the Hardy-Littlewood maximal function of G. Of course the right side is at most C-yojQ(S)ji 1lGl12, and so JF; :S C-y51Q(S)I. Thus (10.35) does follow in this case from Carleson's inequality. Now we want to prove (10.35) for a(2). We have

a(2) :S C

J

L .52k2-t(d+l)2-kd j

(w,l)Ee2

d~J(r) dw.

B(w,2k)

If lw- rl < 2k and (w, k) E e, then D(w)

1

:S D(r) + lw- rl :S D(r) +2k :S D(r) + 2D(w)

if 'Yo is small enough. Thus D(w):::; 2D(r), and so 2._:::; 2C1 -y0 D(r). Hence

I

a(2):::; C

L

wEB(p,2 1+1 )

I

:::;c

62k2-t(d+I)2-kd

2k ~2CnoD(r)

I

dJ.l(r) dw

B(w,2k)

k9

L

.52kTi(d+I)dJ.l(r)

2k ~2C 1 ")'oD(r)

B(p,2 1 + 2 )

J

$C

As before we want to replace dJ.l( r) by dr, and so we decompose

B(p,2H 2 ) using the R;'s. Using (8.8) we obtain a(2)

::S

J

CL

6-ya(diamRi)2-t(d+I)dJ.l(r).

B(p,2 1 +2 )nR;

'

.lf B(p, 21+ 2 ) n Ri # 0, then diamRi

::S iJ?; D(r):::; D(p) + dist(p,Ri) ::S D(p) + 21+ 2 ::S 'i'~t2t+ 1

when (p,£) E a 5 • In this case we obtain from Lemma 9.11 that J.t(B(p,2'+ 2 )

n R;):::; GIB(p, 2~+ 2 ) n 2R,I,

at least if e is small enough. (As before, how small ;o, but the constant doesn't.] This gives a(2)

::S

C

j

~ 1

has to be depends on

6-y0 (diamR;)Tl(d+lldr

B(p,2 1 +2 )n2R;

j

::S C

E

6-yaD(r)2-t(d+I)dr.

B(p,2t+2)

(We have used the fact that the 2R; 's have bounded overlap.) Since D(r)

::S D(p)+ jp-rj ::S ..,;t 21+ 1 if (p, £) E a5 and r E B(p, 2l+ 2 ), 1

We conclude that in particular a(2) :::; Cti-yl. Hence a(2) 2

::S C6 2 -y}

J B(p,2t+2)

D(1·)2-t(d+I)d1·.

Let us plug this into the left side of (10.35). We want to use Fubini to integrate in r last. Because (p, f) E as and r E B(p, 2l+ 2 ) imply that

D( r) ,:S 'Y; 121+1 and r E U3 (since 21 ,:S L ), we get that

j

I:a(2) 2 ,:S Cc5 2 -r!

J U

(p,l)E
a

.:S Cc52 -r!

J

'2:

2-i(d+l)dp D(r)dr

~

2t+l~i'o D(r) B(r,2t+2)

j 2:::

2- 1 D(r)dr .:S C8 2 -roiQ(S)I-

ua 2t+1~i'! D(r) This completes the proof of Lemma 10.31. Combining (10.17) with Lemmas 10.26 and 10.31 we conclude that

J Jo~,l(q)f(q)dq

( 10.36)

I

(p,l)Eus

12

dp .:S (C€ 2

+ C'-rac5 2 )IQ(S)!,

p

with C, C' independent of €, 6, and 9, and C' also independent of 'Yo- We now show that f is smooth on Us, and then look at what happens when you replace f(q) by f(p). Let us check that f satisfies

If! .:S c, IVJI .:S cL- 1

(10.37)

on Us.

The first term on the right side in (10.22) clearly satisfies these conditions. (Don't forget that p. has total mass ,:S CIQ(S)j.) It is not hard to show that the second term also has these properties, using (10.20a)~ If p E U6, 21 ,:S L, then we get

I j o~,i(q)[f(p)- f(q)]dq I .:S

C82 1 L- 1

p

from (10.37), (10.32), and supp o~,i ~ B(p, 2l). Because a 4 (defined in (10.4)) is contained in as n {p E U6} if ko is large enough, we conclude from (10.36) that

j '2: (p,t)Eu4

I

Jo~, 1 (q)dq

2 1

lf(p)l 2 dp .:S (Cf 2 + C'-yac5 2 )IQ(S)I

p

+C

j 2::: 8 2

2 21 L- 2 dp

(p,l)€0"4

.:S [C€ 2 + C'('Yo + 92 )8 2 ]IQ(S)I,

where C' is independent of ")'a, and both C, C' are independent off, h, and (}. This reduces to (10.38)

j

L 2-

21

I

J

vJ(p- q)A(q)dq

2

1

lf(p)l 2 dp

p

(p,t)Eu4

:::; [Cf 2 + C'("Yo + 82)h 2]IQ(S)I, with C, C' as above, because of the definition (10.18) of Q~,t and the vanishing moments assumptions on v. Comparing this with (10.3) we see that Proposition (10.5) will follow if we can show that there is a c2 > 0 so that (10.39)

f(p) ~ C2 1 for all pEP with dist(p, TI(Q(S))):::; 100£.

Fix p. It suffices to show that there is a nonnegative smooth function 17 with integral 1, support contained in the ball with center p and radius

{ C2

(s~; l\7fl + L -

1)} -

1 ,

j

and which satisfies

TJ(q)f(q)dq

~ 202 1 -

Because of (10.37) we can choose TJ so that IITJIIoo :::; C(C2L- 1)d, U'V111loo ::S C(C2L- 1 )d+ 1 • Since d11-(q) is equivalent to dq at the scale of tdiamQ(S) (by (6.4), or Lemma 9.11, applied to Q(S)), we get that

j TJ(q)dll(q) ~ 4C2

1

if c2 is large enough and f is small enough ( f depending on c2, but not vice-versa). Thus it is enough to check that (10.40)

I

J

TJ(q)(g(q) + h(q))dq

I:::; 202 1 •

The contribution of g to (10.40) is certainly less than C2 1 if f is small enough, by Lemma 10.25. For the contribution of h we use the smoothness of 17 and the localization and cancellation properties (10.20) of(:

I j TJ(q)h(q)dq I : :;

mo-l

j L 1(/ * TJ(w)liv/ * 11-(w)ldw ::S C(C2)logs L j L2 lvJ * /1-(w)ldw logs

L

j=O

(w,l)

Ee

m 0 -l

1 L-d- 1

j=O

::S C(C2)

L 21 :5 Cno L

(w,t)

Ee

21L-d-l

J

d11-(r):::; C(C2ho·

Ua

[Remember that e was defined just after (10.24). We have used Fubini in the third inequality.] This is :::; C2 1 if 'Yo is small enough. [It is not hard to compute how C(C2 ) depends on C2 , but we don't need to know the answer.] This proves (10.40), and finishes the proof of Proposition 10.5.

11. The end of the proof that (C2) implies (C4)

To finish the proof we have to show that (2. 7) is true. As we have said before (see Lemma 7.9 and Proposition 10.5), it suffices to show that ~(10.6) implies that S ~ :F1 , which we do in this section. [Recall that :F1 ·\vas defined just before Lemma 7.4.] The idea is to use (10.7) to control ,.he oscillation of V' A in a way that is incompatible with the lower bound on the oscillation of V' A forced by S E :F1 • Assume that (10.6) holds. By Proposition 10.5,

JL

2- 21

I

vi* A(p) 12 dp :s; riQ(S)I

(p,t)Ea~

for j = 0, ... , m 0 - 1, where T = CE2 + C'( 'Yo+ 82 )6 2 , with C, C' as in the proposition. On the other hand, using Lemma 8.22 it is not hard to check tha.t

I

L

2-21' I

v: * A(p) 1 dp :s; CE iQ(S)I 2

2

Uo 2' ~"Yl D(p) l.

for a 1 1 which is much larger than 'Yo2 ( 'Y 1 = (lOOn )- 1 would be O.K.). [One wa.y to do this is to observe that A(p) (A- Ap(p)), where Ap is ~he linear Taylor approximation to A at p, and then estimate brutally using Lemma 8.22.] Thus if we set V = {x E P: dist(x, II(Q(S))) :s; lOOL}, then we can conclude that

vi*

(11.1)

IL V

2 1 ~6L

for j = 0, ... , mo - 1.

2-U

J

vj * A(p)

=vi*

J

2

dp

:s; 2riQ(S)I

vJ's.

To use this estimate we need a reproducing formula involving the As in the discussion between (10.19) and (10.20), if mo is large enough then we can find a function a on P0 that satisfies (10.20) and also mo-l

oo

j=O

l=-oo

f = logs

L L

aJ

* vJ * f

for all functions fin L 2 (P), say, where a~ = sid2-lda(s12- 1p). Of course the convergence is taken in the sense of distributions. In particular we can apply this to the function A. [In dealing with convergence issues it is helpful to observe that we can choose A to have compact support.] We decompose A into two pieces, A= A1 + A2, by mo-l

L L

A1 (p) =logs

j=.O

(11.2)

mo-l

~

+logs

J=O

mo-l

(11.3)

A2(p)=logs

~

J=O

aJ

* vJ * A(p)

2 1 >8L

L

L 21

j

a:(p- q)vJ

* A(q)dq,

2 1 ~8L P\V

~8L

Ja~(p-q)vJ*A(q)dq. y

From (11.1) we get

(11.4)

j

IVA2I 2 $ CriQ(S)I

p

using a duality argument similar to the one in the proof of Lemma 10.25. Here C is a constant that doesn't depend on e, o, -y0 , or 0. (The same will be true of the other constants in this section unless stated otherwise.) We would like to say that A 1 is smoother than A. This is true inside V, but not on P \ V. If we set Vj = {x E P: dist(x, Q(S)) $ 100£2-1}, then we do have that (11.5) To check this it is convenient to let A 11 and A 12 denote the two terms on the right side of (11.2). It is not hard to show that A 12 satisfies (11.5), using IIV All $ Co and the fact that a satisfies (10.20a).

To control {11.5) for A11 it is helpful to make the following observations. We can write mo-l

logs

L L j=O

where h

= [log 2 (9L)]

o:~ * vj = ¢h,

2l>9L

and mo-1

¢ = logs

2: 2: j=O

o:~ *

v:.

2 1 >1

By construction we also have that mo-l

¢

= (Dirac mass at 0) -

logs

L L j=O

2 1 ~1

with this last series converging in the sense of distributions. It is not hard to check that ¢ satisfies (10.20a), using the first formula for ¢(p) when IPI $ 10, and the second when IPI > 10. From here it is easy to see that An = ¢h *A satisfies (11.5). We want to use these estimates on A 1 and A 2 to control how well A ean be approximated by affine functions. First we consider the maximal function

(11.6)

N(A2)(p) =

s~p{IBI-~(IBI- 1

j

IA2- mBA2!)},

B

where the supremum is taken over all balls B that contain p and have radius '$ L, and mBA2 = f A2 • It is well-known and not hard to see that

rm

(11.7)

B

j N(A2)2 $ C j IVA2I 2 $ CriQ(S)l. p

p

[You can use the Poincare inequality

IBI-2.¥

f B

IA2 -mBA21::;

c,~,

f

IV'A21

B

to reduce (11.7) to an estimate on the Hardy-Littlewood maximal function of V' A2 .] We are going to use this together with the following estimate on the oscillation of A 2 on a ball B.

LEMMA 11.8.

Set oscAz =sup \A 2 (p)-mBAz\, and let r denote tberadius B

s; V1,

of B. If B

pEB

tben o~cA2:::; Cr{r- 1 mB(\Az- mBAz\)}m 67fr.

Let q E B be such that

\Az(q)- mBAz\ = oscA2 =: ..\. B

Since !IVA 2 IIL=(B) ~ C6 (because this is true for A, AI), we have \A 2 (p)mBA2 \ ~~when p E B, jp- q\:::; 2 6 . If ~ :::; r, then we get that

&

J

\Az- mBAz\

~ c- 1 ~ ( 2 ~ 6 )

d

B

whence

.Ad+1 :::;

2& 6 ~ r,

C8d J \Az- mBAz\. B

then (Az- mBAz! 2 ~ over a large portion of B, so that mB(\Az- mBAzl) ~ c-•.x. Using IIVAziiL=(B) :::; C6 we also have that If

r- 1 mB(\Az-

mBAzl):::; C6,

so that

C- 1 ..\:::; mB(\A2- mBAzl) = r{r- 1 mB(\Az- m9A2\)} :r.\:-;+;r:h :::; Cr{r- 1 mB(\Az- mBAzl)}

m 6*.

Combining these two cases gives the lemma. Now we are ready to look at how well A is approximated by affine functions. Let B = B(po, r) be contained in V1 • We assume that r:::; r0 L, where r 0 is a small number to be chosen soon.

11.9. Set F = {p E Vz : N(Az)(p) :::; et6}. If B = B(p0 , r) intersects F a.nd r :::; r0 L, then LEMMA

1

(11.10) sup \A(p)- A(po)- V' At(Po) · (p- Po)\ :::; C{ 02i7+IT +roo-t }r6. pEB

Indeed, if p E B, then from (11.5) we obtain

\A(p)- A(po)- VAt(Po) · (p- Po)\ :::; \Az(p)- Az(Po)\

+ \At(P)- At(Po)- VAt(Po) · (p- Po)\

< 2oscA 2 + C68- 1 L- 1 r 2 B

-

.:::; Cr{r- 1 mB(\A2- mBAzl)}

m 6* + C6B-

:::; Cr{N(A2 )(u)}7hom + CMr 1 r 0 r

1

ror

for any u E B. Choosing u E BnF, we get that this is at most Cr0 2 (d~l) 6+ ror' as desired. We shall choose () small enough, and then ro small enough, so that this last quantity is less than 7J r6, for some small 7J that will be chosen soon. Let H B be the d-plane which is the graph of the affine function

cso-l

aB(P)

= A(po) + "l Al(Po) · (p- Po).

If B is as in the lemma, r is the graph of A, and the constants are chosen as we just explained, then r- 1 dist(x,HB)::::;

sup

.(11.11)

TJ6.

zErnrr- 1 (8)

We want to show that these estimates on A are incompatible with

S E Ft . Recall that S E F 1 if

u

(11.12)

Q

I;:: IQ(S)I/2,

QEm 1 (S)

where m1(S) is the set of minimal cubes Q of S such that Angle (PQ,P);:::: 6/2. We are going to show that if Q E m 1 (S), then II(Q) cannot intersect 'F, or even get too close, by comparing the HB 's to the PQ 's. We first need ~~o check that the cubes in m 1 ( S) cannot be too large. ~EMMA 11.13. Given r 0

>0

and a constant M

> 0,

there is an e0

> 0 so

'bat if e::::; e0 then Angle (PQ,P)::::; 6/100 for all Q E S with diamQ;:::: roL/M. This is quite easy: given Q E S, let Q 1 , Q 2 , .•• , QT = Q( S) denote the successive ancestors of Q, i.e., Q1 is the father of Q, Q 2 the father of Q 1 , etc. By definitions, all these cubes must satisfy (6.1), and so Lemma 5.13 gives Angle (PQ•, PQ'+l) ::::; Ce. Thus Angle (PQ,P)::::; C e T ~ Cdog(2L/diamQ), from which the lemma follows easily. LEMMA

11.14. IfQ E m 1 (S), tben dist(II(Q),F) > diamQ.

Suppose not. Pick any XQ E Q, set p 0 = II(xQ) and B = B(po,r), B intersects F, and r :::;: r 0 L because of Lemma 11.13 (if e is sufficiently small). Notice also that B ~ V1 . r :::: 10 diam Q. Then

If x E Q, then

\x- (ll(x), A(II(x)))\

s; CEd(x) $

CE diam Q

by Proposition 8.2 and the definition (7.6) of d(x), and so dist(x,HB) s; c.,hr (if e is small enough) because of (11.11). Invoking Lemma 5.13 again we get that Angle ( PQ, H B) s; C7]h < 1 ~ 0 if we choose 7J (and also f) to be small enough. Let Q* be the largest ancestor of Q such that 10 diam Q* s; r 0 • Let B* be the ball centered at p 0 with radius 10 diam Q•. The same argument gives Angle (PQ·, HB·)::; 1 ~ 0 • FUrthermore, HB· = HB, since the function a B depends only on Po. Thus Angle (PQ,PQ·) s; :0 , and Lemma 11.13 tells us that Angle (PQ•, P) $ 1 g0 (if e is small enough). These two inequalities force Q ft. m 1 (S), and the lemma is proved. We are now at. the final step of the proof that (10.7) is incompatible with S E :F1 • Because of Lemma 11.14, it suffices to show that {11.12) implies that there is a Q E m 1 (S) with dist(ll(Q), F) s; diam Q. (Of course we continue to assume that (10.6) holds.) U Q. For We first choose a convenient covering of the set X = QEmt(S)

each Q E m 1 (S), pick a point XQ E Q, and consider the ball BQ = (xQ, C diam Q). If C is large enough, we can find a subset T of m 1 (S) so that {BQ: Q E T} covers X and the balls B(xQ,3diamQ), Q E T, are pairwise disjoint. (This follows from the well-known argument used in the proof of the covering lemma on p9 of [St].) Hence (11.15)

\XI::;

2.:: IEnBQI::; C 2:(diamQ)d. QET

QET

Because B(xQ, 3 diamQ) are disjoint for Q E T, we can apply Lemma 8.4 (with x = XQ, y = XQ•) to conclude that the balls DQ = PnB(ll( XQ ), diam Q), Q E T, are also disjoint. In particular U DQ ~~ lXI/C.

I

T

All of these balls DQ are contained in V2 , and none of them intersects F, because of Lemma 11.14. Since F = {p E V2 : N(A 2 ) ::5 8t 5}, we conclude from ( 11. 7) that (11.16)

lXI $

cI

u

DQ J$ C\Vz \

T

Fl:;; ce-lcS- 2riQ(S)I.

Recall that r = Ce2 + C'('Yo + fP)8 2 , where C, C' are independent of e, li, and 6, and C' is also independent of ')'o. If we choose 8 small enough, then ")'o and ro, and then f, we get that lXI S ~IQ(S)I, so that (11.12) can't

hold.

Thus we have shown that if S satisfies ( 10.6), then S fl. F 1 , at least if we chose the various parameters correctly. From the arguments in Section 7 it follows that (2. 7) holds, and of course (2.6) was proved in Section 8. This ,tompletes the proof that (C2) implies (C4).

12. The proof that {C3) implies {C4): preliminary discussion

The argument will be very similar to the one used to show that (C2) ,implies (C4). We assume throughout this section and the next two that E satisfies (C3). As we remarked at the beginning of Section 5, E must satisfy the weak geometric lemma. Let t:, 6 be given, as small as we want, with t:/6 as small as we want also. Let k > 0 be large, to be chosen later, but not depending on t: or 6. Let g be the set of cubes for which (6.1) holds, and set B = ~ \ (]. We ·c:an apply Lemma 7.1 to get the stopping-time regions, and to prove that (C4) holds we need only verify (2.7), since the rest of the requirements are .established by Lemma 7.1 and Section 8. Lemma 7.4 tells us once again .that we can reduce (2. 7) to the corresponding estimate for F 1 . To prove the packing condition for F 1 we use a variation of Lemma 7.9, whose particulars are as follows. Let S be one of the stopping-time regions, and set (12.1)

X= {(x,t) E Ex a+: x E kQ(S), k- 1 d(x) :S t :S kL},

where d(x) and L(= diamQ(S)) are as before. To prove the packing condition for F 1 it suffices to show that there is an TJ > 0 such that (12.2)

jj f3I(x,kt)2 d~dt ~ TJIQ(S)I X

at least if we choose k0 , k, t:, 6 correctly. Here TJ is allowed to depend on t:, 6 or k, but not S. The proof of (12.2) will be carried out in two steps, in much the same way as before. In the first step we show how the left side of (12.2) controls

a similar quantity on the graph of A (A as in Section 8). In the second step we use these estimates on A to show that if the left side of (12.2) is small enough, then S ~ JC1.

13. Pushing estimates on {31 ( x, t) from E down to the approximating Lipschitz graph

Let ko be large, to be chosen later; k0 will not depend on 'f, li, and it \ill be chosen before k, in such a way that k will be much bigger than k0 • ~t A, P, II, Ri, Q(i), etc. be as in Section 8, Proposition 8.2 in particular. Given p E U0 and t > 0, set

1)3.1)

-y(p, t) = infCd a

j

~IA(u)a(u)ldu, t

B(p,t)

'rhere the infimum is

taken over all affine functions a : P --+ pJ... It is easy

o see that if we set

l3.2)

J ~dist((u,A(u)),M)du,

i'(p,t) =infrd

t

M

B(p,t)

where the infimum is taken over all d-planes M, then

IJ:3.3)

~i'(p, t) :::; -y(p, t) :::; 2i'(p, t).

'This uses the fact that A is Lipschitz with small norm. Of course i't (p, t) • essentially the same as the analogue of {3 1 ( x, t) for the graph of A. This section is devoted to proving the following estimate.

r

·LEMMA 13.4. Set T = k0 L/10 and

(13.5)

T

=

L-d

1T J-y(p,t)2d~dt. V2

If k / ko is large enough, then T :S C£. 2

(13.6)

dxdt + cL- d jrJ{ {31(x, kt?-t-' X

wl1ere C does not depend on for 6.

[Recall that X was defined in (12.1).] We restricted ourselves to U2 so that B(p, t) stays }nside U1 if p E U2 and t < T. We first estimate I(P, t) for small t. Let R;, I, Io, etc., be as in Section 8, and let /2 denote the set of i E Io such that R; n U2 f. 0. LEMMA

13.7.

The main ingredients for this are Lemma 8.22 and Taylor's theorem. From the latter we get

I(P, t) :S Ct

sup

l'\7 2 A( u )I,

uEB(p,t)

and so using Lemma 8.22 we get {dia.rn Ri

2::: Jo •El2

J

dpdt

I(P, t) 2 - t - :S Cc2

Ri

:S Cc2

{dia.m Ri

2::: Jo I

J

(diamR;)- 2 tdpdt

Ri

2::: j dp :S Cc IQ(S)I :S CLd. 2

I

Ri

Now we need to estimate I(P, t) when p E II(Z) or when p E R; but t > diam R;. Both cases will be covered if we assume that t > D(p)/60, because of (8.8). Let p E U2 and t be given, 1/60D(p) :S t :S T. Choose z E E such that z E Q(S), iP- II(z)l :5 Ct. We want to control1(p, t) in terms of {3 1 (z, / 0 kt). Let Pp,t bead-plane for which the infimum in the definition of {3 1 (z, / 0 kt) is achieved. Taking M = Pp,t in (13.2) we get that (13.8)

1(p,t) :S 2Cd-l

j B(p,t)

dist((u,A(u)),Pp,t)du

:5 a+

2::: iEl(p,t)

a;,

where a = 2cd- 1 JB(p,t)nii(Z) dist((u,A(u)),Pp,t)du, l(p,t) is the set of indices i E Io for which R; n B(p, t) -::j:. 0, and

j

. - 2t-d-! a,-

dist((u,A(u)),Pp,t)du .

B(p,t)nR; Let us estimate a first. If u E B(p,t) n II(Z), then X= (u,A(u)) E z, ud it follows from Lemma 8.4 that jx - zj :::; Ct and that x is the only :point in II- 1 (u) n koQ(S). We can lift the integral defining a from II(Z) to Z ~ E to get that

j

a:::; 2cd-l

£13.9)

dist(x,Pp,t)dx:::; Cf3 1 (z, 110 kt).

rr- 1 (B(p,t))n1: 0 Q(S)

Next we estimate the

a;.

Fix i E I(p, t). We have

(13.10) where bi

= t-d-! JR;nB(p,t)dist((u,A(u)),Pq(i))du

and

c; = cd- 1 (diamR;)d sup{dist(w,Pp,t) :wE Pq(il•

dist( w, Q(i)) :::; C diam Ri}. ,{The set of w's over which this supremum is taken is large enough because dist(R,,II(Qi)):::; CdiamR;.] Because Pq(i) is the graph of B;, (13.11)

b,::::;rd- 1

j

IA(u)-B;(u)idu:=;Ctt-d-l(diamR;)d+I,

R;nB(p,t)

Rince lA( u)- B;( u )I :::; Ce: diam R; by Lemma 8.17 and the definition (8.14) of A. We can control c; as follows. LEMMA

13.12.

c·l
+ ct-d-l (diamR;)d {j2Q(i)j- 1

I

2Q(i)

3

dist(x,Pp,t)i dx}

This lemma is similar to Lemma 5.13, and their proofs are quite similar also. If yo, YI, ... , Yd are the d + 1 points promised by Lemma 5.8 with Q = Q(i), then

To check this we compare Pp,t and Pq(•) with the d-plane Ld generated by Yo, Y1, ... , Yd· The distance of points on Pq(i) to Ld is controlled by sup dist(yj, Pq(•) ). [To see this it is helpful to observe that because this O~j~d

quantity is small compared to diamQ(i), the points z3 in Pq(i) closest to Yi generate Pq(•)·] It is easy to show that sup dist(yj, Pp,t) controls O~j~d

the distance of points on Ld to Pp,t· Combining these two facts gives the inequality. Since Q(i) E S satisfies (6.1) we have

Moreover, this inequality remains true if each Yi is replaced by any Yi E E such that IYj - Yji ~ c;- 1 diamR;, provided Co is large enough. (Indeed, such fiJ's would still satisfy the same properties that the yj's do.) The lemma follows by taking cubic roots of the inequality, averaging over such iii's, and using the regularity assumption (1.5). We want to combine these various inequalities with (13.8) to get a good estimate for -y(p, t). For this it is important that we took a cubic root in Lemma 13.12, instead of just an ordinary average, so that we can control the overlap of the 2Q(i)'s which enters into the sum in i. Let J( i) be the subset of I(p, t) composed of the j's such that diam Q(j) S diamQ(a) and 2Q(j) n 2Q(i)-:/: 0. Set N;(x) = 2:: X•q
l:N;(x)- ~ C for all x (because I:~< oo), and 2

J 2Q(i)

N,(x)

~

L jEJ(t)

I2Q(j)l

~C

L

IRil

~ CIQ(i)l,

jEJ(i)

because the Rj's are disjoint and stay at distance

S C diam R; from II( Q;).

From Holder's inequality we now get that {I2Q(i)j- 1

I dist(x,Pp,t)~N;(x)-fN;(x)fdx} 3 I J

2Q(i)

:S {j2Q(i)j- 1

dist(x,Pp,t)N,(x)- 2 dx}{j2Q(i)j- 1

2Q(i)

:S C(diamRi)-"

N;(x)dx} 2

2Q(i)

j

dist(x,Pp,t)N;(x)- 2 dx.

2Q(i)

~utting

this into Lemma 13.12, and then using (13.8), (13.9), (13.10), and 13.11) we obtain

-y(p,t) SC/31 :}3.13)

(z, 1~kt) + CE

+ c cd-l

L

L

t-<1-l

J

(diamR;)dH

iE!(p,t)

dist(x, Pp,t)N;(x)- 2 dx.

iE!(p,l) 2 Q( •)

Because

L: N;( x )- 2 ,

(13.14)

C

the last term is at most

cd-l

J

dist(x, Pp,t)dx.

U2Q(i) iEl(p,t)

r-et us show

that this can be dominated by /3 1 (z, 110 kt). First note that ~amR; S Ct ifi E I(p,t), because D(u.) :S D(p)+t :S Ct on B(p,t). Thus tiat(II( Q;), II( z )) :S Ct (because oft he various definitions), and this implies lhat Q; ~ B(z,Ct). [Pick x E Q(i). By definition, d(x) :S diamQ(i) :S Ct, ~d so if lx- zl is much bigger than Ct, then Lemma 8.4 implies that

:Because the right side is at most Ct, we get that lx - zj S Ct.] Now it is tmmediate that (13.14) is dominated by C/31 (z, 110 kt), and so (13.15)

-y(p,t)

s C/31 (z, 110kt) +CE cd-l L iEl(p,t)

(diamRi)d+l.

Of course the same argument used to prove (13.15) also works with z replaced by any wEEn B(z, t) and so

f

-y(p,t)'l· 5cCCd

2 1 10 kt) dw

/31 ( w,

B(z,t)nE

(13.16)

L

+C{ECd- 1

(diamR;)t/.+ 1 } 2 .

iE!(p,t)

To finish the proof of Lemma 13.4 we want to use this to estimate

Using {13.16) we have that r 1 S C(a +b), with

J

a= L-d

iT

U2

c-• D(p)

I

cd

/31

(w,

B(z(p,t),t)nE

2:..kt) 2 dw dt dp, 10

t

L

b=t.2L-dj {T r2(d+1){ (diarnR;)d+l}2dtdp, u2 lc-•D(p) iEI(p,t) t where z(p, t) = z is the point in Q( S) we chose shortly after proving Lemma 13.7. Consider a first. By definitions, for any ( w, p, t) that arise in the integral, we have III( w) - PI S Ct and w E ko Q( S). Thus

j

aS L-d

{T

Ji if

k 0 Q(S)

S CL-d

dp)

/31

(w, ]:_kt) 2 dwdt 10 t

EB(II(w),Ct)

k 0 Q(S)

:S CL-rl

j

rt!. (

lc-•D(p)

1 ) {31 ( w, -kt c-'D(p) 10 T

2

dt dwt

f3I(x,kt)2 d:dt.

X

To estimate b recall that diam Ri S Ct if i E I(p, t) (because D( u) S D(p) + t S Ct on B(p, t), and using (8.6)). Thus

2:: (dia.m .R; )d+ iE!(p,t)

1

S Ct

L {diam R,f S Ctd+l. iEl(p,t)

This gives us that

In interchanging the order of the sum and the integral, we have used the facts that dist(p, R;) ~ t when i E I(p, t), and that i E I(p, t) and t ~ c- 1 D(p) imply t ~ c- 1 diam R., and we have let J 1 denote the set of i E I 0 such that Ri n U1 =f 0. From here we easily get that b ~ Ce 2 L-d

L (diam Ri)d+I iE!1

:S Ce2 L -d

J

t>C- 1 diam R;

L (diam Ri)d ~ Cf

2•

iEI1

Combining this with our estimate for a we have Tt

~ CL

-d

jrJ{ f3t(x, kt) 2 -tdxdt + CE 2 . X

This and Lemma 13.7 give us the estimate (13.6) we wanted for r, and so Lemma 13.4 is proved.

14. The end of the proof that (C3) implies (C4)

According to Section 12, it suffices to show that (12.2) holds for 'rf small enough. Thus, for example, it suffices to show that if

jj fJ1(x,kt) 2 d~dt::;

(14.1)

E2

IQ(S)I

X

then S ~ F1. The proof of this is very similar to the argument given in Section 11 in proving that (C2) implies (C4): we use Section 13 to turn (14.1) into an estimate on A, then apply Littlewood-Paley theory to this estimate to show that 'VA doesn't oscillate much, and then check that that prevents many minimal cubes of S from lying in m 1 (S). This step in the proof that (C3) implies (C2) is in fact so similar to the corresponding step for (C2) that we can simply reduce to our earlier argument. Assume that (14.1) holds for a fixed S. By Lemma 13.4,

vi

is as described relatively early in Section 10, then 2-1 1vJ * A(p)l < C-y(p, t) whenever 21 - 1 ::; t ::; 21 , because of the vanishing moments assumption on v. Therefore, just like for (11.1), If

(14.2)

jL V

2- 21 lvJ

* A(p)l 2

::;

CE2 IQ(S)I,

2t~L

where Vis as it was for (11.1). This is even better than (11.1), since there are no 6, "Yo around.

G. DAWD, S. SEMMES

Once we have this version of (11.1), we can use exactly the same argument as in Section 11 to show that S ¢ :F1 if f, 6 are chosen properly. This completes the proof that (C3) implies (C4). Of course if we were to do the argument over again there are some relatively minor changes we could profitably make. For example, we could replace the discrete square function estimate (14.2) by a continuous one, namely .

/ 1L lvt*A(p)l 2 d~:t:::;

C€2 IQ(S)I,

v and then we could use the reproducing formula j

=C

1

dt

00

0

Vt

* Vt * j -t

instead of the more complicated discrete one we used before. There are other simplifications that could be made, stemming from the absence of () in (14.2) and from the stronger nature of -y(p, t) as compared to Vt * A(p), but the main ideas are still the same.

15. (C4) implies (C3)

As promised in Section 1, we shall actually prove that if E satisfies (C4), then {3 (

(15.1)

r

x, t

)2 dxdt t

is a Carleson measure on E X R+ for all r < d~2 if d 2 2, r :S oo if d = 1. The idea is that the existence of a corona decomposition will allow us to reduce to the case of Lipschitz graphs, for which we can use Lemma 10.14. Let "' > 0 be given, small, as in the definition of a corona decomposition. (The smallness of 1J will be a convenience that will not play a major role.) Let B, Q, :F, S E :F be as in Section 2. Given Q E ~'let Q denote the set of (x, t) E Ex R+ such that x E Q and (lOC)- 2 diamQ :S t :S diamQ, where C is as in (2.2). It is easy to check that E x R+ S E :F, set S = U Q.

U Q.

Given

QE.:l.

QES

15.2. Let a(x, t) be a bounded, nonnegative function onE X R+. Then a(x,t)dx~t is a Carleson measure if a(x,t)x5(x,t)d~dt is for each S E :F, witb uniformly bounded norm.

LEMMA

Decompose a into a 0 +at, where ao = axB'

B = U Q.

It is easy to

QEB

check that a0 (x, t)dx/t is a Carleson measure, because of (2.4). On the other hand

by definitions. Fix x E E and R

j LR

L

(15.3)

> 0, and consider a(y, t)xs(y, t)dy

~t.

SE:F yEEnB(z,R)

Let E 1 denote the sum over the S E F for which Q(S) n B(x, R) of 0 and diam Q( S) ~ R, and let :E 2 be the swn over the S's such that S n ( ( B( x, R) n E) x ( 0, R)) of 0 and diam Q( S) > R. We may replace the sum in (15.3) by :E 1 + :E 2 . By assumptions we have that

J

~1 /OR ~ Jo

dydt a(y, t)x;s(y, t)-t-

B(z,R)nE 1 /diam Q(S)

~

L Jo

~

cL

J

dydt a(y, t)x;s(y, t)-t-

Q(S) 1

IQ(S)I::; CRd.

The last inequality comes from (2. 7). To control :E 2 we observe that :E 2 only involves a bounded number of S's. This is because the S's are disjoint, and the ones that arise in :E 2 must each contain a cube Q for which QnB( x, R) of 0 and diam Q ~ R. This and our hypotheses on a yield

L loIR 2

J

&

a(y, t)x;s(y, t)dyt::; C

L

2

Rd::; CRd.

B(z,R)

This proves the lemma. In view of the lemma we need only control the f3r(x, t)'s on each S. Fix S E F, and let r be the Lipschitz graph over the d-plane P promised in (2.6). Let pl. be the (n- d)-plane that passes through the origin and is orthogonal to P, and let A : P --+ pl. be the Lipschitz function such that r = {p+ A(p): pEP}. Let II, ill. be the orthogonal projections onto P, pl. so that X= II(x) + rrl.(x). Define d(x) as in (7.6). Set

(15.4)

for p E P, t > 0, where the infimum is taken over all affine functions a: P--+ pl_. We want to control the f3r's onE in terms of the "Yr's. Let (x, t) E S be given, so that t ~ L = diamQ(S), and let p be any point in P such that IP- II(x)l ~ t. Let a be an affine function for which the infimum in the definition of "Yr(P, lOOt) is attained, and let H = H p, t denote its graph. For y E E, IY- xi ~ t, we have (15.5)

dist(y,H) ~ IIIj_(y)- A(II(y))l + IA(II(y))- a(II(y))l ~

c.,d(y) + IA(II(y))- a(II(y))l.

(This uses dist(y, r) ~ C11d(y) which can be derived from (2.6).) We want to use this to control f3r(x, t) in terms of "Yr(P, lOOt). For r = oo we get that (15.6)

f3co(x,t)

~ C11 (

C 1 d(y)) +"Yoo(p,lOOt)

sup yEB(z,t)nE

for all p E P with IP- II( x) I ~ t. For r < oo we cannot estimate so quickly, because we might get a very singular measure when we push Hausdorff measure onE down toP, and so we don't want to simply integrate (15.5). Instead we have to smear up (15.5) first. Let o E (0, 1) be a small positive number, to be chosen later, and let By denote the ball in P with center II(y) and radius od(y). Because of (15.5) we have for u E By that dist(y,H) ~ c.,d(y) + Cod(y) + IA(u)- a(u)l(We have used here the fact that A is Lipschitz.) Taking averaging over B 11 gives (15.7)

dist(y, HY

~ C(Tl +or d(yy + C[od(y)td

J

rth

lA-

powers and

air du.

B•

[When d(y) = 0 we interpret the average over By as being the value at y. Similar liberal interpretations are required below. Alternatively, one can treat the case d(y) = 0 separately, by simply integrating (15.5).] Hence

J ( 15.8)

dist(y, HY dy

~ C(Tl + oy

B(z,t)nE

J

d(yt dy

B(z,t)nE

+ C

j

B(z,t)nE

[od(y)td

j B~

IA(u)- a(uWdudy,

(Don't forget that y E E, but u E P.) We want to use Fubini to simplify the last term, but first we need some preliminary information. We begin by observing tba.t d(x) :::; Ct because (x, t) E S, and so d(y) ~ Ct if y E B(x, t). In particular, 6d(y) :::; t, if 6 is small enough. Next, let us check that if y, z E B(x, t) n E satisfy By n Bz # 0, then d(y) and d(z) are comparable, and that IY- zl :S d(y). Indeed, we certai~ly have

III(y)- ll(z)l ~ C6(d(y) and also

IY -

(ll(y ), A(II(y)) )I ~

+ d(z)),

c dist(y, r) s

CTJd(y ),

and similarly for z. Hence

IY- zl

~

C(6 + TJ)(d(y)

+ d(z)).

If TJ and 6 are small enough we can conclude that 1

"2d(y)

~

d(z)

~ 2d(y),

because d( ·) is Lipschitz. With these observations we can reverse the order of integration in the last term in (15.8) to obtain

J

[6d(y)rd

B(x,t)nE

J

IA(u)- a(uW dudy

Br

J

<

[od(y)rd dy

du

yEB(x,t)nE uEB,

B(p,lOt)

j

~C

J

IA(u)- a(u)r

IA(u)- a(uWdu.

B(p,lOt)

Hence

f3r(x,

t) ~ (t~ j C

B(z,t)nE

1

(c 1 d(y)r dy) • +C-yr(P, lOOt).

Combining this with (15.6) we get that

(15.9)

f3r(x, t) 5;C

1

(t~ j

(t- 1 d(y)r dy);

B(z,t)nE

+ C pEB(II(z),t) inf 'Yr(P, lOOt) for any r -5; oo, (x, t) E S, t -5; diamQ(S). Let us use this to check that f3r(x, t) 2 x5(x, t)d~dt is a Carleson measure if r < d2!_2 when d;::: 2, r -5; oo when d = 1. Fix z E E, R > 0. To control (15.10)

1R J

dxdt 2 f3r(x, t) Xs(x, t)-t-

B(z,R)nE

we consider separately the contributions coming from the two terms on the right side of (15.9), starting with the second. It is dominated by (15.11)

1R J

1 Xs(x, t) td

B(z,R)nE

J

dt 'Yr(P, lOOt) 2 dpdxt.

B(II(x) ,t)

Of course we want to use Fubini's theorem. It is easy to reduce to the case where z E Q( S) and R -5; diam Q( S). It is also not hard to check that the set of x in Q( S) such that ( x, t) E S and III(x)- pj -5; t (for given p, t) is contained in a ball of radius Ct, and hence has measure -5; Ctd. This uses (2.6) and the fact that d( x) ::; Ct if (x, t) E S. Thus by reversing the order of integration we see that (15.11) is at most

C

1 J R

2 dpdt 'Yr(P, lOOt) - ,

t

0

B(II(z),2R)

and this is -5; CRd by Lemma 10.11. To show that the contribution to (15.10) from the first term on the right side of (15.9) is at most CRd it is enough to prove that for any cube TE S, (15.12)

L Ar(Q) 1Qj::; CjTj 2

QES Q<;,T

if

r

<

= 1), where

d~z (r:::; oo when d

1

A,(Q)

~ (~~~l (d(y)f diamQ)" dy)'

[To see that this is sufficient it is helpful to notice that the first term on the right side of (15.9) is :::; C.>..r(Q) when (x, t) E Q.] To verify (15.12) it is useful to introduce a collection of cubes which is like the set of minimal cubes in S except that it is more regular, in a certain sense, and it is easier to compute d( x) in terms of this collection. Let Q( S) denote the union of the cubes lying in the same generation t:J.j of cubes as Q(S) and which intersect 2Q(S). Let n(S) denote the cubes N ~ Q(S) which are minimal among the cubes that also satisfy 200C 2 diam N 2: inf d( x),

(15.13)

rEN

where Cis as in (2.2). If N is one of these minimal cubes, then none of its children satisfy (15.13), and so 100diamN _::; inf d(x). zEN

This implies that sup d(x):::; 2 inf d(x), zEN

rEN

since d(x) is Lipschitz with norm 1. Notice that d(y) = 0 when y E Q(S) but y does not lie in any of the N's. Of course the N's are disjoint. Let us use these cubes N to prove (15.12). Given Q, let Q denote the union of the cubes in the same generation 6.j as Q that intersect 2Q. For r = oo we have that

L

Aoo(Q) 2 JQj:::; C

QES

L ( QES

Q~T

Q~T

SUP_ N~Q

~::~)' IQI:;;

NEn(S)

C

L L QES

NCQ Q~T NE;:;:(S)

When d = 1 this is at most

c

L (~ NCT NE;;(S)

Q:Q2N

IQI-l) INI 2 :S c

L NCT NE;;(S)

INl :::; CjTj.

:~::;; IQI.

Now suppose that r

< oo.

We may as well assume that r > 2, in which

case l. r

QES QCT

L_

QES

c

IQI

IQI

NCQ NE;;(S)

Q~T

$

( ~)7+1

L L (\~i }:< ;1+

1

)

IQI

QES NCQ Q~T NE;;(S)

$

c

L L

IQI 1 -~-.liNI~+;}.

NCT Q:Q'2N NE;;(S)

If r

< }.!.2 , then 1- j <

~'and so the inner sum converges and this is

$

c

L

INI

$CIT!.

This proves (15.12), which is the last step in the proof that ( C4) implies that (15.1) is a Carleson measure for r < 2 when d ~ 2, r $ oo when d = 1.

l!

16. The main step in the proof that (C4) implies (C5)

PROPOSITION 16.1. Let E be a regular set that satisfies (C4). Then for any e > 0 there is an M > 0 so that for all x E E and r > 0 there is a compact set F ~En B(x, r) and a mapping f: F --t Rd such that

(16.2)

(16.3)

IE n B(x, r) \ Fl :S erd M- 1 lx- Yi :S lf(x)- f(y)l :S Mix-

Yl

for all x, y E F.

Once we prove this proposition all that remains in showing that (C4) implies (C5) is to find a bilipschitz extension of f- 1 : f(F) --t F to a embedding of R d into R n•. This will be taken up in the next section. To prove the proposition we first construct F, and then the definition off will become natural. The general idea of the proof is that for a good region S in Ll as in Section 2 it is easy to choose a map of Q(S) onto Rd that is bilipschitz at scales larger than the minimal cubes of S, because of (2.6) (the approximation of Q(S) by a Lipschitz graph), and we can try to patch these maps together to get f. The total amount of patching required will be controlled by (2.7). Let x E E and r > 0 be given, and let m 0 be such that 2mo :S r < 2mo+l. Let Ro be the union of the cubes in ~mo that intersect B(x, r). Call these cubes (Qo,j), j E J; note that there are a bounded number of them. Decompose the set of cubes Q E ~ such that Q ~ Ro into a bad set and a family of good regions S ~ ~ in such a way that (2.4), (2.5), (2.6) and (2.7) hold, with 77 small, to be chosen later (depending only on the geometry of E, not on e). It is easy to obtain this decomposition from the one provided by (C4).

In the construction of the set F we shall sometimes need to remove from cubes the part which is too close to the boundary. For each Q E ~m set

(16.4)

a(Q) = {x

E Q: dist(x, E \ Q):::; T2m},

where T is a small constant that we shall choose soon (depending on e). Thus a( Q) is the -part of Q which is very close to the boundary, and the measure of a( Q) is controlled by (2.3). Applying (2.3) with a not-too-small choice ofT already gives the existence, for each cube Q E ~' of a "center" CQ E Q such that

(16.5)

dist(cQ,E \ Q) ;::=

~0 diamQ,

with Co independent of e. We still need a little more notation before defining the set F. We call a cube Q ~ Ro a "transition cube" if it is a bad cube, the top cube in one of the good regions S, or a minimal cube in one of the S's. We denote by T the set of transition cubes, and by Tj = T n ~ i the ones of generation j. For each transition cube Q, let f( Q) denote the number of ancestors of Q that are transition cubes. For instance, the Q 0 ,j's are transition cubes such that f( Qo,j) = 0. Our set F is defined by

(16.6)

F=EnB(x,r)n

(u

a(Q))c n (

QET

U Q)<· QET l(Q)~L

where Lis a very large constant whose value will be chosen very soon. Notice that

L

IQI :::; GIRo!,

QET

because of (2.4) and (2.7). Using (2.3) we can choose

L QET

la(Q)I:::; CT-h!Rol < t:rd/2.

T

so small that

Similarly, if L is large enough, then A = because 2: XQ ~Lon A, so that QET

IAI :::; L-l

L

IQI :::;

U

Q satisfies IAI <

ad /2,

QET l(Q)?_L

cL- 1 IRoi-

QET

Thus F satisfies (16.2). As a first approximation of the map f on F we shall define a function g on the set T of transition cubes with values in the set of cubes in Rd. Our mapping will have the following properties: (16.7)

(16.8)

for each Q E Tm,g(Q) is a cube of Rd with diameter c;t(Q)2m, where C 1 is a large constant (to be specified), if Q, Q' E T, Q ~ Q', then g(Q) ~ tg(Q').

Let us start with the cubes Qo,j, j E J. We pick the cubes g( Qo,j) to be cubes of size 2mo, at mutual distances between 2mo and C2mo. This is possible if Cis large enough. In general we define g recursively as follows. Suppose g( Q) has already been defined for some Q E Tm. Assume first that Q is a bad cube or the minimal cube of some good regionS. Let C( Q) denote the set of children in Q, i.e., the set of cubes in 6.m-I which are subcubes of Q. Notice that they are all either bad cubes or top cubes of some other good region, and that f( R) = f( Q) + 1 if R E C( Q). If C 1 is sufficiently large, we can easily choose cubes g(R), R E C( Q), such that each g(R) has diameter c;t(Q)-t2m-t, is contained in kg(Q), and also (16.9)

dist(g(R), g(R')) ~ 2m-tc;t(Q)-I

for all R, R' E C(Q), R =/= R' Assume now that Q is the top cube of a good region S. We want to define g(R) for the minimal cubes R of S, but first we need some more notation. Let fq be as in (2.6), so that fq is a Lipschitz graph over a d-plane Pq, and let IIq be the orthogonal projection onto Pq. Let f/>q be an affine mapping from Pq to Rd such that f/>q(IIq(Q)) ~ fg(Q) and (16.10)

for all p, q E PQ. If R is a minimal cube of S, and m is such that R E Tm, then we take g(R) to be a cube centered at 1/JQ(IIQ(cR)) with diameter C1-i(Q)-l2m, where CR is the "center" of R that we chose before (so that (16.5) holds). Clearly g(R) ~ ty(Q). We need to estimate dist(g(R), g(R')) for any pair R, R' of minimal cubes of S. Observe that

lcR- CR•I 2:

c-

1{

diam R + diamR'

+ dist(R, R')},

because of (16.5). Applying (2.6) to the smallest cube Q' ~ Q that contains CR and has diameter 2: icR - CR• I we get

If "' is small enough this implies that

since rQ is a Lipschitz graph with constant ~ "'over PQ. Combining these estimates with (16.10) we see that if C 1 is large enough, then c;t(Q)-l { diam R

+ diam R' + dist(R, R')}

~ dist(g(R),g(R'))

(16.11)

~

c;t< Q) { diam R + diam R' + dist( R, R')}.

Now that we have constructed g we are ready to define

f

on F. If

x E F, then it can only belong to a finite number of transition cubes; let

Q( x) be the smallest one. By the definitions of F and transition cubes, Q(x) is the top cube of some good region S(x). [Otherwise all the children of Q(x) are also transition cubes, and they are disjoint from F.] We take

with the notations above. Note that f(x) E g(Q(x)). We want to check the bilipschitz condition (16.3). Let x, y E F be given, and let Q 1 be the largest cube in R 0 that contains x but not y. Suppose first that Q 1 lies in a stopping time region S, but is not its top cube Q( S). In this case y E Q( S) too. Suppose further that there are two minimal cubes R, R' of S containing x, y, respectively. Of course

lx- Yl

~ diamR

+ diam R' + dist(R, R'),

but because x ¢: a(R), y

f/: a(R')

(by definition of F), we also have

lx- Yi ~ ~ diamR + ~ diamR' + ~ dist(R, R'). For this pair x, y (16.3) follows from these inequalities, (16.11), and from j(x) E g(R), f(y) E g(R') and l(R) = l(R') S L. If xis not contained in any minimal cube of S, then j(x) = ¢q(IIq(x )), where Q = Q( S) is the top cube of S. The argument used to establish (16.11) also gives c;t(Q)-l {diam R'

+ dist (x, R')}

S dist(f( x ), g( R')) S c;t(Q) { diam R' + dist(x, R')} for such an x, and (16.3) is deduced as before. The same argument works if y is not contained in a minimal cube of S, or if neither x nor y is. We are left now with the case where Q 1 is a transition cube which is not a minimal cube of some good region. This means that the father Q2 of Q 1 is also a transition cube, and it is not the top cube of some good region. Because Q2 contains y, we can bound if(x)- f(y)i from above using f(x), f(y) E g(Q2), and we can bound it from below by applying (16.9) to R = Q 1 and R' = the brother of Q 1 that contains y [since j( x) E g( R), J(y) E g(R')]. These bounds imply (16.3), because

~ diamQ1 S lx- Yi S CdiamQ1 (since x ¢: a(QI), y E Q2 \ Q1). Thus we have shown that F and f satisfy (16.2) and (16.3). If F happens not to be closed, then we can replace it by its closure. This completes the proof of Proposition 16.3.

17. An extension theorem

PROPOSITION

17 .1. Let A ~ R d be a closed set. Suppose that f : A

--+

Rn

satisfies

(17.2)

C0 1 lx- yJ:::; lf(x)- f(y)J:::; CoJx- yJ

for all x, y E A. Assume also that n g: Rd--+ Rn off such that

(17.3)

~

2d + 1. Then there is an extension

M- 1 jx- yJ:::; Jg(x)- g(y)J:::; MJx- yJ

for all x, y E Rd. HereM depends only on d, n, and Co. That (C4) implies (C5) follows from this and Proposition 16.1. (Of course the f here corresponds to the f- 1 from Proposition 16.1.) It is not clear that the condition n ~ 2d + 1 is optimal. If we want g to be a small perturbation of a Lipschitz extension of f that is given in advance, then we do have to have n > 2d. The proposition is not hard to prove using the techniques of [D3], Section 4. (Section 5 of [D3] is relevant to the question of the optimality of n ~ 2d + 1.) Because the proof requires a rather large amount of notations, and no new ideas, we shall omit it here, and content ourselves with a slightly weaker result.

17.4. Let A ~ Rd be closed, and suppose that f : A--+ R" satisfies (17.2). Then there exist an integer m, a constant M, and an extension g : Rd --+ R m off such that (17.3) holds. The constants m, M depend only on d, n, and C0 . PROPOSITION

This result, together with Proposition 16.1, still prove that ( C4) implies (C5), only with a different value of n"'. It is of course still possible to deduce (C1) from this weaker version of (C5).

Let us prove the proposition. By the Whitney extension theorem we know that we can extend f to a map from Rd to Rn which is Lipschitz with norm::::; C~, where C~ depends only on Co and d, and C~ 2: C0 • Let f denote this extension also. We also need a Whitney decomposition of !1 = Rd \A. Let Q;, i E I, be the maximal dyadic cubes in n with

For each i E I choose¢;: Rd--+ Rd+I such that:

(17.5) ¢;is supported in 3Q;, 1¢;1::::; Ct diamQ;, and IV¢;1::::; C1; (17.6)

J¢;(x)- ¢;(y)J2: C! 1 lx- yJ

(17.7)

lt/J;(x)l::::;

(17.8)

~ diamQ;,

J¢;(x)l2: diamQ;,

for x,y E 2Q;; for x

rt 2Q;;

for x E Q,.

The constant C 1 is a geometric constant that depends only on d. It is quite easy to construct these functions ¢;. We added one more coordinate so that (17.8) can be satisfied too. The idea of the proof is to add the functions ¢; to J, to increase f( x) - f(y) when necessary. We shall put the functions ¢; in different dimensions, so that they never interfere with each other, or with f. For each i E I let V( i) denote the set of indices j E I such that

(17.9)

10- 2 diam Q; ::::; diam Qj ::::; 102 diam Q;, and

(17.10)

dist(Q;, Qj)::::; 10 6 (C~) 2 {diam Q;

+ diam Qj}-

Thus j E V(i) if and only if i E V(j), and there is an integer N such that V( i) has less than N elements for each i. Let X be a set with N elements. There is a function a: I--+ X such that a(i) =/:- a(j) whenever j E V(i) but j =f. i. [One way to find such a function is to arrange the elements of I in a sequence and then define a( i) recursively in such a way that this property always holds.] Set m = n + 1 + (d + 1)N, and identify Rm with Rn x R x (Rd+ 1 )x. We define y: Rd--+ Rm by

(17.11)

Yo

= f,

where Yo is the R" coordinate of y,

(17.12) g 1 (x) = dist(x,A), where g 1 denotes the R-coordinate of g,aud

g,(x)

(17.13)

=

L

¢;(x),

a EX

iEI a(i)=a

Clearly g isM-Lipschitz for some M. To check the other part of (17.3) let x, y E Rd be given, x =f y, and let us distinguish a few cases. Suppose first that X and y both lie in n, X E Q;, y E Qj. We may assume that diam Q; ;:::: diam Qi. Case 1 dist(Q;,Qj);:::: 105 (Cb) 2 diamQ;. Thus Q; and Qj are much further away from each other than they are from A (by the way we chose the Q;'s). We shall use (17.2) to find a lower bound for l9o(x)- go(Y)I. Choose u, v E A as close as possible to x, y, so that

lx- ul ::; 3 · 103 diam Q., IY- vi ::; 3 · 103 diam Qj ::; 3 · 10 3 diam Q;. Then lu- vi ;:::: !lx- Yl, and

IJ(x)- f(y)l;:::: IJ(u)- f(v)l-lf(x)- f(u)I-IJ(y)- f(v)l

;:::: C0 1 iu- vi- C~lx- ul- C~IY- vi ;::::

~C0- 1 Ix- yl- 6 · 10 3 C~ diam Q;.

By assumption the last term is at most

1 ~ 0 (Cb)-

1

dist(Q;, Qj)::;

1 ~ 0 (Co)-

1

lx- Yl,

and so IJ(x)- J(y)l;:::: (lOCo)- 1 , as desired. Case 2 dist(Q;, Qj)::; 105 (Cb) 2 diamQ;, and diam Qj::; diam Q,. Thus dist(Q;, Qj) is not too large compared to the distance of Q; to A, but Qj is closer to A than Q; is. In this case we have

i

9I(x);:::: dist(Q;,A);:::: 103 diamQ;, 9I(Y) ::; dist( Qi, A)

+ diam Qj ::; 3 · 103 diam Qi ::; ~ 103 diam Q.,

by definition of our Whitney cubes. This provides an adequate lower bound on lg(x)- g(y)j, because lx- Yl::; 2 ·10 5 (Cb) 2 diamQ;, by assumption.

Case 3 dist(Q., Qi).::; 10 5 (C~Y diamQ1, a.nd diamQi > ~ diamQ •. Thus the distances between Qi, Qi> a.nd A are roughly comparable to each other. Under these conditions we have k E V(i) if 3Q,~: meets Qj; thus 9a(i)(Y) = tPi(y), because of (17.13) and the properties of a(·). We have Ya(i)(x) = tPi(x) for the same reason. We can bound l9a(i)(x)-9a(i)(Y)l from below using (17.6) when y E 2Qi, and (17.7) and (17.8) when y fl. 2Qi. This lower bound is good enough because !x- Yi .::; 2 ·10 5 {C~) 2 diam Qi, by assumption. These three cases take care of the situation where x, y both lie in n. If X E n and y E A, then we can use the same arguments as in Cases 1 and 2 above, depending on whether dist(y, Qi) is larger or smaller than 10 5 (C~) 2 diamQi. When both x andy lie in A we can apply (17.2) directly. This completes the proof of Proposition 17.4.

18. The proof that (C4) implies (C7)

The idea, roughly speaking, is that the corona decomposition permits us to realize E as a subset of a countable union of Lipschitz graphs, with a lot of control on how these graphs fit together, and we shall build onr w-regular mapping by connecting up pieces of these Lipschitz graphs. Before beginning our construction we first address a technical point that will be needed later, concerning the "connectedness" of D.. A finite sequence Q1 , .•. , Qt of cubes in D. will be called a path if for each j, 1 :::; j:::; R.-1, Qi is either a son or the father of Qj+l· A subset of D. is said to be connected if any two points can be joined by a path in the subset. A useful fact which is easy to verify is that D. is connected if and only if every pair of elements has a common ancestor. If D. is the set of dyadic cubes in R n, then D. is not connected; it has 2n components. In general D. will always have only finitely many components. However, it is not hard to modify D. slightly in such a way that D. is connected, as follows. Let D. be as in Section 2, and let Li be the set of subsets of E defined by D.=

= U

-

-

.

D-j, where Q E D-j if Q E D-j and Q does not intersect B(O, 21 ),

J=-oo

or if Q is the union of theRE D.j that intersect B(O, 2i). Clearly each D- 1 is a partition of E, and Li still satisfies (2.2) and (2.3). It is not hard to check that Li also satisfies (2.1); the main point is the trivial fact that if Q E D.(j), Q' E D.( k ), j :::; k, Q ~ Q', and if Q intersects B(O, 2i), then Q' intersects B(O, 2A: ). Of course Li is also connected. In view of this we may as well assume that D. is connected, since we can replace it by Li otherwise. This change in D. will not upset the condition that E admits a corona decomposition. Indeed, we notP.rl in Section 2 that

the existence of a corona decomposition does not depend on the choice as long as (2.1) and (2.2) are satisfied. In this particular ca_::e it is even easier to transfer a corona decomposition for ~ to one for ~- For example, you can take all the cubes in ~ that were changed and put them into 8 without disturbing (2.4), and then take the S's with the changed cubes removed and reorganize them into regions that satisfy (2.5), (2.6), and (2.7). Let us now proceed with the construction of an w-regular mapping whose image contains E. Let 1J > 0 be small, to be chosen later. Its value will only depend on geometric constants. Let 8, g, F, S be as in Section 2. Given S E F, (2.6) tells us that there is a Lipschitz graph r over a d-plane P = Ps which well-approximates E with respect to S. Let II = IIs denote the orthogonal projection onto P, and let r(S) denote the part of r whose projection onto P is the closed ball B(II( x ), 20 diam Q( S) ), where x is any point of Q(S). We observe first that E is contained in the union of the r(S)'s, S E F, except possibly for a set of measure zero. To see this let Qj(Y) denote the element of ~i that contains y. Because of (2.5), Qj(Y) E 8 for at most finitely many j's, except for a set of y's of measure zero. Similarly, (2.7) tells us that for almost all y, Qj(y) is a Q(S) for only finitely many j. Hence for a. a. y there is an S E F such that y E Q( S) and every Q ~ Q( S) with Q 3 y lies in S. For such an y we have y E r(S) because of (2.6). We are going to modify the r(S)'s to make it easier to connect them together. We also introduce an additional dimension and work in R n+l instead of Rn; this will be needed to ensure that the mapping z : Rd --+ Rn+l that we define eventually is indeed regular. (Of course we identify R n with the subspace of R n+l where the last coordinate is zero.) of~

Fix a stopping time region S. In addition to the notation that we've already recalled, let d( x) be as in (7 .6), and let pl. be the ( n - d)-plane that passes through 0 and is orthogonal to P in R n. Let A : P -+ pl. be the function whose graph is r. We shall construct a surface r 1 (S) which is vaguely reminiscent of a pair of trousers with many legs (see Figure 1), and which is better to work with than r(S). Doing this requires quite a bit of notation, unfortunately. Let c1 ( S) denote the ( d - 1)-sphere in P with center II( x) and radius 20 diam Q( S). (Remember that x E Q( S) is the point selected in the definition of r(S).) Also let b1 (S) be the closed d-ball in P bounded by c1 ( S), and let a 1 ( S) be the ( d- 1)-sphere in R n+ 1 that is parallel to c1 ( S)

but which is centered at (II(x),A(II(x)), 2diamQ(S)) E P

X

p..L

X

R ~ Rn+l.

(Here we identify P x pl. with R n via the map that sends (p, q) to p + q.)

Figure 1. A symbolic picture of r 1 (S). The vertical direction contains both p..L and the ( n + 1)th_co-ordinate axis. For each minimal cube Q of S we make the following construction. For each child Q, of Q select a point Xi E Qi at distance ~ 10 diam Q;/Co from E \ Qi. We can do this if Co is large enough, because of (2.3). Let c0 (Q,) be the (d- 1)-sphere contained in P with center II( xi) and radius 2 diam Q;/C0 , and let bo(Q,) be the open ball it encloses.

Let I(S) denote the set of all Q;'s, where Q runs over all minimal cubes of S. If 'I is small enough, we get (using (2.6)) that

(18.1) (18.2)

1 dist (bo(Q;), bo(Qj)) 2: Co [diam Q; + diam Qj] dist (b 0 (Q;), c1(S))

> diam Q(S)

for all i, j E I(S), if:. j. We also let a 0 (Q;) be the (d- 1) sphere in Rn+I that is parallel to ·c0 ( Q;) but is centered at (IT(x;), A(IT(x;)), 2diamQ;). SetD(S)=bi(S)\(

U

bo(R))and

REl(S)

fo(S) = {(u,A(u),d'(u)): u E D(S)} ~ Rn+ 1, where d'(u) = d((u,A(u))), d(x) as in (7.6). Almost every y E E lies in f 0 (S) for some S. Indeed, for almost every y there is an S such that y E Q(S) and such that Q E S whenever Q ~ Q(S), y E Q, and for such an S we have y E rand d(y) = 0. We also have IT(y) E D(S) because of the way we chose the b0 (Q;), and because of (2.6). We define f 1(S) by taking f 0 (S) and adding some pieces to it; to wit,

f1(S) = fo(S) U T1(S) U (

U

To(Q;)) ,

Q;El(S)

where T1(S) and To(R) are defined as follows. We take T1(S) to be the "tube" obtained by joining each point (u,A(u),d'(u)), u E c1(S), to

(u,A(IT(x)), 2diamQ(S)) E a1(S) by a straight line. Similarly, if Q; E I(S), we take To(Q;) to be the tube obtained by joining (u,A(u),d'(u)) to

by a straight line for each u E c0 (Q;). It is not hard to see from our construction that there is a bilipschitz map of D(S) onto f 1(S) which sends c1(S) to a1(S) and co(Q;) to ao(Q;) for all Q; E I(S).

Next we want to do a similar (but simpler) construction for cubes Q in B. To simplify notations we associate to each Q E B a new stopping-time regionS= S(Q), where S has Q as its unique element (and so Q(S) = Q). We let :F denote the union of :F with {S ~ .6.: S = S(Q), Q E B}. Given Q E B, select any x E Q and any d-plane P that passes through x, and let c1 ( S) be the ( d- 1 )-sphere in P with center x and radius 20 diam Q; as before, we take b1(S) to be the closed ball in P enclosed by c1 (S). Let Q; be the children of Q, and let I(S) denote the set.of these Q;. For each Q; E I(S) choose a point x; E b1 (S), and let c 0 ( Q;) be the ( d -I)sphere in P centered at x; with radius 2diamQ;/C0 • The reader will most likely be unsurprised to learn that b0 ( Q;) denotes the open d-ball enclosed by c 0 ( Q;). If Co is large enough, we can choose the x; in such a way that (18.1) and (18.2) hold. [In this case we do not need to take the points x; inside II(Q;); when Q is a bad cube, we don't have to respect the geometry, because the geometry is bad.] We now set D(S) = b1 (S) \ ( U b0 (Q;)) also in this case. We view Q;El(S)

D( S) both as a subset of P and as a subset of R". We take r 1 ( S) to be the translation of D( S) inside R n+l by 2 diam Q( S) in the last variable:

r 1 (S) = {(u,2diamQ(S)) E R"+ 1 : u

E D(S)}.

We let a 1 (S) denote the "exterior" boundary of r 1 (S), and, for Q; E I(S), we take a 0 (Q;) to be the image of c0 (Q;) by the same translation of 2 diam Q( S) in the last variable. Thus, just like when S E :F, r 1 ( S) is a "nice" surface with boundary a1(S)U( U ao(Q;)), and there is a bilipschitz map of D(S) onto r 1 (S). Q;EJ(S)

We can now glue the various r 1 (S)'s together without much difficulty. . ForSE :F let T2(S) be a "tube" that joins the sphere a 1 (S) to a 0 (Q(S)). In writing a 0 (Q(S)) we must keep in mind that Q(S) is not only the top cube in S, but it is also the son of a bad cube, or of a minimal cube of an element of :F. We can choose T2 (S) ~ R"+ 1 in such a way that c- 1 diamQ(S) :S Yn+1 :S CdiamQ(S) for ally E T2 (S), and also so that there is a mapping zs of D(S) onto r 2 (S) = r 1 (S)UT2 (S) with the following properties: (18.3) zs is 1-regular, uniformly ins, and in particular, a.e.;

c- 1 :::; !Vzsl:::; c

( 18.4) the restriction of zs to c1 ( S) is an affine function that sends the sphere c1 ( S) onto a 0 ( Q( S)), and, for each i E I( S), the restriction of zs to co(Qi) is an affine map onto a 0 (Qi)· These affine maps are all compositions of translations, dilations, and rotations. A few comments are perhaps in order concerning (18.3). We can split D(S) into the union of a not-too-large spherical shell near the outer boundary and the remaining subregion of D(S). We can map the latter onto r 1 (S) in a bilipschitz fashion. We can't always map the spherical shell onto T2 (S) in a bilipschitz manner, because it could be that a 1 ( S) and a 0 ( Q( S)) intersect, but we can certainly do it using a 1-regular mapping. Although our definition of w-regular mappings considered only the case of maps defined on all of R d, it can easily be extended to subdomains of Rd. In the case of 1-regular maps, for example, one has to decide if the condition IVzl : : ; C a.e. should be replaced by the requirement that z be Lipschitz on its domain. Fortunately in our case this is not an issue; we can certainly build zs so that it is Lipschitz, and in any case V' z E L 00 implies z is Lipschitz on a domain like D(S). Note that (18.3) implies that T2 (S) is chosen so that r 2 (S) satisfies the regularity condition

whenever X E r 2 ( S) and 0 < R ::::; diam Q( S)' and that diam r 2 ( S) ::::; CdiamQ(S). Set E = U r 2 (S). We certainly have that E \ E has measure 0; we SEF

U r 0 ( S)). Before constructing a parameterse'F ization of E let us check that it satisfies (1.5). We begin by showing that for any x E R n+l and R > 0 we have already checked this for E \ (

(18.5) starting with the case where Xn+l = 0. Let S E F be such that r 2 (S) n B(x, R) ::/= 0. Assume first that S = S(Q) for some Q E B. Then diamQ S CR; otherwise, we have Yn+t > R for all y E r 2 (S), so that y ~ B(x, R). This implies that

Q ~ B(x, CR), and so

IB(x, R)n (

lj

r2(S)) I

~ L

SEF\F

1r2(S)I

~ L

QEB

QEB

Q~B(z,CR)

Q~B(x,CR)

CIQI

~ CRd,

by (2.4). Similarly, we can bound by CRd the total mass of the r 2 (S)'s for which S E F, Q(S) ~ B( x, CR). This leaves theSE F such that r 2(S) n B(x, R) f- 0 but Q(S) ~ B(x, CR). For each such S we certainly have

let us show that there is only a bounded number of these S's. To do this it suffices to show that there is a Q E S such that dist(x, Q) ~ CR and R ~ diam Q ~ C R, because there are only a bounded number of such Q's, and because the S's are disjoint subsets of .6.. For one of these S's we must have diam Q( S) ;::: R, since otherwise Q(S) ~ B(x, CR) (if C is large enough). Because of this we are reduced to finding Q E S with dist(x,Q) ~ CR and diamQ ~ CR; we can replace Q by an ancestor if necessary to get diam Q ;::: R. We may as well assume that diam Q(S);::: CR, since otherwise we can take Q = Q(S). To find such a Q we use the assumption that r2(S) n B(x, R) f- 0. If R- 1 diamQ(S) is large enough we have T2 (S) n B(x,R) = 0, and so B(x, R) must intersect r 1 (S). Suppose that B( x, R) intersects r 0 (5), and that y E R n+l lies in the intersection. Write y = (y', Yn+l), y' E R n. Then IYn+II ~ R, and Yn+l = d(y'), where d(·) is as in (7.6) (for this 5). Thus d( y') ::::; R, y' E B( x, R) n R n, and the definition of d( y') provides us with the Q that we want. Suppose now that there is a y = (y', Yn+I) in B(x, R) n T 1 (5). By definition T1 ( S) is a union of line segments, and there is a >. E [0, 1J such that y = >.(u,A(u),d'(u))

+ (1- >.)(u,A(IT(xs)),

2diamQ(S))

for some u E c1 (S), where xs denotes the point selected in the definition of r(5) (which we called x before). Because IYn+ll ::=:;Rand R- 1 diamQ(5) is big, we must have that d'(u) ~Rand that 1- >.::::; R(2 diam Q(S))- 1 .

This implies that IY' - ( u, A( u)) I ::; C R, by using the fact that IA(u)- A(II(xs))l::; 20diamQ(S) in the above formula for y. Since d'( u) = d( u, A( u)) ::; R and d( ·) is Lipschitz, we have that d(y') ::; CR, and so we are in the same situation as before. The only remaining possibility is that there is a y = (y', Yn+I) in B(x, R)nT0 (Q;) for some Q; E J(S). By definition y lies on the line segment that joins (u,A(u),d'(u)) to (u,A(II(x;)),2diamQ;) for some u E c0 (Q;). If diam Q; ::; CR, then we can take Q = Q;; if not, the same argument as the one we just used implies that d(y') ::; CR, and this again provides us with the sort of Q that we want. Combining all these cases we get that (18.5) holds when Xn+l = 0. Let us show that this is still true if Xn+l f. We may as well assume that Xn+l > 0, since Yn+l 2:: 0 whenever y E E. We can also require that Xn+l > 2R, since the other possibility can easily be reduced to the case where Xn+l = 0. Write x = (x',xn+d· LetS E F be such that f2(S) intersects B(x, R). If S = S(Q) for some Q E B, then diamQ "'Xn+l, and dist(x',Q)::; Cxn+I, and there are only a bounded number of Q's like that. If E F, then diam Q( S) 2:: c-l Xn+l' and an argument like the one we gave in the Xn+l = 0 case shows that there can only be a bounded number of these S's as well. (For each such S you prove that there is a Q inS with diam Q"' Xn+l and dist(x', Q)::; Cxn+I·) Since

2·

s

for any S E :F, we get that (18.5) is true in this case also. Altogether now we have proved (18.5) for all x E Rn+l. Actually, our proof gives a bit more: for each x E an+l and R > 0 we have (18.6)

L

IB(x, R)

n r2(S)I::;

SE7f

This controls the overlapping of the f 2 (S)'s. Let's check the lower bound, i.e., that (18. 7)

CRd.

whenever x E E, R > 0. Notice that {y E E : zero. Indeed, we have

Yn+l

= 0} is exactly E modulo a set of measure

EnRn=

U(r 0 (S)nRn) SEJ:

and ro(S) n Rn ~ E by definition, and we have already seen that E \ E has measure zero. From this it follows that (18. 7) holds if Xn+l = O, since we are assuming that E satisfies (1.5). Suppose now that x = (x',xn+I), Xn+l > 0. ChooseS E :F so that X E r2(S). By construction we have Xn+l

+ dist(x', Q(S))::; C diam Q(S).

If R is large enough compared to diam Q(S), then there is a pointy E Q(S) such that y E B(x, ~R). Hence B(x, R) 2 B(y, ~R), and since y E E and E \ E has measure zero, we can derive (18.7) from (1.5) again. If R ::; C diam Q( S), then we have

This completes the proof that E satisfies (1.5). We now construct our parameterization z( ·) of E. For each S E F we are going to choose a mapping hs: Ds-+ Rd which is the composition of a translation, rotation, and a dilation. We shall define z(·) on hs(D(S)) by z(x) = zs o h8 1 (x), where zs: D(S)-+ r 2 (S) is the map we chose earlier (satisfying (18.3) and (18.4)). We need to choose the hs's in such a way that we can do this coherently. Fix So E :F, and let hs0 be any map of D(S0 ) into Rd that is the composition of a translation, rotation, and a nontrivial dilation. For each Q, E /(So) there is exactly one choice of hs(Q;) [S(Q;) is the element ofF whose top cube is Q;] such that hs0 (co(Q;)) = hs(Q;)(c 1 (S(Q;))) and such that zs(Q;) o hslQ;) agrees with zs0 o hs: on that set. By repeating this procedure we choose hs for all S E F with Q(S) ~ Q(S0 ). We can also run this process backwards to choose hs for all S E F with Q(S) 2 Q(So), and then run it forwards again to get all S E F. We do indeed reach all S' E F, because of the connectedness of/}...

Set

n= U

hs(Ds). (Remember that each Ds is closed, by construc-

SEF

tion.) Thus f2 ~ Rd, and we can define z: f2-+ Rn+I by z(x) = zsoh5 1 (x) if x E hs(D(S)). This is well-defined by construction. In particular, the interiors of the hs(Ds) are pairwise disjoint. We need to understand n better. By definitions, hs(Ds) = f3s \ /S, where f3s is a closed ball and /S is a countable union of open balls. One can check that /S. ~ / 0 f3s; this follows from the corresponding inclusion back on Ds, and the fact that hs does not distort relative size. Hence if B is a ball in /s, lOB ~ f3s. Our construction also gives that for S, S' E F,

f3s ~ f3s' when Q(S) ~ Q(S'),

(18.8) f3s

n f3s'

=

0 when Q(S) n Q(S')

=

0.

These observations imply that f3s ~ / 0 f3s' when S, S' E F, Q(S) ~ Q( S'), and S f. S', and also that

(18.9) It now follows that

U

f3s

= Rd.

Indeed, if Sj is any sequence in F such

SEF

that Q(S;) ~ Q(Sj+I), Sj f. S;+b for each j, then f3si 2 (10)i- 1 f3s1 • Next let us show that M = Rd \ n has measure zero. Let S E F be given. Let S1 ,k be an enumeration of the S(Q)'s with Q E I(S). Repeat this process; if we have chosen Sj,k E F, k = 1, ... , N(j ), for a given j, we let Sj+l,l be an enumeration of the S(Q)'s that arise from Q E I(Sj,k), k = 1, ... , N(j). This process may terminate after a finite number of steps. Set /31·, k = f3sJ.•""'•. From (18.9) we obtain

(18.10)

In particular,

Q(y/3j,k) has measure zero. This set contains Mnf3s, and

so we conclude that M has measure zero, since Sis arbitrary. We can extend z(·) so that it is defined on all of Rd, not just n. Fix x E M, and choose S E F so that x E f3s. Let Sj,k be as above, and for j = 1, 2, ... , let k(j) be the index such that x E f3j,k(j)· (Notice that there must be infinitely many Sj,k's that contain x if x EM.) Then Q(Sj,k(j)) is

a decreasing sequence of closed sets whose diameters tend to zero, and so their intersection consists of a single point, which we take to be z(x ). Observe that z(M) has measure zero; if y E z(M), then y E Q(S) for infinitely many S E :F, and the set of such y has measure zero because of (2.4) and (2.7). Let us check that this extension of z( ·) to R d is continuous. By construction, (18.11)

diamr 2(S)::; CdiamQ(S),dist(r2(S),Q(S))::; CdiamQ(S).

Combining this with (18.8) it is easy to see that (18.12)

diamz(,Bs)::; C diam Q(S), dist(z(,Bs), Q(S))::; C diamQ(S).

From this it follows easily that z( ·) is continuous even at points of M. This is a pretty good time to show that z(Rd) 2 E. We already know that z(f!) = E contains almost every point in E; the same argument shows that z(,Bs \ M) = u{r2(S') : Q(S') ~ Q(S)} contains almost all points in Q( S). Hence z(7J s) contains all points with positive lower density in Q(S), because z(fis) is compact. From here and (1.5) it follows easily that z(Rd) = Uz(Ps) 2 E. It remains to show that z(·) is w-regular for some Arweight won Rd. By definitions z(.) is locally Lipschitz on n, and so V' z( X) is defined almost everywhere. Let us show that for any x E E and any R > 0 we have (18.13)

c-

1

/B(x, R) n E/

J

::;

/Y'z(y)/ddy::; C/B(x, R) n E/.

z- 1 (B(z,R))

Notice first that we don't have to worry about M or z( M), since they bot~ have measure zero. We know that the analogue of (18.13) holds for z, E replaced by zs a h5 1 , r 2 (S), because each zs is 1-regular, and because hs is just a composition of a translation, rotation, and a dilation. Hence

J

/Y'z(y)/ddy =

L

J

SET z- 1 (B(z,R))n(/1s\"Ys))

z-l(B(z,R))

~

L SEF

/B(x, R) n rz(S)/.

The desired estimate now follows from (18.6) and (18.7). Next we want to check that w(x) = I'Vz(x)ld is an A 1 -weight. Let B = B(x, R) be a given ball in Rd. Choose S so that B ~ Ps and so that Q( S) is minimal with this property. Let 6( S) E R+ denote the factor by which hs dilates; this makes sense, because hs is a composition of a translation, rotation, and dilation. Observe that I'Vz(x)l ~ {C6(S))- 1 a.e. on f3s.

(18.14)

To see this notice that (18.15) whenever S E :F and Q E I(S), because h-s(co(Q)) = hs(Q)(c1 (S(Q)) and c0 (Q) is much smaller than c1 (S(Q)), although not excessively smaller. In particular S(S') ~ S(S) if Q(S') ~ Q(S), and (18.14) now follows from this, (18.3), and the definition of z [z = zs o h$ 1 on hs(D(S))]. To finish the proof that w is an A1 -weight we need to show that (18.16) In view of (18.13) and (18.5) we need only show that z(B) is contained in a ball of radius C6(S)- 1 R. Set as= f3s \"''sand Bo = B n {as U ( U as(Q;))}. Of course B Q;EI(S)

intersects as, since otherwise S is not minimal. Let y0 be a point in this intersection. We know that z is Lipschitz on B 0 , with norm ~ CS(S)-I, because of (18.3), the remarks concerning (18.3) given after (18.4), and (18.15). Hence z(Bo) ~ B(y 0 , CS(S)- 1 R). Suppose that B \ B 0 is not empty. Let Qi E I( S) and Q 1 ,j E I(S( Qi)) be such that B intersects as(Qt,j)· Since as(Qt,j) ~ / 0 f3s(Q;) and B intersects the complement of f3s(Q;)• we have that

R

~~

radius f3s(Q•>

Because we also have diamz(f3s(Q;)) that

~

~ c- 1 6(S) diam Qi. C diam Qi from (18.12), we conclude

z(f3s(Q,)) ~ {y: dist(y, z(B 0 )) ~ CS(S)- 1 R} ~ B(y 0 , C'5(S)- 1 R)

if C' is large enough. It now follows that z(B) ~ B(y0 , C'b(S)- 1 R). This proves (18.16), and hence that w E A1. Combining (18.16) and (18.14) we get that w(B)l is comparable to b(S)- 1 R, and putting this fact together with the one we just proved gives (18.17)

diamz(B) $ Cw(B)l

for all balls B ~ Rd. This implies that z has a locally integrable distributional gradient on Rd, not just on n. Indeed, the distributional directional derivatives of z can be given as weak limits of C 1 (z(x +tv)- z(x)), and (18.17) implies that for 0 < t $ 1 these functions are uniformly dominated by sup O
(R j w)! 1d

B(z,R)

Thus the distributional directional derivatives must be locally integrable. This argument also gives /V'zi S Cwl a.e., but of course /Y'z/d = w a.e. by definition of w. This completes the proof that (C4) implies (C7), and hence of the theorem stated in Section 1.

19. A variant of (C2) and (C3)

Set 1

(19.1)

syr(x,

t) = (t;d j

j

(t- 1 dist(2y- w, E)Y dydw) r:

B(z,t)nE B(z,t)nE

for 1 .::; r .::; oo. This measures the extent to which E is symmetric about each of its points. In particular, if syr(x, t) = 0 for all x, t, then E must be symmetric about all of its elements. The variant of (C2) and (C3) that we shall consider is the following: (C8) sy 1 (x,t) 2 d~dt is a Carleson measure onE x R+· This condition is equivalent to the others, and this is still true if we replace r = 1 by any r, 1 :::=; r < d'2_!2 (1 :::=; r :::=; oo if d = 1). We are going to indicate the proofs of these results in this section. One can think of sy 1 ( x, t) as a geometrical version of an average of second differences of a function. When E is a Lipschitz graph it is not hard to make precise the relationship between sy 1 ( x, t) and averages of second differences of the function being graphed. Let us first check that (C8), with r = 1, implies our local symmetry condition (LS) from Section 4. Let r > 0 be given, as in Definition 4.2. Suppose that Q E R( r ), so that there are y, w E 2Q such that dist(2y- w, E)~ r diam Q. If we can show that this implies that

(19.2)

for any x E Q and any t such that 3 diam Q ~ t ~ 4 diam Q, then it is easy to derive ( 4.3) from (C8). Fix any such x and t. It is easy to check (19.2) using the definition (19.1) and the fact that dist(2y' -

w', E) 2:

T

10 diam Q

for any y' E B(y, 1 ~ 0 rt), w' E B(w, 1 ~ 0 rt). From here it is not too difficult (but rather tedious) to prove that (C8) with r = 1 implies (C4) by modifying the arguments used to show that (C2) and (C3) each imply (C4). We omit the details, because they are somewhat messy, and because we now know that (LS) itself implies (C1)-(C7), as mentioned just before Lemma 5.13. Assume now that E satisfies (C4) and consequently the conclusion of Proposition 5.5 as well, and let us show that ( C8) must hold, and even the stronger version where r = 1 is replaced by r < d2_!2 (r ~ oo if d = 1). Let "' > 0 be small and let k > 0 be large, to be specified later. Let B, Q, :F, S E :F be as in the definition of a corona decomposition. Let Q(T'f, k) be as defined in the beginning of Section 6. We want to modify the corona decomposition slightly, so that all the good cubes also lie in Q(T'f, k). Thus we replace B by B U (.6. \ Q(T'f, k)) and g by g n Q(T'f, k). By assumptions (2.4) is not disturbed, but (2.5)-(2.7) notice the change. However, iffor each S E :F you subdivideS n Q("', k) into maximal regions that satisfy (2.5), then the resulting family of regions also satisfies (2.6) and (2. 7). Thus we may as well assume that our corona decomposition has g ~ Q(.,.,, k). The argument we use to show that (C8) holds is quite similar to the one in Section 15. We shall use the same notations also, and the reader may find it helpful to review that section. By Lemma 15.2 it suffices to show that, for r < d2_!2, (19.3)

dxdt 2 syr(x, t) Xs(x, t)-t-

is a Carleson measure for each S E :F, with norm bounded independently of S. We may as well replaceS here by S1, where S1 = {(x,t) E S: t ~ 10- 3 diam Q(S)}; the contribution to (19.3) coming from S \ S 1 can easily be controlled using syr(x, t) ~C.

Fix S E F and r < d2_!2. As in Section 15 we let r be the Lipschitz graph over ad-plane P as promised in (2.6). We also let pl_ be the ( n- d)plane that passes through the origin and which is orthogonal toP, and we let A : P ---4 pl_ be the function whose graph is r. We denote by II and II j_ the usual projections of R" onto P and Pl_, and we define d(x) and D(p) by (7.6) and (8.1), as always. In order to show that (19.3) is a Carleson measure we are going to estimate syr(x, t) in terms of an Lr-average of second differences of A, plus error terms arising from the approximation of E by r. Fix x, t E S1. Given y E R", set y = (II(y), A(II(y))) E r. If y E 2Q(S) we have IY- ill:::; Cryd(y), because of (2.6). Hence

syr(x,t):::;

(c

2d

j

j

(C 1 dist(2y- w,E)r dydw);

B(x,t)nE B(x,t)nE

(19.4)

l

+ c71

j

(cd

1

(t- d(y)f dy) •

B(x,t)nE

The first term on the right side of (19.4) can be dominated by 1

(19.5)

(

c2d

1 1

(t- 1 dist(2q- s, E)r d>.(q)d>.(s)) •

B(p,2t)nP B(p,2t)nP

wherep = II(x), q = (q,A(q)), 8 = (s,A(s)), and). is the measure obtained by pushing Hd j2 Q(S) down to P using II. We want to replace ). by Lebesgue measure in (19.5). Of course A is comparable to Lebesgue measure on II(Z), and so we need only look at P \ II(Z), which is the set of points q E P such that D(q) > 0. In this region we have that ). is comparable to Lebesgue measure at the scale of 5ryD(q), at least if you stay close to II(Q(S)). More precisely, we claim that (19.6) for all q E P such that (19.7)

dist(q, II(Q(S))):::; 110 diam Q(S).

(Actually, we need (19.7) only for the first inequality in (19.6).) Before checking (19.6) let us record and verify the useful fact that (19.8)

c-

1

d(u) :5 D(II(u)) :5 d(u)

when u E 2Q(S).

We have seen this sort of thing before, in Lemma 8.21, but the circumstances were somewhat different there. The second inequality in (19.8) is a direct consequence of the definitions, and so we need only concern ourselves with the first. Let u E 2Q(S) be given, and choose v E Q(S) such that

I II(u)- II(v) I

+ d(v) :5 2D(II(u)).

This we can do, because of the definitions of D(·) and d(·) in (8.1) and (7.6). We can control lu- vi by projecting onto r, as follows:

lu- vi :5 lu- (II(u))"l + I(II(u))"- (II(v))"l

+ lv- (II(v))"l

:5 c.,d(u) + CIII(u)- II(v)l + c.,d(v) :5 C11d(u) + CD(II(u)). It is easy to obtain the first half of (19.8) from these inequalities together with d(u)- d(v) :5 lu- vi, if 11 is small enough. Now let us check the right side of (19.6). It suffices to show that

(19.9)

rr- 1 (B(q, 511 D(q)) n 2Q(S)) ~ B(ii, C11 D(q)).

Let u be an element of the left side of (19.9). Then u E 2Q(S), and so

lu- (II(u))"l :5 C11d(u). From (19.8) we have

d(u) :5 CD(II(u))

:5 CD(q) + CIII(u)- ql :5 CD(q), since u lies in the left side of (19.9). Thus

lu - iii :5 lu - (II( u ))"I + I(II( u ))" - iii

:5 c.,D(q) + CIII(u)- qj

::; c11 D(q).

This proves (19.9), and also the right side of (19.6). Next we verify the left side of (19.6). By definition of D(q) we can find a Q E S such that

(19.10)

dist(q,II(Q))

+ diamQ :S 2D(q).

By replacing Q by an ancestor if necessary we may also assume that

D(q) :S CdiamQ.

(19.11)

Because Q E S, Q is a good cube, and in particular Q E Q(ry, k). Thus Q satisfies (6.2), with f. replaced by"'· We want to use this to produce a point in tQ(S) whose projection onto Plies near q. We shall then use that and the regularity of E to get the first inequality in (19.6). Let us begin by observing that Angle(P, PQ) :S Cry.

(19.12)

This is a minor variation of Lemma 5.13; the point is that every element of Q is within Cry diam Q of both PQ and r, and r is a Lipschitz graph over P with constant :S 1/· The argument used to prove Lemma 5.13 can also be applied in this case. Let w be the element of PQ such that II( w) = q, and let Q be the projection of Q onto PQ. Then

(19.13)

dist(w,Q) :S (1

+ Cry)dist(q,II(Q)) + CrydiamQ.

Indeed, let y E Q be such that iq- II(y)i = dist(q,II(Q)), and let ii be the projection of y onto PQ. Then iY- iii :S rydiamQ, and dist( w, Q) :S iw- iii (by (19.12))

:S (1 + C11)iii(w)- II(ii)i

+ Cry){iq- II(y)i + iii(y)- II(ii)i} :S (1 + C11){ dist( q, II( Q)) + "'diam Q}. :S (1

In particular we have dist(w,Q) :S CdiamQ, by (19.10) and (19.11). Because w E PQ and (6.2) holds with f. replaced by ry, we get that there is a point u E E such that

(19.14)

iu- wi :S 11diamQ.

(This is the place where we need k to be reasonably large.) From (19.14) and (19.10) we obtain III(u)- ql ~ 2ryD(q),

(19.15) and hence

rr- 1 (B(q, 5ryD(q))) 2 B(u, ryD(q)). To prove the first inequality in (19.6) it is enough to show that

B( u, ryD( q)) n E

(19.16)

~

2Q( S),

since >.(B(q, 5ryD(q))) = III- 1 (B(q, 5ryD(q))) n 2Q(S)I (by definition of>.). To prove (19.16) it suffices to check that dist( u, Q( S)) ~ ~ diam Q( S) if ry is small enough, since ryD(q) ~ ~~rydiamQ(S) ~ tdiamQ(S) if ry is small enough (and if q satisfies (19.7)). Using (19.14) and (19.13) we have

+ dist(w, Q(S)) ~ ry diam Q + dist(w, Q) ~ 2ry diam Q + dist( w, Q) ~ Cry diam Q + (1 +Cry) dist( q, II( Q)).

dist(u, Q(S)) ~ lu- wl

This is less than ~ diam Q(S) if ry is small enough and q satisfies (19.7). [We are using here the fact that Q ~ Q(S).] This finishes the proof of (19.6). Let us use this to analyze (19.5). It is not hard to show that

(19.5)

1

~ c (c d j

j

2

(c 1 dist(2q- s, E)r dqds)-;:-

B(p,at)nP B(p,3t)nP 1

(19.17)

+C(cd

J

(ryC 1 D(q)yd>.(q))-;:-

B(p,3t)nP

We shall only sketch the argument. The first step is to find a covering of II(2Q(S)) by II(Z) and a countable family of balls that have bounded overlap and which are of the form B(a, 5ryD(a)). This is similar to the story about the R;'s in Section 8, and it is not difficult. To derive (19.17)

you use this covering to break up the integrals in (19.5), and then you control the pieces separately using the following three facts. First, the oscillation of dist(2q- s, E) in q or s over a ball B( a, 5TJD( a)) is bounded by C71D(a). Second, we know that>. and Lebesgue measure are comparable on II(Z), and that they give the balls B(a, 5TJD(a)) comparable mass. Third, since (x,t) E S we have Ct ~ d(x) ~ D(p). This implies that if B(p,2t) intersects B(a, 5TJD(a)), then Ct ~ D(a). This ensures that our covering is thick enough to be useful for controlling the integrals over B(p, 2t). More precisely, it implies that B(a,5TJD(a)) ~ B(p,3t) if B(a,5TJD(a)) intersects B(p, 2t) and TJ is small enough. The last term in (19.17) is dominated by

To show this it is enough to check that

rr- 1 (B(p, lOt)) n 2Q(S)

(19.18)

~

B(x, Ct),

because of (19.8) and the definition of>.. The proof of (19.18) is similar to that of (19.9), and we omit the details. It is important to use the fact that Ct ~ d( x ), which holds because ( x, t) E S. Putting these inequalities back into (19.4) we obtain

syr(Y, t)

~ C (C

2d

j

l.

J

(t- 1 dist(2q- s, E)r dqds) •

B(p,3t)nP B(p,3t)nP l

(19.19)

+ CTJ

(cd j

1

(t- d(y)rdy)"

B(x,Ct)nE

To deal with this we observe that dist(2q- s,E) ~ l2q- s- (2q- s)"l

+ dist((2q- s)",E).

Of course l2q- s- (2q- s)"'l = I2A(q)- A(s)- A(2q- s)l, and so its contribution to the first term on the right of (19.24) is at most 1

J

(t- 1 IA(2q- s) + A(s)- 2A(q)lf dqds)

~

B(p,3t)nP

~

~

C-yr(P, 6t)

C

inf

B(IT(.r),t)

'Yr( ·,lOt),

where 'Yr(P, t) is as in (15.4). Thus

syr(x,t)~C

1

inf

B(ll(.r),t)

j (t- 1 dist(u,E)rdu)~

'Yr(·,lOt)+C(cd

B(p,9t)nP 1

j

+Cry (cd

(19.20)

~

(t- 1 d(y)ydy)

B(.r,Ct)nE

Let us show that the middle term on the right can be controlled by the last term. To see this we first check that 1

( cd

j

(t- 1 dist(u,E)rdu)

~

B(p,9t)nP

(19.21)

1

~ C (cd j

~

(t- 1 dist(u,E)Yd.\(u))

B(p,10t)nP 1

+ c (cd

j

(ryc 1 D(u)rd).(u))

~

B(p,10t)nP

This can be proved using the same sort of covering argument as used to get (19.17): dq and d.\.(q) are comparable on the scale of 5ryD(u), and dist( q, E) oscillates by at most CryD( u) on a ball of radius 5ryD( u ), because

it is Lipschitz. It is also important that D(p) :::; d( x) :::; Ct and that t < 10-3 diamQ(S), which are true because (x, t) ESt. The second conditio~ is needed to ensure that ( 19. 7) holds in the relevant cases. Next we convert the integrals on the right side of (19.21) into integrals on E. Using (19.18) and the definition of,\ we obtain (19.22)

left side of (19.21) 1

j

:::; C (rd

(t- 1 dist((II(y), A(II(y))), E)Y dy) r:

B(z,Ct)n2Q(S) 1

+

c (rd

j

(ryr 1 d(y)fdy) •

B(z,Ct)nE

We have also used the fact that D(II(y)) :::; d(y). The first term on the right is dominated by the second, because of (2.6), and because all the relevant y's lie in E. From (19.22) it follows that the middle term on the right side of (19.20) is indeed dominated by the last term, so that ~

(19.23) syr(x,t) :S C

inf

B(II(z),t)

'Yr(·, IOt)+Cry (cd

J

(t- 1 d(y)ydy) •

B(z,Ct)nE

The remainder of the proof that ( 19.17) is a Carleson measure with bounded norm if r < 2 is exactly like the corresponding step in Section 15 (beginning at (15.9)). This finishes the proof of the result stated at the beginning of the section, that ( C8) is equivalent to ( C1 )-( C7), even if r = 1 is replaced by any r such that 1 :S r < 2 .

/'!.

l!

20. A counterexample

Although the weak geometric lemma (5.2) is a useful auxiliary condition, it is not strong enough to imply rectifiability properties of the set in question. We shall now construct a 1-dimensional set E in R 2 which satisfies the weak geometric lemma but not much else. On the other hand, we shall show in [DS3] that a d-dimensional regular set that satisfies the weak geometric lemma and another geometrical condition (big projections on some d-planes) has big pieces of Lipschitz graphs, and satisfies (C1 )-(C8) in particular. We use a minor modification of the Van Koch snowflake. The set E will be obtained as the limit of a sequence En of sets, with each En being the union of 4n line segments of length 4 -n. Given a sequence {an} of small real numbers we construct the En's recursively according to the following recipe. We take E 0 to be the unit interval on the x-axis. Suppose En-l has been constructed. To construct En we replace each line segment L of En-l by four segments £1, L2, La, L 4 with the following properties. (See Figure 2.)

= 1, ... , 4.

(20.1)

The length of L; is 4-n, i

(20.2)

The endpoint of L; is the initial point of Li+ 1 , i = 1, 2, 3.

(20.3)

The L;'s make the angles 0, an, with L.

71"-

an, and 0, respectively,

The midpoint of L is also the midpoint of the segment that joins the initial point of L 1 to the endpoint of £ 4 . The an's are allowed to be quite arbitrary, for the moment anyway, although we do ask that they be small enough so that the various segments do not cross each other. (ian!< 1 ~ 0 for all n will do.) (20.4)

. . .~. .L . . ~--~·-··· Figure 2

Let E be the set obtained in the limit from the En's, using the Hausdorff metric, for instance. It is not hard to see that E is (locally) regular; if B is a ball centered on E with radius 4 -n, then the total length of En n B is about 4 -n, and taking further generations does not alter this significantly. It is also not hard to check that E satisfies the weak geometric lemma if and only if Cl'n tends to 0. If I:G; < oo, then E is contained in a curve of finite length: the distance from the initial point of one of the segments of En-! to the initial point of the corresponding piece of En is about G~ 4 -n, and similarly for the endpoints, and so you can connect the various pieces of E together and get a curve which contains E and has finite length (and is even chord-arc). If I:G~ = oo, then the curve you get from the method just described has infinite length. Moreover, any curve that contains E must have infinite length. This follows from the theorem of Peter Jones in [J3] and the fact that the measure p. = /3 1 (x, t) 2 d"/t on EX R+ [as in (C3)] satisfies

p.(B

X

(0, R))

= 00

for any ball centered on E and any R > 0. Even more is true. If I:G~ = +oo, then Jr n EJ = 0 for any rectifiable curve r, so that E is totally unrectifiable in the sense of geometric measure theory. It suffices to check this for Lipschitz graphs, because any rectifiable curve is contained in a countable union of C 1 curves, except perhaps for a set of length zero. (See [Fe], for example.) Suppose that r is a Lipschitz graph and IE n rJ = T > 0. For each n, E is naturally divided into 4" parts E(n, i), 1 ~ i ~ n. This is easily seen from the construction of E; these 4n parts of E correspond to the 4"

components of En.

Because~

If n E(n, i)l = r, there is ani such that

• If n E(n, i)l;:::

;4-n =

riE(n, i)l.

Since any E( n, i) can be obtained from any E( n, j) by a rigid motion, we conclude that for each n, j there is a Lipschitz graph r n,j such that

Ifn,j n E(n,j)l;::: riE(n,j)l. In other words, we can use the self-similarity of E to pass from the positivity of IEnfl toE having big pieces of Lipschitz graphs. In particular E must satisfy (C6), which implies that E is contained in a regular curve and that the measure p. is a Carleson measure, both of which we know to be false. This proves that IE n fl = 0 for all Lipschitz graphs f.

21. Some open problems

There are three omissions in the theorem that are particularly glaring. The first is that we don't know if we can restrict ourselves to a "small" collection of kernels in (C1), e.g., to K(x) = i = 1, ... , n. A partial result in this direction is given in [Fg]. A related result is in [Ma2]. We also don't know so much about restricting ourselves to small sets of 1/J's in (C2). Our arguments in Sections 4,5, and 6 (for proving Proposition 6.5) relied heavily on much flexibility in the choice of 1/J's. However, we really didn't need so many 1/J's in Sections 9, 10, and 11, although it simplified the proof substantially. We used the fact that we could choose 1/J's that satisfy (10.1) and {10.2) in order to obtain (10.9) and (10.10). We could have used much less special (and much smaller) families of 1jJ 's to get versions of (10.9) and (10.10) with e2 replaced by e2 + 64 • Roughly speaking, the reason for this is that you can split 1/J( (p, A(p))- ( q, A( q))) into the part that is linear in A and the remainder, and the latter is controlled by Cli2 using the fact that the Lipschitz norm of A is ~ Cli. The second omission is that we do not know whether "E has BPLG" is equivalent to our other conditions. (Of course it implies (C6).) In some sense (C5) and (C7) are not so far from this, since we know from [D4] that images of Rd under bilipschitz or w-regular maps in Rn have BPLG. The third is that the theorem does not say anything about the case when E is not regular. On the other hand, the theorem of Jones [J3] characterizing the subsets of planar curves with finite length in terms of quadratic conditions on the {300 's does not need the set to be regular. Of course for this it helps that you are working with {300 instead of /3q for q < oo, and, as we've pointed out, Jones and Fang found a counterexample to show that such quadratic conditions on the {3 00 's need not hold for higherdimensional Lipschitz graphs.

rxf#r,

A natural conjecture concerning non-regular sets that might be provable using current technology is that qualitative versions of ( Cl )-( C3) imply that E is rectifiable. For example, if Hd(E) < oo and if sup I •>O

j

K(x- y)J(y)dy I

En{lx-yl>•}

is finite a.e. onE fq_r all K as before and all fin some reasonably rich class (e.g., bounded measurable functions with compact support), then is it true that E is rectifiable? Mattila (Ma2] has obtained a result related to this problem. Roughly speaking he gives a characterization of rectifiability of one-dimensional sets in the plane in terms of the existence of principal values for the Cauchy kernel. The natural qualitative versions of (C2) and (C3) are given in terms of the finiteness a.e. of square functions. For (C2) this would be the requirement that for all 'ljl as in ( C2),

J j 'ljl~_(yI

t=:;o

z)dz

1

2

dy < oo

E

for a.e. x E E. The counterpart to (C3) is

for a.e. x E E. One can formulate a qualitative version of (C8) similarly. This possible relationship between the finiteness a.e. of these square functions and the rectifiability of E is analogous to the classical results that characterize the existence of limits or derivatives a.e. by the finiteness a.e. of square functions, as in Chapters 7 and 8 of (St]. There are already results in geometric measure theory that are reminiscent of the characterizations of a.e. existence of limits or derivatives in terms finiteness a.e. of maximal functions. For example, there are results that relate rectifiability (which is equivalent to the existence a. e. of approximate tangent planes) to the existence of cones at a.e. point in the set which contain most of the mass of the set asymptotically as you shrink down to the given point. There are other questions like these which show up in connection with harmonic measure estimates and the Ahlfors distortion theorem. (See

Chapter 1 of [B], especially p 33-34, or [BCGJ].) One formulation goes as follows. Let r be a Jordan curve in the plane, not necessarily rectifiable, with complementary components f2 1, f22. Given z E r, t > 0, let 6i( z, t) denote the length of the largest component of aB( z, t) n n,, i = 1, 2. Thus if r is a line, 6;(z, t) = ?r, and in general

e(z, t) =max l1r- 6i(z, t)l •=1,2

measures how close r is to being a line. Let E be a closed subset of r of finite length. If

1 1

0

dt e(z,t) 2 - < oo t

a.e. on E, does that imply that E is rectifiable, as a !-dimensional set in the plane? Notice that e(z,t) is quite similar to syq(x,t) (defined in Section 19); syq(x, t) is an integrated version of e(z, t). However, we do not know how to show that quadratic conditions on e( x, t) imply rectifiability properties of E, even if we make quantitative versions of these assumptions, i.e., that E is regular and e(z, t) 2 dz~t is a Carleson measure onE x R+. The problem is that e( z, t) is not stable enough to apply the methods of this paper. Another basic issue is to find and understand more simple geometrical sufficient conditions onE for singular integrals to be bounded on E. Some things are known (see [D4], [DJ], [D53], [51, 3, 5]), but it is quite easy to generate questions of this type that we do not know how to answer. Here is an example. Let E be a d-dimensional regular set in R n. Suppose that there is a constant C so that for each x E E, R > 0 there is a relatively open subset U of E which is homeomorphic to a ball in Rd and satisfies En B(x,R) ~ U ~En B(x, CR). Does this imply that ( Cl) holds for E? This can be thought of as a higher-dimensional version of the chordarc condition for curves. When d = 2 it is shown in [52] that if you make a priori smoothness assumptions about E, then there is a quasisymmetric parameterization of E (by a plane) which is w-regular for some w E A=, with estimates that do not depend on the a priori assumptions. We don't know anything for d > 2; for d = 2 we also don't have any direct geometrical understanding of this condition. (The result was proved by using

uniformization to get a conformal map from the plane to E, and then estimating extremal length.) A productive method for generating questions of this sort is to take a book on geometric measure theory ([Fe] for instance, or [Ma]), look at some of the known results concerning rectifiability, and try to find quantitative versions. For example, it is known that A is rectifiable if and only if Hd(IIvB) > 0 for all measurable subsets B of A with Hd(B) > 0, and almost all d-plane& V, where IIvB denotes the orthogonal projection of B onto V. We would like to have a version of this theorem with estimates. Another group of problems pertains to (C7) and the existence of good parameterizations. An obvious question is whether in (C7) we can do any better than A 1 weights, e.g., is (C7) equivalent to being able to find a !-regular mapping whose image contains E? At least one of the authors thinks that the answer should be no, but neither knows how to prove it. We don't know any good geometrical invariants to help us distinguish between !-regular and w-regular mappings. We observed in the introduction that (C7) is equivalent to the version of itself with A 1 replaced by A 00 • Is there a more direct way to see this? One could hope that an w-regular mapping for w E A 00 could be "reparameterized" somehow to get an w-mapping, w E A 1 . An optimist might hope that this could be done by composing with a quasiconformal mapping on Rd. This leads us to the old and difficult problem of understanding which A 00 weights can arise as the Jacobian of quasiconformal mappings, modulo multiplication by a function which is bounded and bounded away from zero. The paper [DS2] is related to this problem. One can also ask for a characterization of the Aoo weights w for which there is an w-regular mapping. This is probably easier than the corresponding problem for quasiconformal mappings. We do know some partial results for this question, including the fact that there does exist an w-regular mapping when w is an A 1 weight.

BIBLIOGRAPHY [B) C.J. Bishop, Harmonic measures supported on Chicago, 1987.

cun~es,

Dissertation, Univ. of

(BCGJ) _ _ _ , L. Carleson, J.B. Garnett and P.W. Jones, Harmonic measures supported on curves, Pac. J. Math. 138 (1989), 233-236. (C1) F.M. Christ, CBMS Lectures, 1989. [C2) , A T(b) theorem with remarks on analytic capacdy and the Cauchy mtegral, to appear. (CW] R.R. Coifman and G. Weiss, Extensions of Hardy spaces and their uses in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. (CDM] R.R. Coifman, G. David andY. Meyer, La solut&on des conJectures de Calderon, Adv. in Math. 48 (1983), 144-148. [CMM] R.R. Coifman, A. Mcintosh and Y. Meyer, L 'mtegrale de Cauchy definit un operateur borne sur L 2 pour les courbes lipsch\tziennes, Ann. of Math. 116 (1982), 361-388. [D1] G. David, Operateurs mtegraux singuliers sur certaines courbes du plan complexe, Ann. Sci. Ec. Norm. Sup. 1 T (1984), 1'57-189 [D2] , Une minoration de la norme de l'operateur de Cauchy sur les graphes l1pschdnennes, Trans. Amer. Math. Soc. 302 (1987), 741-750. [D3] , Operateurs d'integrale singul1ere sur les surfaces regulieres, Ann. Sci. Ec. Norm. Sup. (4) 21 (1988), 225-258. [D4] , Morceaux de graphes lipsch1tziennes et mtegrales singulieres sur un surface, Rev. Mat. lberoamericana 4 (1988), 73-114. [D5] , Wavelets and Singular Integrals on Curves and Surfaces, to appear (Lecture Notes in Math, Springer-Verlag). [DJ] _ _ _ and D. Jerison, Lipschitz approximations to hypersurfaces, harmonic measure, and singular integrals, Indiana J. Math 39 (1990), 831-845. [DS1] _ _ _ and S. Semmes, Harmonic analysis and the geometry of subsets of R", to appear in the proceedings of the conference held in honor of J .L. Rubio de Francia (EI Escorial, Spain, June, 1989). (DS2] , Strong A 00 -weighl.s, Sobolev inequalities, and quasiconformal mappings, in Analysis and Partial Differential Equations, edited by C. Sadosky, Lecture Notes in Pure and Applied Math., vol 122, Marcel Dekker, 1990.

[DS3] __________ , Quant1tatwe rechfiab1ldy and Lipsch1tz mappmgs, prepru [Do] J.R. Dorronsoro, A charactenzation of potent1al spaces, Proc. Amer. Math. Soc. 95 (1985), 21-31. [Fe] H. Federer, Geometric ~.;[easure Theory, Springer- Verlag, 1969. [Fg] X. Fang, The Cauchy mtegral of Calderon and analytic capacdy, Thesis, Yale, 1990. [FI] K.J. Falconer, The Geometrv or Fractal Sets, Cambridge University Press, 1984. [G 1] J. Garnett, Pos1tive length but zero analytic capac1ty, Pro c. Amer. Math. Soc. 21 (1970), 696-699. [G2] , Analytic C<~pacitv and Measure, Lecture Notes in Math. 297, Springer- Verlag, 1972. [G3J _____ , Bounded Analytic Functions, Academic Press, 1981. [GJ] _ _ _ and P.W. Jones, The corona theorem for Denjoy domains, Acta. Math. 155 (1985), 29-40. [J1] P.W. Jones, Square funchons, Cauchy integrals, analyt1c capacity, and harmomc measure, in Harmonic Analvsis and Partial Differential Eguations, edited by J. Garcia-Cuerva, Lecture Notes m Math. 1384, Springer- Verlag, 1989. [J2] , Lipschitz and b1-L1psch1tz functions, Rev. Mat. lberoamericana 4 (1988), 115-122. [J3] , Rectifiable sets and the travellmg salesman problem, Invent. Math. 102 (1990), 1-15. [JL] J.L. Journe, Calderon-Zvgmund Operators, Pseudo-Differential Operators and the Cauchy Integral or Calderon, Lecture Notes in Math. 994, Springer- Verlag, 1983. [Ma] P. Mattila, Lecture Notes on Geometric Measure Theorv, Departmento de Matematicas, Universidad de Extremadura, 1986. [Ma2] - - - - - · Cauchy singular mtegrals and rect1fiab1ldy of measures in the plane, preprint. [Ma3] , A class of sets with posdwe length and zero analytic capac1ty, Ann. Acad. Sci. Fenn. 10 (1985), 387-395. [Mu] T. Murai, A Real- Variable Method for the Cauchy Transrorm. and Analytic Capacity, Lecture Notes in Math. 1307, Springer-Verlag, 1988. [S1] S. Semmes, Chord-arc surfaces w1.th small constant I, to appear in Adv. in Math. [S2] - - - - - , Chord-arc surfaces with small constant II: good parameterizat1.ons, ibid. [S3] , A criterion for the boundedness of singular mtegrals on hypersurfaces, Trans. Amer. Math. Soc. 311 (1989), 501-513. [S4] , Analys1.s vs geometry on a class of rectifiable hypersurfaces in R", to appear, Indiana Math. Journal.

J{ESUM.E

L'objet de ce texte est l'etude de relations entre certaines proprietes analytiques d'un sous-ensemble E de lRn (notamment, Ia continuite-L2 d'operateurs defmis par des noyaux singuliers comme le noyau de Cauchy ou le potentiel de double-couche) et des proprietes plus geometriques de E . Nous supposerons toujours que E est un ensemble de dimension d regulier au sens d'Ahlfors, c-a-d. tel que, pour toute boule B(x,r) centree sur E , Ia mesure de Hausdorff d-dimensionelle de EnB(x,r) est comprise entre C-1 rd et C rd . Le resultat principal est l'equivalence de diverses conditions, analytiques ou geometriques, portant surE. La premiere condition est Ia continuite sur L 2(E) de l'operateur d'integrale singuliere defini par tout noyau K(x-y) , ou K est impaire et a des derivees d'ordre j inferieures a c lx-yl-d-j . La plupart des conditions geometriques peuvent etre vues comme des formes plus fortes, et quantifiees, de rectifiabilite. Par exemple, l'une des conditions equivalentes est que E est contenu dans une surface r admettant un parametrage "co-regulier" (en dimension d=l, cela veut dire que r est une courbe reguliere au sens d'Ahlfors ; dire que E est rectifiable signifierait seulement que E est contenu, a un ensemble de longueur nulle pres, dans une union denombrable de courbes rectifiables). D'au~res conditions sont obtenues en mesurant, de diverses manieres, l'ecart entre l'inttrsection de E avec chaque boule centree sur E et un plan affine de dimension d , et en demandant une certaine integrabilite du resultat obtenu. Ce point de vue est inspire du resultat de P. Jones sur le probleme du voyageur de commerce. On peut aussi voir certaines des conditions equivalentes comme des analogues de conditions de Littlewood-Paley, ou de differences symetriques, utilisees pour decrire Ia regularite des fonctions. Les techniques utilisees sont des techniques de variable reelle. La partie la plus delicate de Ia demonstration repose sur un argument de temps d'arret tres proche de Ia construction dite "de la couronne" introduite par L. Carleson, ou I'on cherche a bien approximer E par des graphes lipschitziens sur des regions de ExlR + les plus grandes possibles.