Transference of scale-invariant estimates from Lipschitz to Non-tangentially accessible to Uniformly rectifiable domains
aa r X i v : . [ m a t h . A P ] A p r TRANSFERENCE OF SCALE-INVARIANT ESTIMATES FROM LIPSCHITZ TONON-TANGENTIALLY ACCESSIBLE TO UNIFORMLY RECTIFIABLE DOMAINS
STEVE HOFMANN, JOS ´E MAR´IA MARTELL, AND SVITLANA MAYBORODAA bstract . In relatively nice geometric settings, in particular, on Lipschitz domains, absolute con-tinuity of elliptic measure with respect to the Lebesgue measure is equivalent to Carleson measureestimates for solutions, to square function estimates, to ε -approximability, for any second order el-liptic PDE. In more general situations, notably, in a domain with a uniformly rectifiable boundary,absolute continuity of elliptic measure with respect to the Lebesgue measure may fail, already forthe Laplacian. In the present paper the authors demonstrate that nonetheless, Carleson measure es-timates for solutions, square function estimates, and ε -approximability remain valid. Moreover, thepaper o ff ers a general real-variable transference principle of certain scale-invariant estimates fromLipschitz to NTA to uniformly rectifiable domains, not restricted to harmonic functions or even tosolutions of elliptic equations. In particular, this allows one to deduce the first bounds for higherorder systems on uniformly rectifiable domains, in the setting where the elliptic measure does notexist, and to treat subharmonic functions. C ontents
1. Introduction 22. Preliminaries and relevant results from [HMM] 6Case ADR 11Case UR 11Case NTA 133. Transference of Carleson measure estimates from NTA to Uniformly Rectifiabledomains 144. John-Nirenberg inequality and transference of Carleson measure estimates fromLipschitz to Uniformly Rectifiable domains 155. A < N bounds: good- λ arguments 206. N < S bounds: from Lipschitz to NTA domains 277. From N < S bounds on NTA domains to ε -approximability in a complement of a URset 368. Applications: solutions of divergence form elliptic equations with bounded measurablecoe ffi cients 37 Date : April 30, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Carleson measures, square functions, non-tangential maximal functions, ε -approximability,uniform rectifiability, harmonic functions.The first author was supported by NSF grant DMS-1664047. The second author acknowledges financial support fromthe Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excel-lence in R&D” (SEV-2015-0554). He also acknowledges that the research leading to these results has received fundingfrom the European Research Council under the European Union’s Seventh Framework Programme (FP7 / / ERC agreement no. 615112 HAPDEGMT. The third author was supported in part by the NSF INSPIRE Award DMS-1344235, NSF CAREER Award DMS-1220089, NSF-DMS 1839077, Simons Fellowship, and the Simons Foundationgrant 563916, SM.. ffi cients satisfying a Carlesonmeasure condition 378.2. Higher order elliptic equations and systems with constant coe ffi cients 39Appendix A. Sawtooths have UR boundaries 40References 491. I ntroduction In the setting of a Lipschitz domain Ω ⊂ R n + , n ≥
1, for any divergence form elliptic operator L = − div A ∇ with bounded measurable coe ffi cients, the following are equivalent:( i ) Every bounded solution u , of the equation Lu = Ω , satisfies the Carleson measure estimate (see Definition 1.8 with F = |∇ u | / k u k L ∞ ( Ω ) ).( ii ) Every bounded solution u , of the equation Lu = Ω , is ε -approximable , for every ε > iii ) The elliptic measure associated to L , ω L , is (quantitatively) absolutely continuous with respectto the Lebesgue measure, ω L ∈ A ∞ ( σ ) on ∂ Ω .( iv ) Uniform Square function / Non-tangential maximal function (“ S / N ”) estimates hold locally in“sawtooth” subdomains of Ω (see Definition 1.14 and the discussion following the Definition).Historically, Dahlberg [Da3] obtained an extension Garnett’s ε -approximability result, observingthat ( iv ) implies ( ii ) . The explicit connection of ε -approximability with the A ∞ property of har-monic measure, i.e., that ( ii ) = ⇒ ( iii ), appears in [KKoPT] (where this implication is establishednot only for the Laplacian, but for general divergence form elliptic operators). That ( iii ) implies( iv ) is proved for harmonic functions in [Da2] , and, for null solutions of general divergence formelliptic operators, in [DJK]. Finally, Kenig, Kirchheim, Pipher and Toro [KKiPT] have recentlyshown that ( i ) implies ( iii ), whereas, on the other hand, ( i ) may be seen, via good-lambda and John-Nirenberg arguments, to be equivalent to the local version of one direction of ( iv ) (the “ S < N ”direction) .The main goal of the present paper is to show that while ( iii ) may fail on general uniformlyrectifiable domains even for harmonic functions [BJ] or might be not applicable in the absence ofa suitable concept of elliptic measure (e.g., for systems), ( i ) , ( ii ) and ( iv ) carry over from Lipschitzdomains to uniformly rectifiable sets by a purely real variable mechanism. But let us start withmore historical context.Past several decades brought to the center of attention uniformly rectifiable sets as the mostgeneral geometric setting in which meaningful analytic properties continue to hold. It was shownin the beginning of 90’s that uniform rectifiability of a set E is equivalent to boundedness of allsingular integral operators with odd kernels in L ( E ) [DS1], and, much more recently, that uniformrectifiability is equivalent to boundedness of the Riesz transform in L ( E ) (see [MMV] for the case n =
1, and [NToV] in general).However, it seemed to be vital for the estimates on solutions of elliptic PDEs in a domain Ω that, in addition to uniform rectifiability of E = ∂ Ω , Ω possesses some additional topologicalfeatures, ensuring a reasonably nice approach to the boundary. In particular, it has been known This implication holds more generally for null solutions of divergence form elliptic equations, see [KKoPT] and[HKMP]. And thus all three properties hold for harmonic functions in Lipschitz domains, by the result of [Da1]. We will prove this fact in much bigger generality in this paper.
RANSFERENCE OF ESTIMATES 3 that ( i ) , ( iii ) , ( iv ) hold for harmonic functions on non-tangentially accessible domains which satisfyan interior and exterior corkscrew condition (quantitative openness) and Harnack chains condition(quantitative connectedness) – see [JK], [DJK]. At the same time, the counterexample of Bishopand Jones [BJ] showed that absolute continuity of harmonic measure with respect to the Lebesguemeasure ( iii ) may fail on a general set with a uniformly rectifiable boundary: they construct a onedimensional (uniformly) rectifiable set E in the complex plane, for which harmonic measure withrespect to Ω = C \ E , is singular with respect to Hausdor ff H measure on E . In [HMM] the au-thors proved that, in spite of Bishop-Jones counterexamples, Carleson measure estimates ( i ) and ε -approximability ( ii ) for harmonic functions remain valid on all domains with a uniformly rectifi-able boundary and shortly thereafter it was shown that, at least in the presence of interior corkscrewpoints, each of these properties is necessary and su ffi cient for uniform rectifiability [GMT]. For thesake of completeness, we also want to point out that, in the absence of any additional topologicalassumptions, absolute continuity of the harmonic measure ω with respect to the Hausdor ff measureon E ⊂ ∂ Ω implies that ω | E is rectifiable [AHM3TV].The present paper introduces a new transference mechanism, which shows that a passage fromscale-invariant estimates, such as a Carleson measure bound or square function estimates / non-tangential maximal function estimates, on Lipschitz domains to analogous results on non-tangen-tially accessible domains and further to the same bounds on all sets with uniformly rectifiableboundaries is, in fact, a real variable phenomenon. That is, whenever one has suitable bounds fora given function on Lipschitz domains, they automatically carry over to uniformly rectifiable sets.This immediately entails a series of new results in very general PDE settings (for solutions of sec-ond order elliptic PDEs with coe ffi cients satisfying a Carleson measure condition, for solutions ofhigher order systems, for subharmonic functions), but clearly the power of having a general, purelyreal-variable scheme, goes beyond these applications. Let us now discuss the details. Definition 1.1. ( ADR ) (aka
Ahlfors-David regular ). We say that a set E ⊂ R n + is n -dimensionalADR (or simply ADR) if it is closed, and if there is some uniform constant C ≥ C − r n ≤ σ (cid:0) ∆ ( x , r ) (cid:1) ≤ C r n , ∀ r ∈ (0 , diam( E )) , x ∈ E , where diam( E ) may be infinite. Here, ∆ ( x , r ) : = E ∩ B ( x , r ) is the “surface ball” of radius r , and σ : = H n | E is the “surface measure” on E , where H n denotes n -dimensional Hausdor ff measure. Definition 1.3. ( UR ) (aka uniformly rectifiable ). An n -dimensional ADR (hence closed) set E ⊂ R n + is n -dimensional UR (or simply UR) if and only if it contains “Big Pieces of Lipschitz Images”of R n (“BPLI”). This means that there are positive constants θ and M , such that for each x ∈ E andeach r ∈ (0 , diam( E )), there is a Lipschitz mapping ρ = ρ x , r : R n → R n + , with Lipschitz constantno larger than M , such that H n (cid:16) E ∩ B ( x , r ) ∩ ρ (cid:0) { z ∈ R n : | z | < r } (cid:1) (cid:17) ≥ θ r n . Note that, in particular, a UR set is closed by definition, so that Ω : = R n + \ E is open, but neednot be connected.We recall that n -dimensional rectifiable sets are characterized by the property that they can becovered, up to a set of H n measure 0, by a countable union of Lipschitz images of R n ; we observethat BPLI is a quantitative version of this fact. Definition 1.4. ( “UR character” ). Given a UR set E ⊂ R n + , its “UR character” is just thepair of constants ( θ, M ) involved in the definition of uniform rectifiability, along with the ADRconstant; or equivalently, the quantitative bounds involved in any particular characterization ofuniform rectifiability. STEVE HOFMANN, JOS ´E MAR´IA MARTELL, AND SVITLANA MAYBORODA
It is worth mentioning that there exist sets that are ADR (and that even form the boundary ofan open set satisfying interior Corkscrew and Harnack Chain conditions), but that are totally non-rectifiable (e.g., see the construction of Garnett’s “4-corners Cantor set” in [DS2, Chapter1]).
Definition 1.5. ( Corkscrew condition ). Following [JK], we say that an open set Ω ⊂ R n + satisfiesthe “Corkscrew condition” if for some uniform constant c > ∆ : =∆ ( x , r ) = B ( x , r ) ∩ ∂ Ω , with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ), there is a ball B ( X ∆ , cr ) ⊂ B ( x , r ) ∩ Ω .The point X ∆ ⊂ Ω is called a “Corkscrew point” relative to ∆ . We note that we may allow r < C diam( ∂ Ω ) for any fixed C , simply by adjusting the constant c . Definition 1.6. ( Harnack Chain condition ). Again following [JK], we say that Ω satisfies theHarnack Chain condition if there is a uniform constant C such that for every ρ > , Λ ≥
1, andevery pair of points X , X ′ ∈ Ω with dist( X , ∂ Ω ) , ≥ ρ , dist( X ′ , ∂ Ω ) ≥ ρ and | X − X ′ | < Λ ρ , thereis a chain of open balls B , . . . , B N ⊂ Ω , N ≤ C ( Λ ), with X ∈ B , X ′ ∈ B N , B k ∩ B k + , Ø and C − diam( B k ) ≤ dist( B k , ∂ Ω ) ≤ C diam( B k ) . The chain of balls is called a “Harnack Chain”.
Definition 1.7. ( NTA ). Again following [JK], we say that an open set Ω ⊂ R n + is NTA (“Non-tangentially accessible”) if it satisfies the Harnack Chain condition, and if both Ω and Ω ext : = R n + \ Ω satisfy the Corkscrew condition.As we pointed out above and as can be seen from the definitions, non-tangentially accessibledomains possess certain quantitative topological features. One can characterize an NTA domainwith an ADR boundary in terms close to (1.3), but ensuring Big Pieces of Lipschitz Subdomains,rather than Big Pieces of Lipschitz Images (see Proposition 4.8), the crucial di ff erence being thatin some sense, a nice access to the boundary of a Lipschitz domain is partially retained, contrary tothe general UR case.Finally, let us define the scale-invariant estimates at the center of this paper. Definition 1.8.
Let E ⊂ R n + be an n -dimensional ADR set and let F ∈ L ( R n + \ E ). We say that F satisfies the Carleson measure estimate (CME) on R n + \ E if there exists a constant C > x ∈ E , < r < ∞ r n Z Z B ( x , r ) | F ( Y ) | δ ( Y ) dY ≤ C . Similarly, we say that F ∈ L ( D ) satisfies the Carleson measure estimate in some open set D ⊂ R n + with ∂ D being n -dimensional ADR if there exists a constant C > x ∈ ∂ D , < r < ∞ r n Z Z B ( x , r ) ∩ D | F ( Y ) | δ ( Y ) dY ≤ C , where δ ( Y ) = dist( Y , ∂ D ).More generally, if E is the boundary of some open set D ⊂ R n + , we say that a given propertystated on R n + \ E is satisfied on D if the function in question is supported on D . Definition 1.11.
Let Ω : = R n + \ E , where E ⊂ R n + is an n -dimensional ADR set (hence closed);thus Ω is open, but need not be a connected domain. Let u ∈ L ∞ ( Ω ), with k u k ∞ ≤
1, and let ε ∈ (0 , u is ε - approximable , if there is a constant C ε , and a function ϕ = ϕ ε ∈ W , ( Ω )satisfying(1.12) k u − ϕ k L ∞ ( Ω ) < ε , and(1.13) sup x ∈ E , < r < ∞ r n Z Z B ( x , r ) |∇ ϕ ( Y ) | dY ≤ C ε . RANSFERENCE OF ESTIMATES 5
Definition 1.14. ( Area integral and non-tangential maximal function ). Let E ⊂ R n + be an n -dimensional ADR set, as in (1.1). For a continuous function H in R n + \ E , we define the “non-tangential maximal function” as N ∗ H ( x ) : = sup Y ∈ Γ ( x ) | H ( Y ) | , x ∈ E and for G ∈ L ( R n + \ E ), we define the area integral A ( G ), as follows:(1.15) A G ( x ) : = (cid:18)Z Z Γ ( x ) | G ( Y ) | δ ( Y ) − n dY (cid:19) / , x ∈ E , with δ ( Y ) = dist( Y , E ), as before. In the body of the paper we will always use these definitions withdyadic cones Γ ( x ), x ∈ ∂ Ω , which will be introduced later – see (2.22). Equivalently though, thereader can think instead of the traditional cones(1.16) Γ Ω ( x ) : = { Y ∈ Ω : | Y − x | ≤ (1 + κ ) dist( Y , ∂ Ω ) } , x ∈ ∂ Ω , for some κ > Theorem 1.17.
Let E be a n-dimensional UR set, and let Ω : = R n + \ E.If for some F ∈ L ( Ω ) for every bounded NTA subdomain D ⊂ Ω with an ADR boundary theCarleson measure estimate (1.10) is satisfied with a constant depending on n and the NTA / ADRconstants of D only, then the Carleson measure estimate holds on Ω as well, i.e., (1.9) is satisfied,with the constant depending on n and the UR character of E only.Furthermore, given an NTA domain D ⊂ R n + with an ADR boundary E = ∂ D and F ∈ L ( D ) which satisfies (3.2) , if F satisfies the Carleson measure estimate (1.10) on all bounded Lipschitzsubdomains of D with the constant depending on the Lipschitz constants of the underlying domainsonly, then F satisfies the Carleson measure estimate (1.10) in D as well, with the bound dependingon the constant in (3.2) , the NTA constants of D and the ADR constants of ∂ D only.
We do not explain in details condition (3.2) now, but let us mention that generally it is a harmlessbound on interior cubes, which, in the context of solutions, is a simple consequence of Caccioppoli’sinequality.
Proof.
The Theorem is a combination of (reduced versions of) Theorem 3.3 and Theorem 4.10proved in the body of the paper. (cid:3)
Secondly, in the class of Lipschitz domains, or in the class of NTA domains with ADR bound-aries, or in the class of sets with UR boundaries, the Carleson measure estimates are equivalent tolocal and global area integral bounds (AKA square function estimates).
Theorem 1.18.
Let Σ be a subclass of ADR domains in R n + with the property that if D belongs to Σ then all its local sawtooth subdomains belong to Σ (for example, a class of ADR subdomains of acertain set, or a class of bounded NTA subdomains with ADR boundaries, or a class of sets with URboundaries, with uniform relevant geometric constants). Let G ∈ L ( D ) and H ∈ C ( D ) ∩ L ∞ ( D ) for all D ∈ Σ .Then the Carleson measure estimate (1.10) is satisfied for F = G / k H k L ∞ ( D ) for all D ∈ Σ if andonly if (1.19) kA G k L q ( ∂ D ) ≤ C k N ∗ H k L q ( ∂ D ) , for all D ∈ Σ ,for some < q < ∞ if and only if (1.19) holds for all < q < ∞ . STEVE HOFMANN, JOS ´E MAR´IA MARTELL, AND SVITLANA MAYBORODA
The Theorem is a particular case of Theorem 5.7, which actually contains considerably moredetailed statements, as well as equivalence to local area integral bounds.Finally, we discuss the transference for the converse bounds on non-tangential maximal functionin terms of the square function and their connection with ε -approximability. In this context, onehas to tie up explicitly the arguments of A and N ∗ . Theorem 1.20.
Let E ⊂ R n + be an n-dimensional UR set, Ω = R n + \ E, and suppose thatu ∈ W , ( Ω ) ∩ C ( Ω ) satisfies (7.2) . Assume, in addition, that F = |∇ u | / k u k L ∞ ( Ω ) satisfies theCarleson measure estimate (1.9) . If for every bounded NTA subdomain Ω ′ ⊂ Ω with an ADRboundary (1.21) (cid:13)(cid:13) N ∗ ( u − u ( X +Ω ′ )) (cid:13)(cid:13) L ( Ω ′ ) ≤ C kA ( ∇ u ) k L ( ∂ Ω ′ ) , holds with a constant depending on n, the NTA constants of Ω ′ and the ADR constants of ∂ Ω ′ only,then u is ε -approximable on Ω , with the implicit constants depending on n and the UR character ofE only. Here, X +Ω ′ is any interior corkscrew point of Ω ′ at the scale of diam( Ω ′ ) . The Theorem is a combination of (the reduced versions of) Theorem 7.1 and 6.2. The interiorbound (7.2) is, again, a fairly harmless prerequisite which follows from known interior estimatesin the context of solutions of elliptic PDEs. We remark that the estimate (1.21) itself would notmake much sense for general UR sets, because of topological obstructions (there is no preferredcomponent for a corkscrew point in such a general context), and for that reason we pass directly to ε -approximability. 2. P reliminaries and relevant results from [HMM]We start with some further notation and definitions. • We use the letters c , C to denote harmless positive constants, not necessarily the same at eachoccurrence, which depend only on dimension and the constants appearing in the hypotheses ofthe theorems (which we refer to as the “allowable parameters”). We shall also sometimes write a . b and a ≈ b to mean, respectively, that a ≤ Cb and 0 < c ≤ a / b ≤ C , where the constants c and C are as above, unless explicitly noted to the contrary. At times, we shall designate by M a particular constant whose value will remain unchanged throughout the proof of a given lemmaor proposition, but which may have a di ff erent value during the proof of a di ff erent lemma orproposition. • Given a closed set E ⊂ R n + , we shall use lower case letters x , y , z , etc., to denote points on E ,and capital letters X , Y , Z , etc., to denote generic points in R n + (especially those in R n + \ E ). • The open ( n + r will be denoted B ( x , r ) when the center x lies on E , or B ( X , r ) when the center X ∈ R n + \ E . A “surface ball” is denoted ∆ ( x , r ) : = B ( x , r ) ∩ E where unless otherwise specified we implicitly assume that x ∈ E . • Given a Euclidean ball B or surface ball ∆ , its radius will be denoted r B or r ∆ , respectively. • Given a Euclidean or surface ball B = B ( X , r ) or ∆ = ∆ ( x , r ), its concentric dilate by a factor of κ > κ B : = B ( X , κ r ) or κ ∆ : = ∆ ( x , κ r ) . • Given a (fixed) closed set E ⊂ R n + , for X ∈ R n + , we set δ ( X ) : = dist( X , E ). • We let H n denote n -dimensional Hausdor ff measure, and let σ : = H n (cid:12)(cid:12) E denote the “surfacemeasure” on E . • We will also work with open sets Ω ⊂ R n + in which case the previous notations and definitionseasily adapt by letting E : = ∂ Ω . RANSFERENCE OF ESTIMATES 7 • For a Borel set A ⊂ R n + , we let 1 A denote the usual indicator function of A , i.e. 1 A ( x ) = x ∈ A , and 1 A ( x ) = x < A . • For a Borel set A ⊂ R n + , we let int( A ) denote the interior of A . • Given a Borel measure µ , and a Borel set A , with positive and finite µ measure, we set > A f d µ : = µ ( A ) − R A f d µ . • We shall use the letter I (and sometimes J ) to denote a closed ( n + ℓ ( I ) denote the side lengthof I . If ℓ ( I ) = − k , then we set k I : = k . Given an ADR set E ⊂ R n + , we use Q to denote adyadic “cube” on E . The latter exist (cf. [DS1], [Chr]), and enjoy certain properties which weenumerate in Lemma 2.1 below. Lemma 2.1. ( Existence and properties of the “dyadic grid” ) [DS1, DS2], [Chr].
Suppose thatE ⊂ R n + is an n-dimensional ADR set. Then there exist constants a > , γ > and C < ∞ ,depending only on dimension and the ADR constant, such that for each k ∈ Z , there is a collectionof Borel sets (“cubes”) D k : = { Q kj ⊂ E : j ∈ I k } , where I k denotes some (possibly finite) index set depending on k, satisfying ( i ) E = ∪ j Q kj for each k ∈ Z . ( ii ) If m ≥ k then either Q mi ⊂ Q kj or Q mi ∩ Q kj = Ø . ( iii ) For each ( j , k ) and each m < k, there is a unique i such that Q kj ⊂ Q mi . ( iv ) diam (cid:0) Q kj (cid:1) ≤ C − k . ( v ) Each Q kj contains some “surface ball” ∆ (cid:0) x kj , a − k (cid:1) : = B (cid:0) x kj , a − k (cid:1) ∩ E. ( vi ) H n (cid:0)(cid:8) x ∈ Q kj : dist( x , E \ Q kj ) ≤ ̺ − k (cid:9)(cid:1) ≤ C ̺ γ H n (cid:0) Q kj (cid:1) , for all k , j and for all ̺ ∈ (0 , a ) . A few remarks are in order concerning this lemma. • In the setting of a general space of homogeneous type, this lemma has been proved by Christ[Chr], with the dyadic parameter 1 / δ ∈ (0 , δ = / • For our purposes, we may ignore those k ∈ Z such that 2 − k & diam( E ), in the case that the latteris finite. • We shall denote by D = D ( E ) the collection of all relevant Q kj , i.e., D : = ∪ k D k , where, if diam( E ) is finite, the union runs over those k such that 2 − k . diam( E ). • For a dyadic cube Q ∈ D k , we shall set ℓ ( Q ) = − k , and we shall refer to this quantity as the“length” of Q . Evidently, ℓ ( Q ) ≈ diam( Q ) . • For a dyadic cube Q ∈ D , we let k ( Q ) denote the “dyadic generation” to which Q belongs, i.e.,we set k = k ( Q ) if Q ∈ D k ; thus, ℓ ( Q ) = − k ( Q ) . • Properties ( iv ) and ( v ) imply that for each cube Q ∈ D , there is a point x Q ∈ E , a Euclidean ball B ( x Q , r ) and a surface ball ∆ ( x Q , r ) : = B ( x Q , r ) ∩ E such that c ℓ ( Q ) ≤ r ≤ ℓ ( Q ) for some uniformconstant 0 < c < ∆ ( x Q , r ) ⊂ Q ⊂ ∆ ( x Q , Cr ) , STEVE HOFMANN, JOS ´E MAR´IA MARTELL, AND SVITLANA MAYBORODA for some uniform constant C . We shall denote this ball and surface ball by(2.3) B Q : = B ( x Q , r ) , ∆ Q : = ∆ ( x Q , r ) , and we shall refer to the point x Q as the “center” of Q .At this stage we would like to recall some results from [HMM]. Many of them have been statedfor harmonic functions, but here we would like to highlight a more general point of view. We firstgive a definition to then continue with some key geometric lemmas from [HMM]. Definition 2.4. [DS2]. Let S ⊂ D ( E ). We say that S is “coherent” if the following conditions hold:( a ) S contains a unique maximal element denoted by Q ( S ) which contains all other elementsof S as subsets.( b ) If Q belongs to S , and if Q ⊂ e Q ⊂ Q ( S ), then e Q ∈ S .( c ) Given a cube Q ∈ S , either all of its children belong to S , or none of them do.We say that S is “semi-coherent” if only conditions ( a ) and ( b ) hold. Lemma 2.5 (The bilateral “corona decomposition”, [HMM]) . Suppose that E ⊂ R n + is n-dimen-sional UR. Then given any positive constants η ≪ and K ≫ , there is a disjoint decomposition D ( E ) = G ∪ B , satisfying the following properties. (1)
The “Good” collection G is further subdivided into disjoint stopping time regimes, suchthat each such regime S is coherent (cf. Definition 2.4). (2) The “Bad” cubes, as well as the maximal cubes Q ( S ) satisfy a Carleson packing condition: X Q ′ ⊂ Q , Q ′ ∈B σ ( Q ′ ) + X S : Q ( S ) ⊂ Q σ (cid:0) Q ( S ) (cid:1) ≤ C η, K σ ( Q ) , ∀ Q ∈ D ( E ) . (3) For each S , there is a Lipschitz graph Γ S , with Lipschitz constant at most η , such that, forevery Q ∈ S , (2.6) sup x ∈ ∆ ∗ Q dist( x , Γ S ) + sup y ∈ B ∗ Q ∩ Γ S dist( y , E ) < η ℓ ( Q ) , where B ∗ Q : = B ( x Q , K ℓ ( Q )) and ∆ ∗ Q : = B ∗ Q ∩ E. Now we construct, for each stopping time regime S in Lemma 2.5, a pair of NTA domains Ω ± S ,with ADR boundaries, which provide a good approximation to E , at the scales within S , in someappropriate sense. To be a bit more precise, Ω S : = Ω + S ∪ Ω − S will be constructed as a sawtooth regionrelative to some family of dyadic cubes, and the nature of this construction will be essential to thedyadic analysis that we will use below. We first discuss some preliminary matters.Let W = W ( R n + \ E ) denote a collection of (closed) dyadic Whitney cubes of R n + \ E , so thatthe cubes in W form a pairwise non-overlapping covering of R n + \ E , which satisfy(2.7) 4 diam( I ) ≤ dist(4 I , E ) ≤ dist( I , E ) ≤
40 diam( I ) , ∀ I ∈ W (just dyadically divide the standard Whitney cubes, as constructed in [Ste, Chapter VI], into cubeswith side length 1 / /
4) diam( I ) ≤ diam( I ) ≤ I ) , whenever I and I touch.Let E be an n -dimensional ADR set and pick two parameters η ≪ K ≫
1. Define(2.8) W Q : = (cid:8) I ∈ W : η / ℓ ( Q ) ≤ ℓ ( I ) ≤ K / ℓ ( Q ) , dist( I , Q ) ≤ K / ℓ ( Q ) (cid:9) . RANSFERENCE OF ESTIMATES 9
Remark . We note that W Q is non-empty, provided that we choose η small enough, and K largeenough, depending only on dimension and the ADR constant of E . Indeed, given an n -dimensionalADR set E , and given Q ∈ D ( E ), consider the ball B Q = B ( x Q , r ), as defined in (2.2)-(2.3), with r ≈ ℓ ( Q ), so that ∆ Q = B Q ∩ E ⊂ Q . By [HM, Lemma 5.3] , we have that for some C = C ( n , ADR ), (cid:12)(cid:12) { Y ∈ R n + \ E : δ ( Y ) < ǫ r } ∩ B Q (cid:12)(cid:12) ≤ C ǫ r n + , for every 0 < ǫ <
1. Consequently, fixing 0 < ǫ < X Q ∈ B Q , with δ ( X Q ) ≥ ǫ r . Thus, B ( X Q , ǫ r / ⊂ B Q \ E . We shall refer to this point X Q as a “Corkscrew point”relative to Q . Now observe that X Q belongs to some Whitney cube I ∈ W , which will belong to W Q , for η small enough and K large enough.Next, we choose a small parameter τ >
0, so that for any I ∈ W , and any τ ∈ (0 , τ ], theconcentric dilate I ∗ ( τ ) : = (1 + τ ) I still satisfies the Whitney property(2.10) diam I ≈ diam I ∗ ( τ ) ≈ dist (cid:0) I ∗ ( τ ) , E (cid:1) ≈ dist( I , E ) , < τ ≤ τ . Moreover, for τ ≤ τ small enough, and for any I , J ∈ W , we have that I ∗ ( τ ) meets J ∗ ( τ ) if andonly if I and J have a boundary point in common, and that, if I , J , then I ∗ ( τ ) misses (3 / J .At this point the discussion splits into a few special cases depending whether we have some extrainformation about E . The main idea consists in constructing some kind of “Whitney regions” whichwill allow us to introduce some “Carleson boxes” and “sawtooth subdomains”. The construction ofthe Whitney regions depends very much on the background assumptions, having extra informationabout E will allow us to augment the collections W Q so that we gain some connectivity on thecorresponding Whitney regions and hence the resulting subdomains would have better properties.We consider three cases. In the first one we assume only that E is ADR (but is not UR) and we set W Q = W Q (here we do not gain any connectivity). The second case deals with E being UR, inwhich case we can invoke Lemma 2.5 and use the Lipschitz graphs associated to the good regimesso that the augmented collection W Q creates two nice Whitney regions one lying respectively aboveand below the Lipschitz graph. Finally, when E is the boundary of D a bounded NTA domain withADR boundary (hence E is UR) we are just interested on keeping the Whitney regions inside D andwe can augment W Q using that D is Harnack chain connected so that the resulting collections W Q give some Whitney regions which are indeed bounded NTA domains with ADR boundary.In order to keep a unified presentation let us assume that for every Q ∈ D we are given W Q ⊃W Q (below we will give the specific definition in each di ff erent case) and a constant C ≥ C − η / ℓ ( Q ) ≤ ℓ ( I ) ≤ CK / ℓ ( Q ) , ∀ I ∈ W Q , dist( I , Q ) ≤ CK / ℓ ( Q ) , ∀ I ∈ W Q . Fix 0 < τ ≤ τ / Q ∈ D ( E ), we may define an associated Whitneyregion U Q (not necessarily connected), as follows:(2.12) U Q = U Q ,τ : = [ I ∈W Q I ∗ ( τ )For later use, it is also convenient to introduce some fattened version of U Q (2.13) b U Q = U Q , τ : = [ I ∈W Q I ∗ (2 τ ) . When the particular choice of τ ∈ (0 , τ ] is not important, for the sake of notational convenience,we may simply write I ∗ and U Q in place of I ∗ ( τ ) and U Q ,τ . We may also define the
Carleson box relative to Q ∈ D ( E ), by(2.14) T Q = T Q ,τ : = int [ Q ′ ∈ D Q U Q ,τ , where(2.15) D Q : = (cid:8) Q ′ ∈ D ( E ) : Q ′ ⊂ Q (cid:9) . Let us note that we may choose K large enough so that, for every Q ,(2.16) T Q ,τ ⊂ T Q ,τ ⊂ B ∗ Q : = B (cid:0) x Q , K ℓ ( Q ) (cid:1) . For future reference, we also introduce dyadic sawtooth regions as follows. Given a family F ofdisjoint cubes { Q j } ⊂ D , we define the global discretized sawtooth relative to F by(2.17) D F : = D \ [ F D Q j , i.e., D F is the collection of all Q ∈ D that are not contained in any Q j ∈ F . Given some fixed cube Q , the local discretized sawtooth relative to F by(2.18) D F , Q : = D Q \ [ F D Q j = D F ∩ D Q . Note that we can also allow F to be empty in which case D Ø = D and D Ø , Q = D Q .Similarly, we may define geometric sawtooth regions as follows. Given a family F ⊂ D ofdisjoint cubes as before we define the global sawtooth and the local sawtooth relative to F byrespectively(2.19) Ω F : = int (cid:18) [ Q ′ ∈ D F U Q ′ (cid:19) , Ω F , Q : = int (cid:18) [ Q ′ ∈ D F , Q U Q ′ (cid:19) . Notice that Ω Ø , Q = T Q . For the sake of notational convenience, we set(2.20) W F : = [ Q ′ ∈ D F W Q ′ , W F , Q : = [ Q ′ ∈ D F , Q W Q ′ , so that in particular, we may write(2.21) Ω F , Q = int (cid:18) [ I ∈ W F , Q I ∗ (cid:19) . Finally, for every x ∈ E , we define non-tangential approach regions, cones , as(2.22) Γ ( x ) = [ Q ∈ D ( E ): Q ∋ x U Q . Their local versions are given by(2.23) Γ Q ( x ) = [ Q ′ ∈ D Q : Q ′ ∋ x U Q ′ , x ∈ Q . We shall often change the “aperture” of cones, Carleson boxes, sawtooth regions, by either using b U Q = U Q , τ (cf. (2.13)) in place of U Q in the definitions or by changing η and K . The corre-sponding larger sets will be always distinguished by a “widehat”, and within the same notation, theaperture can become larger from line to line as long as τ , η and K (or κ ) depend only on allow-able geometric characteristics, that is, ADR, UR, NTA constants (depending on the case). Standardreal variable arguments show that the L p norms of non-tangential maximal functions defined withdi ff erent apertures are equivalent, and the same applies to area integrals and square functions. RANSFERENCE OF ESTIMATES 11
Case ADR.
Here we assume that the set E under consideration is merely ADR, but possibly notUR. Let us set W Q = W Q as defined in (2.8). By definition (cf. (2.8)) we clearly have (2.11) with C = Proposition 2.24 ([HMM, Proposition A.2]) . Let E ⊂ R n + be a closed n-dimensional ADR set(which may be UR, or not; if so, the sawtooth regions and Carleson boxes are built using theaugmented collections W Q to be constructed momentarily). Then all dyadic local sawtooths Ω F , Q and all Carleson boxes T Q have n-dimensional ADR boundaries. In all cases, the implicit constantsare uniform and depend only on dimension, the ADR constant of E and the parameters η , K, and τ . Case UR.
Here we further assume that E is UR and make the corresponding bilateral corona de-composition of Lemma 2.5 with η ≪ K ≫
1. Given Q ∈ D ( E ), for this choice of η and K , weset as above B ∗ Q : = B ( x Q , K ℓ ( Q )), where we recall that x Q is the center of Q (see (2.2)-(2.3)). Fora fixed stopping time regime S , we choose a co-ordinate system so that Γ S = { ( z , ϕ S ( z )) : z ∈ R n } ,where ϕ S : R n −→ R is a Lipschitz function with k ϕ k Lip ≤ η . Claim . If Q ∈ S , and I ∈ W Q , then I lies either above or below Γ S .Moreover, dist( I , Γ S ) ≥ η / ℓ ( Q ) (and therefore, by (2.6), dist( I , Γ S ) ≈ dist( I , E ), with implicitconstants that may depend on η and K ).Next, given Q ∈ S , we augment W Q . We split W Q = W , + Q ∪ W , − Q , where I ∈ W , + Q if I liesabove Γ S , and I ∈ W , − Q if I lies below Γ S . Choosing K large and η small enough, by (2.6), wemay assume that both W , ± Q are non-empty. We focus on W , + Q , as the construction for W , − Q is thesame. For each I ∈ W , + Q , let X I denote the center of I . Fix one particular I ∈ W , + Q , with center X + Q : = X I . Let e Q denote the dyadic parent of Q , unless Q = Q ( S ); in the latter case we simply set e Q = Q . Note that e Q ∈ S , by the coherency of S . By Claim 2.25, for each I in W , + Q , or in W , + e Q ,we have dist( I , E ) ≈ dist( I , Q ) ≈ dist( I , Γ S ) , where the implicit constants may depend on η and K . Thus, for each such I , we may fix a Harnackchain, call it H I , relative to the Lipschitz domain Ω +Γ S : = (cid:8) ( x , t ) ∈ R n + : t > ϕ S ( x ) (cid:9) , connecting X I to X + Q . By the bilateral approximation condition (2.6), the definition of W Q , and thefact that K / ≪ K , we may construct this Harnack Chain so that it consists of a bounded numberof balls (depending on η and K ), and stays a distance at least c η / ℓ ( Q ) away from Γ S and from E .We let W ∗ , + Q denote the set of all J ∈ W which meet at least one of the Harnack chains H I , with I ∈ W , + Q ∪ W , + e Q (or simply I ∈ W , + Q , if Q = Q ( S )), i.e., W ∗ , + Q : = n J ∈ W : ∃ I ∈ W , + Q ∪ W , + e Q for which H I ∩ J , Ø o , where as above, e Q is the dyadic parent of Q , unless Q = Q ( S ), in which case we simply set e Q = Q (so the union is redundant). We observe that, in particular, each I ∈ W , + Q ∪ W , + e Q meets H I , bydefinition, and therefore(2.26) W , + Q ∪ W , + e Q ⊂ W ∗ , + Q . Of course, we may construct W ∗ , − Q analogously. We then set W ∗ Q : = W ∗ , + Q ∪ W ∗ , − Q . It follows from the construction of the augmented collections W ∗ , ± Q that there are uniform constants c and C such that(2.27) c η / ℓ ( Q ) ≤ ℓ ( I ) ≤ CK / ℓ ( Q ) , ∀ I ∈ W ∗ Q , dist( I , Q ) ≤ CK / ℓ ( Q ) , ∀ I ∈ W ∗ Q . It is convenient at this point to introduce some additional terminology.
Definition 2.28.
Given Q ∈ G , and hence in some S , we shall refer to the point X + Q specified above,as the “center” of U + Q (similarly, the analogous point X − Q , lying below Γ S , is the “center” of U − Q ).We also set Y ± Q : = X ± e Q , and we call this point the “modified center” of U ± Q , where as above e Q is thedyadic parent of Q , unless Q = Q ( S ), in which case Q = e Q , and Y ± Q = X ± Q .Observe that W ∗ , ± Q and hence also W ∗ Q have been defined for any Q that belongs to some stop-ping time regime S , that is, for any Q belonging to the “good” collection G of Lemma 2.5. On theother hand, we have defined W Q for arbitrary Q ∈ D ( E ). We now set(2.29) W Q : = ( W ∗ Q , Q ∈ G , W Q , Q ∈ B , and for Q ∈ G we shall henceforth simply write W ± Q in place of W ∗ , ± Q . Notice that by (2.8) when Q ∈ B and by (2.27) when Q ∈ G we clearly obtain (2.11) with C depending on UR character of E .Given an arbitrary Q ∈ D ( E ) and 0 < τ ≤ τ /
4, we may define an associated Whitney region U Q (not necessarily connected) as in (2.12) or the fattened version of b U Q as in (2.13). In the presentsituation, if Q ∈ G , then U Q splits into exactly two connected components(2.30) U ± Q = U ± Q ,τ : = [ I ∈W ± Q I ∗ ( τ ) . We note that for Q ∈ G , each U ± Q is Harnack chain connected, by construction (with constantsdepending on the implicit parameters τ, η and K ); moreover, for a fixed stopping time regime S , if Q ′ is a child of Q , with both Q ′ , Q ∈ S , then U + Q ′ ∪ U + Q is Harnack Chain connected, and similarlyfor U − Q ′ ∪ U − Q .We may also define the Carleson boxes T Q , global and local sawtooth regions Ω F , Ω F , Q , cones Γ , and local cones Γ Q as in (2.14) (2.19), (2.22), and (2.23). Remark . We recall that, by construction (cf. (2.26), (2.29)), given Q ∈ G W , ± e Q ⊂ W Q ,and therefore Y ± Q ∈ U ± Q ∩ U ± e Q . Moreover, since Y ± Q is the center of some I ∈ W , ± e Q , we have thatdist( Y ± Q , ∂ U ± Q ) ≈ dist( Y ± Q , ∂ U ± e Q ) ≈ ℓ ( Q ) (with implicit constants possibly depending on η and / or K ) Remark . Given a stopping time regime S as in Lemma 2.5, for any semi-coherent subregime(cf. Definition 2.4) S ′ ⊂ S (including, of course, S itself), we now set(2.33) Ω ± S ′ = int [ Q ∈ S ′ U ± Q , and let Ω S ′ : = Ω + S ′ ∪ Ω − S ′ . Note that implicitly, Ω S ′ depends upon τ (since U ± Q has such dependence).When it is necessary to consider the value of τ explicitly, we shall write Ω S ′ ( τ ).The main geometric lemma for the previous sawtooth regions is the following. RANSFERENCE OF ESTIMATES 13
Lemma 2.34 ([HMM, Lemma 3.24]) . Let S be a given stopping time regime as in Lemma 2.5, andlet S ′ be any nonempty, semi-coherent subregime of S . Then for < τ ≤ τ , with τ small enough,each of Ω ± S ′ is an NTA domain with ADR boundary with character depending only on n , τ, η, K, andthe ADR / UR constants for E.
Case NTA.
Here we assume that E is ADR and is the boundary of D , a bounded NTA. This is,strictly speaking, a sub-case of the Case UR above, but the extra assumption that E is a boundary ofsome bounded NTA makes the construction simpler. In this case, we are basically in the situationwhich is equivalent to being within one regimen S , at least as far as the construction of W Q isconcerned.Let D be a bounded NTA with ADR boundary and write E = ∂ D . Define W as above, butin this case we only keep those Whitney cubes contained in D (that is we are doing a Whitneydecomposition of D rather than that of R n + \ E ). Let W Q be as defined in (2.8) (once again,considering only the Whitney cubes in D ). Next, given any Q ∈ D ( E ), augment W Q to W ∗ Q asdone in [HM, Section 3] using the fact that one can construct a Harnack chain to connect X Q (acorkscrew point relative to Q ) with any of the centers of the Whitney cubes in W Q . Notice thatin the case when E is UR and Q ∈ S we have used a similar idea, the main di ff erence is that theHarnack chain in that case comes from the fact that Ω +Γ S is a Lipschitz domain, whereas here suchproperty comes from the assumption that D is NTA and hence the Harnack chain condition holds.With the appropriate choice of a su ffi ciently small η and a su ffi ciently large K depending on n , theNTA constants of D only, we can guarantee the same key properties for the resulting augmented W ∗ Q . In particular, (2.11) holds, the cubes in W Q ∪ W e Q are contained in W ∗ Q together with theassociated Harnack chains, the corkscrew points X Q and X e Q are contained in W ∗ Q , and others. Thenone set W Q = W ∗ Q and uses (2.12)–(2.23) to define Whitney regions, Carleson boxes, sawtoothregions, cones, in the very same way as in the UR case and, respectively, satisfying the sameproperties.We observe that from [HM, Lemma 3.61] it follows that all Carleson boxes, all sawtooth re-gions and local sawtooth regions have ADR boundary and satisfy the (interior) Harnack chain andCorkscrew condition. We claim that the exterior Corkscrew condition holds as well. Let D ⋆ be oneof these subdomains and take x ⋆ ∈ ∂ D ⋆ and 0 < r < diam( ∂ D ⋆ ). By construction ∂ D ⋆ ⊂ D andwe consider two cases 0 ≤ δ ( x ⋆ ) ≤ r / δ ( x ⋆ ) > r /
2. In the first scenario we pick x ∈ ∂ D so that | x ⋆ − x | = δ ( x ⋆ ) ≤ r / x = x ⋆ if x ⋆ ∈ ∂ D ∩ ∂ D ⋆ ). Since D is an NTA do-main it satisfies the exterior Corkscrew condition, hence we can find X ∈ D ext = R n + \ D so that B ( X , c r / ⊂ B ( x , r / ∩ D ext where c is the exterior corkscrew constant. Note that D ⋆ ⊂ D ,hence B ( X , c r / ⊂ ( D ⋆ ) ext . Also, B ( X , c r / ⊂ B ( x , r / ⊂ B ( x ⋆ , r ). This shows that X is anexterior corkscrew point relative to the surface ball B ( x ⋆ , r ) ∩ ∂ D ⋆ for the domain D ⋆ with constant c /
2. Consider next the case on which δ ( x ⋆ ) > r /
2. Note that in particular x ⋆ ∈ Ω and there-fore we can find two Whitney cubes I , J ∈ W so that x ∈ ∂ I ∗ ∩ J , ∂ I ∩ ∂ J , Ø, int( I ∗ ) ⊂ D ⋆ and J is a Whitney cube which does not belong to any of the W Q that define D ⋆ . Note that ℓ ( J ) ≥ δ ( x ⋆ ) / C > r / (2 C ) for some uniform constant C ≥
1, that I ∗ misses J as observed beforeand that the center of J satisfies X ( J ) ∈ ( D ⋆ ) ext . It is then clear that the open segment joining x ⋆ with X ( J ) is contained in ( D ⋆ ) ext and we pick X in that segment so that | X − x ⋆ | = r / (8 C ) and hence B ( X , r / (16 C )) ⊂ B ( x ⋆ , r ) ∩ D ⋆ . This shows that X is an exterior corkscrew point relative to thesurface ball B ( x ⋆ , r ) ∩ ∂ D ⋆ for the domain D ⋆ with constant 1 / (16 C ). Therefore, we have shownthat D satisfies the exterior Corkscrew condition. Remark . When Ω is an NTA domain with ADR boundary, or more generally, an open set Ω with a UR or even ADR boundary E = ∂ Ω , we will also use a non-dyadic definition of cones (1.16).It is straightforward to see that given η and K as above there exists κ such that dyadic cones Γ ( x )are contained in Γ Ω ( x ) for all x ∈ ∂ Ω . Vice versa, given κ >
0, there exist η and K such that Γ Ω ( x )are contained in Γ ( x ) for all x ∈ ∂ Ω .
3. T ransference of C arleson measure estimates from NTA to U niformly R ectifiable domains Let us now discuss the “transference” mechanism allowing one to pass from the Carleson mea-sure estimates on NTA domains to those for domains with UR boundaries. They are due to [HMM],although there the discussion is formally confined to the case of harmonic functions.When u is a bounded solution of a second order elliptic PDE, e.g., a harmonic function in R n + \ E or in a domain Ω , for reasonably nice E and Ω , one expects (1.9)–(1.10) with F = |∇ u | / k u k L ∞ ( Ω ) , andfor a solution of a 2 m -th order elliptic PDE, m ∈ N , we will be aiming at F = |∇ m u | / k∇ m − u k L ∞ ( Ω ) .We shall come back to this point with more details in Section 8 and for now try to keep the discus-sion general for as long as possible. Remark . There is a slightly glitchy notation point here. For homogeneity reasons one couldprefer to normalize so that F = δ |∇ u | / k u k L ∞ ( Ω ) . However, making the function F and later on G and H in Section 5 depend on the domain (via distance) has its own dangers and kills the beauty of thegenerality here.Recall now the dyadic grid in Lemma 2.1 and the Whitney regions U Q from (2.12). Since everyWhitney region is contained in a ball (of a possibly larger but proportional to the scale radius) by(2.16), a necessary condition for (1.9) is that(3.2) sup Q ∈ D ( E ) σ ( Q ) Z Z b U Q | F ( Y ) | δ ( Y ) dY ≤ C . This will be a starting assumption in most of our statements, which however in all applications tothe CME for solutions of elliptic PDEs will be automatically fulfilled by Caccioppoli’s inequality.We shall discuss this in more details together with the corresponding applications.While stated exclusively for harmonic functions, the main result in [HMM] can be reformulatedas follows.
Theorem 3.3.
Let E ⊂ R n + be an n-dimensional UR set, and suppose that F ∈ L ( R n + \ E ) satisfies (3.2) .If for every Ω ± S defined by (2.33) (with S ′ = S ) we have (3.4) sup x ∈ ∂ Ω ± S , < r < ∞ r n Z Z B ( x , r ) ∩ Ω ± S | F ( Y ) | dist( Y , ∂ Ω ± S ) dY ≤ C , for some C > , then (3.5) sup x ∈ E , < r < ∞ r n Z Z B ( x , r ) | F ( Y ) | δ ( Y ) dY ≤ C , with C > depending on C , the constant in (3.2) , n, the ADR / UR constants of E, and the choice of η, K , τ only.In particular, if for some F ∈ L ( R n + \ E ) for every bounded NTA subdomain Ω ⊂ R n + \ Ewith an ADR boundary (3.6) sup x ∈ ∂ Ω , < r < ∞ r n Z Z B ( x , r ) ∩ Ω | F ( Y ) | dist( Y , ∂ Ω ) dY ≤ C , with a constant C > depending on n, and the NTA / ADR constants of Ω only, then (3.5) holds,and C > depends on n , η, τ, K, and the ADR / UR constants of E only.
Note that under the assumption (3.6) pertaining to all bounded NTA subdomains of R n + \ E , thecondition (3.2) is automatically satisfied. RANSFERENCE OF ESTIMATES 15
4. J ohn -N irenberg inequality and transference of C arleson measure estimates from L ipschitzto U niformly R ectifiable domains We start with a following version of the John-Nirenberg inequality. It is a suitable modificationof Lemma 10.1 in [HMa] which, in turn, was inspired by Lemma 2.14 in [AHLT].
Lemma 4.1.
Let E ⊂ R n + be an n-dimensional ADR set. Suppose there exist numbers < α < and < N < ∞ such that for some function F ∈ L ( R n + \ E ) and every Q ⊂ D ( E )(4.2) σ (cid:26) x ∈ Q : (cid:16) Z Z Γ Q ( x ) | F ( Y ) | δ ( Y ) − n dY (cid:17) / > N (cid:27) ≤ ασ ( Q ) . Then there exists C > such that (4.3) sup Q ∈ D ( E ) σ ( Q ) Z Q (cid:18)Z Z Γ Q ( x ) | F ( Y ) | δ ( Y ) − n dY (cid:19) p / d σ ( x ) ≤ C , for all p ∈ (0 , ∞ ) .Proof. Fix Q ∈ D ( E ) and denote the set on the left-hand side of (4.2) by E N , Q , so that σ ( E N , Q ) ≤ ασ ( Q ). We can assume that σ ( E N , Q ) ,
0, for, otherwise, the contribution of Q into (4.3) is N p (which can be absorbed in C ). Clearly, also E N , Q , Q since α <
1. Moreover, by outer regularity ofthe measure, we can find a set e E N , Q such that E N , Q ⊂ e E N , Q ⊂ Q , e E N , Q is relatively open in Q , and σ ( e E N , Q ) ≤ + α σ ( Q ) . Thus, one can build a collection of (pairwise disjoint) maximal dyadic cubes { Q j } j ⊂ D Q \ Q with S j Q j = e E N , Q . Fix one of the maximal cubes Q j . By maximality, for every P ⊃ Q j , P , Q j , P ∈ D Q , (and since { Q j } j ⊂ D Q \ Q , at least one such P always exists) there exists x ′ ∈ P such that (cid:16) Z Z Γ P ( x ′ ) | F ( Y ) | δ ( Y ) − n dY (cid:17) / ≤ N . Hence, in particular, for every P ∈ D Q \ D Q j , P ⊃ Q j , P , Q j (and again, at least one such P alwaysexists) we have(4.4) (cid:16) Z Z U P | F ( Y ) | δ ( Y ) − n dY (cid:17) / ≤ N . Now let M Q ( k ) : = sup Q ′ ∈ D Q σ ( Q ′ ) Z Q ′ (cid:18)Z Z Γ Q ′ , k ( x ) | F ( Y ) | δ ( Y ) − n dY (cid:19) p / d σ ( x ) , where, much as before,(4.5) Γ Q , k ( x ) = [ P ∈ D Q : P ∋ x − k ≤ ℓ ( P ) ≤ k U P , x ∈ Q , and M Q ( k ) : = Γ Q ′ , k ( x ) = Ø for all x ∈ Q ′ , Q ′ ∈ D Q . Then(4.6) Z Q (cid:18)Z Z Γ Q , k ( x ) | F ( Y ) | δ ( Y ) − n dY (cid:19) p / d σ ( x ) ≤ Z Q \ e E N , Q (cid:18)Z Z Γ Q , k ( x ) | F ( Y ) | δ ( Y ) − n dY (cid:19) p / d σ ( x ) + X j Z Q j (cid:18)Z Z Γ Q , k ( x ) | F ( Y ) | δ ( Y ) − n dY (cid:19) p / d σ ( x ) ≤ N p σ ( Q ) + X j Z Q j (cid:18)Z Z Γ Qj , k ( x ) | F ( Y ) | δ ( Y ) − n dY (cid:19) p / d σ ( x ) + X j Z Q j (cid:18)Z Z Γ Q , k ( x ) \ Γ Qj , k ( x ) | F ( Y ) | δ ( Y ) − n dY (cid:19) p / d σ ( x ) ≤ N p σ ( Q ) + X j σ ( Q j ) M Q ( k ) + X j Z Q j (cid:18)Z Z Γ Q , k ( x ) \ Γ Qj , k ( x ) | F ( Y ) | δ ( Y ) − n dY (cid:19) p / d σ ( x ) ≤ N p σ ( Q ) + + α σ ( Q ) M Q ( k ) + X j Z Q j (cid:18)Z Z Γ Q , k ( x ) \ Γ Qj , k ( x ) | F ( Y ) | δ ( Y ) − n dY (cid:19) p / d σ ( x ) . Note that, by definition, Γ Q , k ( x ) \ Γ Q j , k ( x ) = [ P ∈ D Q \ D Qj : P ∋ x − k ≤ ℓ ( P ) ≤ k U P , x ∈ Q j , and observe that the conditions x ∈ Q j and x ∈ P , P ∈ D Q \ D Q j , guarantee that P ) Q j . Indeed, by (ii) of Lemma 2.1, if there is a point x ∈ P ∩ Q j , then either P ) Q j or P ⊂ Q j and the latter is notpossible since P ∈ D Q \ D Q j . Thus, (4.4) applies, and the last term on the right-hand side of (4.6) isbounded by + α σ ( Q ) N p . All in all, (4.6) demonstrates that M Q ( k ) ≤ CN p , for all k ∈ N , and thus, letting k → ∞ , we arrive atsup Q ′ ∈ D Q σ ( Q ′ ) Z Q ′ (cid:18)Z Z Γ Q ′ ( x ) | F ( Y ) | δ ( Y ) − − n dY (cid:19) p / d σ ( x ) ≤ C N p , for all Q ∈ D ( E ) which finishes the proof of the Lemma. (cid:3) At this point we are ready to address the transference of the Carleson measure condition fromLipschitz to NTA domains. We shall use the fact that NTA domains with ADR boundaries containbig pieces of Lipschitz subdomains due to [DJ]. To be more precise, the following holds.
Definition 4.7.
We say that the domain Ω ⊂ R n + is a Lipschitz graph domain if there is someLipschitz function ψ : R n R and some coordinate system such that Ω = { ( x ′ , t ) : x ′ ∈ R n , t > ψ ( x ′ ) } . We refer to M = k∇ ψ k L ∞ ( R n ) as the Lipschitz constant of Ω . RANSFERENCE OF ESTIMATES 17
The open connected set Ω is said to be a bounded Lipschitz domain if there is some positivescale r = r Ω , some constants M > c ≥
1, and some finite set { x j } mj = of points with x j ∈ ∂ Ω ,such that the following conditions hold. First, ∂ Ω ⊂ m [ j = B ( x j , r j ) for some r j with 1 c r Ω < r j < c r Ω . Second, for each x j there is some Lipschitz graph domain V j , with x j ∈ ∂ V j and with Lipschitzconstant at most M , such that Z j ∩ Ω = Z j ∩ V j where Z j is a cylinder of height (8 + M ) r j , radius 2 r j , and with axis parallel to the t -axis (in thecoordinates associated with V j ).We refer to the triple ( M , m , c ) as the Lipschitz character of Ω . Proposition 4.8 ([DJ]) . Given D ⊂ R n + , a bounded NTA with ADR boundary, there exist constantsC ≥ and < θ < such that for every surface ball ∆ ( x , r ) = B ( x , r ) ∩ ∂ D, x ∈ ∂ D, r < diam( ∂ D ) ,there exists a bounded Lipschitz domain Ω ′ for which we have the following conditions: (1) H n ( ∂ Ω ∩ ∂ Ω ′ ∩ B ( x , r )) ≥ θ H n ( ∆ ( x , r )) ≈ θ r n . (2) There exists X ∆ so that B ( X ∆ , r / C ) ⊂ B ( x , r ) ∩ D ∩ Ω ′ . (3) Ω ′ ⊂ Ω ∩ B ( x , r ) .The Lipschitz character of Ω ′ as well as < θ < and C ≥ depend on n, the NTA constants of D,and the ADR constant of ∂ D only (and are independent of x , r). We remark that in [DJ], Proposition 4.8 is proved under a weaker assumption than that of NTA,namely, only an interior corkscrew condition, and a “weak exterior corkscrew condition” whichentails exterior disks rather than exterior balls, and with no hypothesis of Harnack chains. Later on,in [Bad], existence of big pieces of Lipschitz subdomains was also proved for usual NTA domains,with no upper ADR assumption on ∂ Ω (the lower ADR bound holds automatically in the presenceof a two-sided corkscrew condition, by virtue of the relative isoperimetric inequality). For theapplications that we have in mind here, neither amelioration is significant, and we will simply workwith the NTA domains in the sense of Definition 1.7 with ADR boundaries.For future reference we also would like to provide the following corollary. Corollary 4.9.
Given a bounded NTA domain D ⊂ R n + , with an ADR boundary E = ∂ D, thereexist constants C > and < θ < such that for every Q ∈ D ( E ) there exists a bounded Lipschitzdomain Ω Q for which we have the following: (1) σ ( ∂ Ω Q ∩ Q ) ≥ θ σ ( Q ) ≈ θℓ ( Q ) n . (2) For every Q ′ ∈ D ( Q ) such that there exists a point y Q ′ ∈ Q ′ ∩ ∂ Ω Q it follows that thedomain Ω Q contains a corkscrew point Y Q ′ relative to B ( y Q ′ , r ) ∩ ∂ Ω Q , r ≈ ℓ ( Q ′ ) , and thedomain Ω Q , and furthermore, with the appropriate choice of η and K in (2.8) , we haveB ( Y Q ′ , c ℓ ( Q ′ )) ⊂ U Q ′ . (3) Ω Q ⊂ Ω ∩ B ( x Q , C ℓ ( Q )) .The Lipschitz character of Ω Q as well as < θ < , c , C > , and the constants implicitly usedin the statement that “ Ω Q contains a corkscrew point Y Q ′ relative to B ( y Q ′ , r ) ∩ ∂ Ω Q , r ≈ ℓ ( Q ′ ) ”depend on n, the NTA constants of D and the ADR constants of E = ∂ D only (uniformly in Q, Q ′ ).Proof. The corollary follows directly from Proposition 4.8. Indeed, for any Q ∈ D ( E ) there exists ∆ ( x , r ) ⊂ Q , x ∈ Q , r ≈ ℓ ( Q ). One can build a Lipschitz domain from Proposition 4.8 correspondingto this ∆ ( x , r ), and then the conditions (1), (3) in Proposition 4.8 entail (1) and (3) in Corollary 4.9,respectively. The condition (2) in Corollary 4.9 follows from the fact that a Lipschitz domain Ω Q is,in particular, an NTA domain, and hence, it has a corkscrew point corresponding to B ( y Q ′ , r ) ∩ ∂ Ω Q as long as r < diam( ∂ Ω Q ). Using the fact that Ω Q ⊂ D , one can easily see that Y Q is also acorkscrew point of D , relative to B ( y Q ′ , r ) ∩ E , r ≈ ℓ ( Q ′ ). It remains to observe that a suitablechoice of η and K (uniform in Q ′ ) ensures that such a corkscrew point always belongs to U Q ′ andmoreover, B ( Y Q ′ , c ℓ ( Q ′ )) ⊂ U Q ′ , for some uniform constant c depending on n , the NTA constantsof D and the ADR constants of E = ∂ D only. (cid:3) Theorem 4.10.
Given an NTA domain D ⊂ R n + with an ADR boundary E = ∂ D and F ∈ L ( D ) which satisfies (3.2) , the following holds. If F satisfies the Carleson measure estimate (1.10) on allbounded Lipschitz subdomains of D with the constant C = C depending on the Lipschitz constantsof the underlying domains only, then F satisfies the Carleson measure estimate (1.10) in D aswell, with the bound depending on C , the constant in (3.2) , the NTA constants of D and the ADRconstants of ∂ D only.
Let us remark that in the course of the proof we ensure a suitable choice of a (su ffi ciently small) η and a (su ffi ciently large) K is (2.8) which strictly speaking a ff ect the constant in (3.2). However,as all choices depend on the NTA constants of E and ADR constants of E = ∂ D only, this does nota ff ect the result as stated above. Proof.
First of all, given that F satisfies (3.2), the same argument as in [HMM] allows one to reducematters to proving that(4.11) sup Q ∈ D ( E ) σ ( Q ) Z Z T Q | F ( X ) | δ ( X ) dX ≤ C . Note that here and below, δ = δ E denotes distance to E = ∂ D ; the distance to the subdomains willbe distinguished by the corresponding subscript. Furthermore, due to John-Nirenberg Lemma 4.1,it is in fact su ffi cient to show that there exist numbers 0 < α < < N < ∞ such that for every Q ⊂ D ( E )(4.12) σ (cid:26) x ∈ Q : (cid:16) Z Z Γ Q ( x ) | F ( Y ) | δ ( Y ) − n dY (cid:17) / > N (cid:27) ≤ ασ ( Q ) . Fix some Q ⊂ D ( E ). According to Corollary 4.9 (along with the inner regularity property of themeasure) there exists a bounded Lipschitz domain Ω Q and a closed set F Q ⊂ ∂ Ω Q ∩ Q such that σ ( F Q ) ≥ θ σ ( Q ), and the Lipschitz character of Ω Q as well as 0 < θ < n , the NTAconstants of D and the ADR constants of E only (uniformly in Q ). The domain Ω Q further satisfiesproperties (1)–(3) in Corollary 4.9.Now let us take a relatively open set Q \ F Q , single out the collection of maximal disjoint cubes F = { Q j } such that Q \ F Q = S j Q j , and build the corresponding sawtooth region Ω F , Q .By Tchebyshev inequality, it is su ffi cient to prove that(4.13) 1 σ ( Q ) Z Z Ω F , Q | F ( X ) | δ ( X ) dX ≤ C , in order to conclude (4.12). Indeed,(4.14) σ (cid:26) x ∈ F Q : (cid:16) Z Z Γ Q ( x ) | F ( Y ) | δ ( Y ) − n dY (cid:17) / > N (cid:27) ≤ N Z F Q Z Z Γ Q ( x ) | F ( Y ) | δ ( Y ) − n dY ≈ N Z Z S Q ′∈ D Q : Q ′∩ FQ , ∅ U Q ′ | F ( Y ) | δ ( Y ) dY . RANSFERENCE OF ESTIMATES 19
Now recall that by construction dyadic “cubes” are either contained in each other or do not intersect.Hence, if Q ′ ∈ D Q is such that Q ′ ∩ F Q , ∅ , we have Q ′ ∈ D Q \ S F D Q j , for, otherwise, Q ′ ∈ S F D Q j and hence Q ′ ⊂ Q \ F Q . Thus, the right-hand side of (4.14) is bounded by CN Z Z Ω F , Q | F ( Y ) | δ ( Y ) dY , so that (4.13), in conjunction with σ ( Q \ F Q ) ≤ (1 − θ ) σ ( Q ) yield the desired result (4.12).Hence, it remains to show (4.13) with a constant C depending on C , the constant in (3.2), theNTA constants of D and the ADR constants of ∂ D only.To this end, let us write(4.15) Z Z Ω F , Q | F ( X ) | δ ( X ) dX = X Q ′ ∈ D F , Q Z Z U Q ′ | F ( Y ) | δ ( Y ) dY and split this sum according to whether dist( U Q ′ , E ) ≤ ε dist( U Q ′ , ∂ Ω Q ) or else dist( U Q ′ , E ) > ε dist( U Q ′ , ∂ Ω Q ), for some small ε > U Q ′ can intersect ∂ Ω Q . We also record that ℓ ( Q ′ ) ≈ dist( U Q ′ , E ) by definitions (see (2.10), (2.11), and (2.8)).Let us start with Case I: (4.16) Q ′ ∈ D F , Q : dist( U Q ′ , E ) ≤ ε dist( U Q ′ , ∂ Ω Q ) . We claim that in this scenario(4.17) ℓ ( Q ′ ) ≈ dist( U Q ′ , E ) ≈ dist( U Q ′ , F Q ) ≈ dist( U Q ′ , ∂ Ω Q ) . As discussed above, the first equivalence follows from definitions. Now,(4.18) ℓ ( Q ′ ) ≈ dist( U Q ′ , E ) & dist( U Q ′ , Q ′ ) . This is because for any I ∈ W Q ′ we have ℓ ( Q ′ ) & dist( I , Q ′ ) by (2.11) and (2.8) and hence, ℓ ( Q ′ ) & dist( I ∗ ( τ ) , Q ′ ) as well. Next, note that there exists y ∈ Q ′ such that y ∈ F Q . Indeed, if not, Q ′ ⊂ Q \ F Q and hence, Q ′ ⊂ Q j for some j which is a contradiction with Q ′ ∈ D Q \ S F D Q j .Hence,(4.19) dist( U Q ′ , F Q ) ≤ dist( U Q ′ , y ) . dist( U Q ′ , Q ′ ) + ℓ ( Q ′ ) . ℓ ( Q ′ ) . In addition,(4.20) dist( U Q ′ , F Q ) ≥ dist( U Q ′ , E )for trivial reasons ( F Q ⊂ E ). Thus, combining (4.18)–(4.20), we have proved the second equiva-lence in (4.17).As for the third one, we have(4.21) dist( U Q ′ , F Q ) ≥ dist( U Q ′ , ∂ Ω Q )once again due to the fact that F Q ⊂ ∂ Ω Q . This, in combination with the current assumption (4.16)finally finishes the proof of (4.17).In particular, we conclude that for every Y ∈ U Q ′ with U Q ′ satisfying (4.16) we have δ ( Y ) = dist( Y , E ) . ℓ ( Q ′ ) + dist( U Q ′ , ∂ Ω Q ) . dist( U Q ′ , ∂ Ω Q ) ≤ dist( Y , ∂ Ω Q ) . Note also that the condition Q ′ ∩ F Q , ∅ proved above implies that, according to Corollary 4.9, Ω Q contains a corresponding corkscrew point, which is in turn contained in U Q ′ . Hence, Ω Q ∩ U Q ′ , ∅ ,and then due to (4.17), U Q ′ ⊂ Ω Q in this case.With this at hand, the part of the sum on the right-hand side of (4.15) corresponding to Case Ican be bounded as follows: X Q ′ ∈ D F , Q : dist( U Q ′ , E ) ≤ ε dist( U Q ′ , ∂ Ω Q ) Z Z U Q ′ | F ( Y ) | δ ( Y ) dY . X Q ′ ∈ D F , Q : dist( U Q ′ , E ) ≤ ε dist( U Q ′ , ∂ Ω Q ) , U Q ′ ⊂ Ω Q Z Z U Q ′ | F ( Y ) | dist( Y , ∂ Ω Q ) dY . Z Z Ω Q | F ( Y ) | dist( Y , ∂ Ω Q ) dY ≤ C σ ( Q ) , where we used finite overlap property of U Q ’s in the next-to-the-last inequality and the assumptionthat CME holds on all Lipschitz subdomains of D in the last one. Note that Ω Q ⊂ B ( x Q , C ℓ ( Q )) forsome uniform constant C , which justifies the bound by σ ( Q ). Case II: (4.22) Q ′ ∈ D F , Q : dist( U Q ′ , E ) > ε dist( U Q ′ , ∂ Ω Q ) . We shall demonstrate a packing condition on the cubes satisfying (4.22). Indeed, recall from abovethat ℓ ( Q ′ ) ≈ dist( U Q ′ , E ), so that in particular,(4.23) ℓ ( Q ′ ) & ε dist( U Q ′ , ∂ Ω Q ) . It follows that for a suitably small ε depending on the implicit constant in (4.23) and τ , we canensure that fattened regions b U Q ′ corresponding to U Q ′ from (4.22) necessarily intersect ∂ Ω Q and,moreover, σ ( b U Q ′ ∩ ∂ Ω Q ) ≈ ℓ ( Q ′ ), while b U Q ′ ’s still have finite overlap. Since the Lipschitz characterof ∂ Ω Q is controlled, σ ( ∂ Ω Q ) ≈ σ ( Q ) with some uniform in Q constants. Thus, all in all,(4.24) X Q ′ ∈ D F , Q : dist( U Q ′ , E ) > ε dist( U Q ′ ,∂ Ω Q ) σ ( Q ′ ) . σ ( Q ) . However, then(4.25) X Q ′ ∈ D F , Q : dist( U Q ′ , E ) > ε dist( U Q ′ ,∂ Ω Q ) Z Z U Q ′ | F ( Y ) | δ ( Y ) dY . X Q ′ ∈ D F , Q : dist( U Q ′ , E ) > ε dist( U Q ′ ,∂ Ω Q ) σ ( Q ′ ) , simply invoking (3.2). Then, combining (4.24)–(4.25) we finish the argument for Case II. (cid:3) A < N bounds : good - λ arguments Recall now definitions of the area integral, square function, and non-tangential maximal functionfrom Definition 1.14. We point out that we work with the dyadic cones which were not defined inthe introduction but rather in Section 4.When u is a solution of a second order elliptic PDE, e.g., a harmonic function in R n + \ E , onenormally works with the square function(5.1) S u ( x ) : = A ( |∇ u | ) , x ∈ E , and for a solution of a 2 m -th order elliptic PDE, m ∈ N , we will be interested in(5.2) S m u ( x ) : = A ( |∇ m u | ) , x ∈ E . We shall come back to this point with more details in Section 8 and for now try to keep thediscussion general for as long as possible. This is the same normalization as in the previous sections,and thus is also susceptible to the issue raised in Remark 3.1.
RANSFERENCE OF ESTIMATES 21 By b A , b S , b N ∗ we denote the area integral, square function, and the non-tangential maximal func-tion defined using the family b Γ in place of Γ . Note that according to these definitions, the conesare unbounded when E is unbounded. On the other hand, when E is bounded, so are the cones, allbeing contained in a C diam ( E )-neighborhood of E . We note also that when E is bounded, thereexists a cube Q ∈ D ( E ) such that Q = E and for any Q ∈ D ( E ) we have Q ∈ D Q . It is, however,particularly useful to work with local versions. To this end, by A Q , S Q , N Q ∗ we denote the areaintegral, square function, and the non-tangential maximal function defined using the family Γ Q inplace of Γ , and similarly for b A Q , b S Q , b N Q ∗ . Definition 5.3.
Let E ⊂ R n + be an n -dimensional ADR set, as in (1.1). By M D = M D E we denotethe dyadic Hardy-Littlewood maximal function on E , that is, for f ∈ L ( E ) M D f ( x ) = sup Q ∈ D ( E ): x ∈ Q ? Q | f ( y ) | d σ ( y ) , and we also write M D p f = M D ( | f | p ) p . Definition 5.4. (“ A / N ” estimates ). Let E ⊂ R n + be an n -dimensional ADR set, G ∈ L ( R n + \ E ), H ∈ C ( R n + \ E ). We say that “ A < N ” estimates hold for G , H in L q ( E ) if(5.5) kA G k L q ( E ) ≤ C k b N ∗ H k L q ( E ) , for some q ∈ (1 , ∞ ), and some uniform constant C . Same conventions apply if G and H aresupported on a subset D ⊂ Ω with an ADR boundary E = ∂ D . The L p norms are taken with respectto “surface measure” σ : = H n | E .Similarly, we will say that “ A Q < N Q ” estimates hold for G , H in L q ( E ) if(5.6) kA Q G k L q ( Q ) ≤ C k b N Q ∗ H k L q ( Q ) , for all Q ∈ D ( E ) , for some q ∈ (1 , ∞ ). Theorem 5.7.
Let E be an n-dimensional ADR set in R n + , Ω = R n + \ E, G ∈ L ( Ω ) , H ∈ C ( Ω ) ∩ L ∞ ( Ω ) . Given < q < ∞ , consider the following statements: ( A ) Carleson measure estimate (1.10) holds for F = G / k H k L ∞ ( Ω ) in Ω ; ( A loc ) Carleson measure estimate (1.10) holds on every (bounded) local sawtooth subdomain b Ω F , Q for F = G / k H k L ∞ ( b Ω F , Q ) , for any Q ∈ D ( E ) and any pairwise disjoint family of cubes F ∈ D ( E ) ; ( B ) q A < N on L q ( E ) holds for G and H, in the sense of Definition 5.4, i.e., (5.5) is valid; ( B loc ) q A Q < N Q on L q ( E ) holds for G and H, in the sense of Definition 5.4, i.e., (5.5) is valid; ( G λ ) q for every ε, γ > and for all α > σ { x ∈ E : A G ( x ) > (1 + ε ) α, b N ∗ H ( x ) ≤ γα } ≤ C ( γ/ε ) σ { x ∈ E : M D q ( A G )( x ) > α } ;( G λ loc ) q for every ε, γ > and for all α > σ { x ∈ Q : A Q G ( x ) > (1 + ε ) α, b N Q ∗ H ( x ) ≤ γα }≤ C ( γ/ε ) σ { x ∈ Q : M D q ( A Q G )( x ) > α } , for any Q ∈ D ( E ) . Consider, in addition, a condition (5.10) 1 σ ( Q ) (cid:18)Z Z U Q | G ( Y ) | δ ( Y ) dY (cid:19) / ≤ C k H k L ∞ ( b U Q ) , for all Q ∈ D ( E ) . Then ( A loc ) = ⇒ ( G λ ) q , for all < q < ∞ , (5.11) [( A loc )&(5.10)] = ⇒ ( B ) q , for all < q < ∞ , (5.12) ( A loc ) = ⇒ ( G λ loc ) q , for all < q < ∞ , (5.13) [( A loc )&(5.10)] = ⇒ ( B loc ) q , for all < q < ∞ , (5.14) ( B loc ) q = ⇒ ( B ) q , for all < q < ∞ , (5.15) (cid:0) ( B loc ) q for some < q < ∞ (cid:1) = ⇒ ( A ) . (5.16) In particular, if there exists a subclass of ADR domains, Σ , such that Ω ∈ Σ ; all (bounded) localsawtooth subdomains b Ω F , Q are in Σ , for any Q ∈ D ( E ) and any pairwise disjoint family of cubes F ∈ D ( E ) ; all local sawtooth subdomains of each of b Ω F , Q are also in Σ , etc., and G ∈ L ( Ω ) ,H ∈ C ( Ω ) ∩ L ∞ ( Ω ) are such that (5.10) holds for any domain in Σ then ( A ) on every D ∈ Σ (5.17) ⇐⇒ ( B loc ) q on the boundary of every D ∈ Σ for some < q < ∞⇐⇒ ( B loc ) q on the boundary of every D ∈ Σ for all < q < ∞⇐⇒ ( B ) q on the boundary of every D ∈ Σ for some < q < ∞⇐⇒ ( B ) q on the boundary of every D ∈ Σ for all < q < ∞ , with the understanding that all implicit constants in the statements above are uniform within Σ . An example of a class Σ as above (which is used in the present paper) is the class of uniformlyrectifiable domains with uniformly controlled ADR and UR constants. One has to point out thatthe definition of the sawtooth regions and with it, the meaning of the statements above, is slightlydi ff erent depending on whether the involved domains are just ADR or UR as well (or even NTA).For that reason, the reader will see statements like “the constant depends on the ADR constants of E (or the UR character if E is UR)”. We consider, however, the resulting dependence of either ADRconstants or UR (or NTA) character harmless.We remark that the assumption (5.10) is only needed to justify finiteness of some integrals in A < N arguments. If, e.g., it is known a priori that the L q norm of A is finite (or even thatthe L q norm of a certain truncated from above and below version of A is finite), then the resultof Theorem 5.7 carries over without (5.10). However, in all practical applications to solutions ofelliptic PDEs (5.10) is easily justified by Caccioppoli’s inequality. Remark that it is an analogue ofthe assumption that F = G / k H k L ∞ ( Ω ) satisfies (3.2), which is used in Theorem 3.3. In fact, the latterfollows from (5.10).Finally, a combination of (5.14) and (5.15) of course absorbs (5.12), but we will prove the latterearlier, based on (5.11), and then use an analogous argument towards (5.14). Proof of Theorem 5.7.
Step I: ( A loc ) = ⇒ ( G λ ) p , for all < p < ∞ . We start by proving that in theassumptions of the Theorem (with or without (5.10) at this stage) the statement ( A ) loc implies thatfor any 0 < p < ∞ , for every ε, γ > α > σ { x ∈ E : A G ( x ) > (1 + ε ) α, b N ∗ H ( x ) ≤ γα } ≤ C ( γ/ε ) σ { x ∈ E : M D p ( A G )( x ) > α } , with the constant C depending on the ADR constants of E (or the UR character if E is UR) and theconstant in ( A ) loc only. RANSFERENCE OF ESTIMATES 23
We can assume that the set on the right-hand side of (5.8) is not empty (otherwise A G ( x ) ≤ α for a.e. x ∈ E and the left-hand side of (5.8) has measure zero, as desired). We can also assume thatit is finite, even if E is unbounded (for, otherwise, there is nothing to prove).Thus, one can then build a collection of maximal dyadic cubes comprising the set { x ∈ E : M D p ( A G )( x ) > α } , following the Calder´on-Zygmund decomposition argument and extracting thecubes maximal with respect to the property (cid:16) > Q |A G | p d σ (cid:17) / p > α . One can check that the unionof such cubes is equal to { x ∈ E : M D p ( A G )( x ) > α } . In our assumptions, a maximal cube alwaysexists.Let us denote by Q one of these maximal cubes. We will prove that for every such Q we have(5.19) σ { x ∈ Q : A G ( x ) > (1 + ε ) α, b N ∗ H ( x ) ≤ γα } ≤ C ( γ/ε ) σ ( Q ) , with the constant C depending on the ADR constants of E (or UR character if E is UR) and theconstants in A loc only.Let us temporarily separate the cases. We note that if E is bounded, then E itself is the largestcube in D ( E ), and in this case we set E = Q . Case I of Step I: E is unbounded, or E is bounded and (cid:16) > Q |A G | p d σ (cid:17) / p ≤ α .In this case the considered collection of maximal cubes does not cover the entire E , so that Q has a parent e Q ∈ D ( E ), e Q , Q . Indeed, when E is bounded, such a property is guaranteed by thecondition (cid:16) > Q |A G | p d σ (cid:17) / p ≤ α . When E is unbounded, the existence of a parent e Q ∈ D ( E ), e Q , Q , is clear from the implicit assumption that σ { x ∈ E : M D p ( A G ( x )) > α } < ∞ (for, otherwisethere would be nothing to prove).Therefore, since Q is maximal, there exists an e x belonging to e Q , such that M D p ( A G )( e x ) ≤ α andhence, (cid:16) > e Q |A G | p d σ (cid:17) / p ≤ α . Then there exists z ∈ e Q such that A G ( z ) ≤ α .Thus, if we denote (cf. (2.23)) Γ ( x ) = Γ Q ( x ) = [ Q ′ ∈ D Q : Q ′ ∋ x U Q ′ , Γ ( x ) = [ Q ′ ∈ D ( E ) \ D Q : Q ′ ∋ x U Q ′ , x ∈ Q , and by A and A the corresponding portions of the square function, then(5.20) A G ( x ) ≤ α, for every x ∈ Q , since for every x ∈ Q we have A G ( x ) ≤ A G ( z ) . This follows from the properties (ii) and (iii) ofthe dyadic decomposition, see Lemma 2.1. Thus, for every x belonging to the set on the left-handside of (5.19) we have A G ( x ) > εα and in particular, it is su ffi cient to prove that(5.21) σ { x ∈ Q : A G ( x ) > εα, b N ∗ H ( x ) ≤ γα } ≤ C ( γ/ε ) σ ( Q ) . Case II of Step I: E is bounded (so that E = Q ) and (cid:16) > Q |A G | p d σ (cid:17) / p > α .In this case, by definition, { x ∈ E : M D p ( A G )( x ) > α } = E = Q , and therefore, again bydefinitions, A = A = A . Hence, in this case (5.19) reduces to (5.21) trivially.Thus, the two cases are now merged and we concentrate on proving (5.21). To this end, let usdenote by F the set { x ∈ Q : b N ∗ H ( x ) ≤ γα } . If σ ( F ) =
0, there is nothing to prove.Otherwise, if σ ( F ) >
0, we subdivide Q dyadically and stop the first time that Q ′ ∩ F = Ø . If onenever stops, we set F = Ø, otherwise we let
F ⊂ D Q \ { Q } be the family of stopping cubes whichis maximal by construction.The sawtooth regions Ω F and Ω F , Q retain the same significance as in (2.19) and by b Ω F and b Ω F , Q we denote analogous sawtooth regions defined with b U Q ′ in place of U Q ′ . Of course, in the case that F is empty, then Ω F , Q = T Q and b Ω F , Q = b T Q are just the corresponding Carleson boxesassociated to Q .Observe that by construction | H ( X ) | ≤ γα for every X ∈ b Ω F , Q . Indeed, if X ∈ b Ω F , Q then X ∈ b U Q ′ for some Q ′ ∈ D F , Q , where we recall that by definition, D F , Q is comprised of those dyadic sub-cubes of Q that are not contained in any Q j ∈ F . Thus, such a Q ′ necessarily contains a point from F . Now, let z ∈ Q ′ ∩ F . By definition, b N ∗ H ( z ) ≤ γα , and therefore, | H ( X ) | ≤ γα for every X ∈ b U Q ′ ,as desired.Next,(5.22) σ { x ∈ F : A G ( x ) > εα } . ( εα ) − Z F ( A G ( x )) d σ ( x ) = ( εα ) − Z F Z Z Γ ( x ) | G ( Y ) | δ ( Y ) − n dY d σ ( x ) ≈ ( εα ) − Z F X Q ′ ∈ D Q : x ∈ Q ′ Z Z U Q ′ | G ( Y ) | δ ( Y ) − n dY d σ ( x ) . Any Q ′ ∈ D Q which contains points of F must be an element of D Q ∩ D F . Hence, the expressionabove is bounded modulo a multiplicative constant by(5.23) ( εα ) − X Q ′ ∈ D Q ∩ D F Z Q ′ dx Z Z U Q ′ | G ( Y ) | δ ( Y ) − n dY d σ ( x ) . ( εα ) − X Q ′ ∈ D Q ∩ D F Z Z U Q ′ | G ( Y ) | δ ( Y ) dY . Observe, however, that for every Y ∈ U Q ′ as above δ ( Y ) = dist( Y , E ) ≈ ℓ ( Q ′ ) ≈ dist( Y , ∂ b Ω F , Q )since, as explained above, b Ω F , Q is comprised of fattened Whitney regions b U Q ′ . Using the boundedoverlap of the Whitney regions, we have(5.24) σ { x ∈ F : A G ( x ) > εα } . ( εα ) − Z Z S Q ′∈ D Q ∩ D F U Q ′ | G ( Y ) | dist( Y , ∂ b Ω F , Q ) dY . ( εα ) − Z Z b Ω F , Q | G ( Y ) | dist( Y , ∂ b Ω F , Q ) dY . At this stage, we recall two facts. First, | H ( X ) | ≤ γα for every X ∈ b Ω F , Q and hence, k H k L ∞ ( b Ω F , Q ) ≤ γα. Secondly, by definition b Ω F , Q is a subset of B ( x , C ℓ ( Q )) for some x ∈ ∂ b Ω F , Q and C dependingon the ADR constants (or UR character if E is UR) only. It follows then from ( A ) loc that we canbound the right-hand side of (5.24) by C ( εα ) − ( γα ) σ ( Q ), as desired. Step II: [( A loc )&(5.10)] = ⇒ ( B ) q , for all < q < ∞ . Due to Step I, we have at hand good- λ inequalities ( G λ ) p , for all 0 < p < ∞ . Assuming that the left-hand side of (5.5) is finite , the proofof (5.5) would be a standard argument using the L q boundedness of the Hardy-Littlewood maximaloperator M D p , p < q . Let us recall it for future reference. We have kA G k qL q ( E ) = (1 + ε ) q Z ∞ q α q σ { x ∈ E : A G ( x ) > (1 + ε ) α } d αα (5.25) ≤ (1 + ε ) q Z ∞ q α q σ { x ∈ E : A G ( x ) > (1 + ε ) α, b N ∗ H ( x ) ≤ γα } d αα RANSFERENCE OF ESTIMATES 25 + (cid:18) + εγ (cid:19) q k b N ∗ H k qL q ( E ) ≤ C (cid:16) γε (cid:17) (1 + ε ) q Z ∞ q α q σ { x ∈ E : M D p ( A G )( x ) > α } d αα + (cid:18) + εγ (cid:19) q k b N ∗ H k qL q ( E ) ≤ C (cid:16) γε (cid:17) (1 + ε ) q k M D p ( A G ) k qL q ( E ) + (cid:18) + εγ (cid:19) q k b N ∗ H k qL q ( E ) . (cid:16) γε (cid:17) (1 + ε ) q kA G k qL q ( E ) + (cid:18) + εγ (cid:19) q k b N ∗ H k qL q ( E ) . Note that we used Step I in the second inequality above. Then, assuming that kA G k L q ( E ) < ∞ andchoosing γ su ffi ciently small to ensure that the constant on front of kA G k qL q ( E ) on the right-handside of (5.25) is less than 1, we can “hide” the corresponding term on the left-hand side of (5.25)and conclude (5.5).Let us now run the argument without an a priori assumption of finiteness of the L q -norm of thesquare function. It is here that we use (5.10) for the first (and the only) time.To this end, we observe that it is enough to establish a refined version of the good-lambda in-equality, involving truncated square functions. For every k ∈ N we define the truncated cones (cf.(4.5)) Γ k ( x ) = [ Q ′ ∈ D ( E ): Q ′ ∋ x − k ≤ ℓ ( Q ′ ) ≤ k U Q ′ , x ∈ E , and write A k for the area integral defined with Γ k in place of Γ ( x ). Note that if E is bounded, thetruncation of cones from above is invisible when 2 k > ℓ ( Q ) ≈ diam( E ).We have the following analogue of (5.18) for A k . Take any 0 < p < ∞ . Then ( A ) loc implies thatfor every ε, γ >
0, and for all α > σ { x ∈ E : A k G ( x ) > (1 + ε ) α, b N ∗ H ( x ) ≤ γα } ≤ C ( γ/ε ) σ { x ∈ E : M D p ( A k G )( x ) > α } , where C is independent of k ∈ N .This is proved following line-by-line the argument of Step I and systematically changing A to A k etc. We only mention that at the final step, the appropriate analogue of (5.22) becomes(5.27) σ { x ∈ F : A k G ( x ) > εα } . ( εα ) − Z F X Q ′ ∈ D Q : x ∈ Q ′ , − k ≤ ℓ ( Q ′ ) ≤ k Z Z U Q ′ | G ( Y ) | δ ( Y ) − n dY d σ ( x ) . Here, A k corresponds to the integration over Γ k ( x ) : = Γ Q , k ( x ) (see (4.5)). At this point we canremove the restriction 2 − k ≤ ℓ ( Q ′ ) ≤ k , dominating the right-hand side of (5.27) by the right-handside of (5.22) and finish the argument as in Step I.Now, if k b N ∗ H k L q ( E ) < ∞ (and otherwise there is nothing to prove) then k S k u k L q ( E ) is qualitativelyfinite (albeit with the norm depending on k ) by (5.10). Therefore, we can apply the argument in(5.25) to conclude that (5.26) implies(5.28) kA k G k L q ( E ) ≤ C k N ∗ H k L q ( E ) , where C is independent of k (since C in the good- λ estimate (5.26) was independent of k and weused finiteness of kA k G k L q ( E ) only qualitatively). Now one can pass to the limit as k → ∞ andconclude ( B ) q , as desired. Step III: ( A ) loc implies ( G λ loc ) p , and [( A loc )&(5.10)] imply ( B loc ) q for all < p , q < ∞ . The argument follows line-by-line Steps I and II for the case of a bounded E . Further details areleft to the interested reader. Step IV: ( B loc ) q implies ( B ) q , for all < q < ∞ . Recall that when E is bounded, we assign E = Q and hence, ( B ) q is a particular case of ( B loc ) q .When E is unbounded, we proceed as follows. Given k ≫ A k a truncated squarefunctions where the dyadic cones incorporate the restriction that the cubes involved satisfies ℓ ( Q ) ≤ k . Clearly A k G ր A G . Clearly, ( A k G ) 1 Q = A Q G for every Q ∈ D − k . In each such Q , theestimate (5.6) holds uniformly on Q . With this in hand we can sum over those cubes and conclude A k G is uniformly controlled by H . This an the monotone convergence theorem gives the desiredestimate. Step V: validity of ( B loc ) q for some < q < ∞ implies ( A ) . Assume that ( B loc ) q , for some0 < q < ∞ holds. Fix the corresponding constant C from (5.6). Upon renormalization e G : = G (cid:0) C k H k L ∞ ( Ω ) (cid:1) − (with C coming from (5.6)), we have(5.29) ? Q (cid:16) A Q e G ( x ) (cid:17) q d σ ( x ) ≤ , for all Q ∈ D ( E ) , in particular, (4.2) is verified for F = e G uniformly on all Q ∈ D ( E ). It follows that (4.3) holds withthe same F for 0 < p < ∞ . The case p = q = σ ( Q ) Z Z T Q | e G ( Y ) | δ ( Y ) dY ≈ ? Q (cid:16) A Q e G ( x ) (cid:17) d σ ( x ) , and then the statement for any other q ≥ < q ≤ Step VI: the proof of (5.17). Fix any domain D ∈ Σ . The fact that the first line (5.17) for all D ∈ Σ ,and, in particular, for all sawtooth subdomains of D , implies any other line, is a consequence of(5.12) and (5.14) applied to G χ D and H χ D in place of G and H . The third line implies the secondon any D and similarly the fifth line implies the fourth for trivial reasons. The second one impliesfirst on any D ∈ Σ by (5.16).It remains to show that the validity of ( B ) q on the boundary of every D ∈ Σ implies the validityof ( A ) on Ω (and similarly for any subdomain of Ω in Σ ). To this end, recall that b Ω Ø , Q = b T Q ∈ Σ byour assumptions. And thus, with the same e G as in (5.29), ( B ) q on b T Q ’s implies that kA Q e G k L q ( ∂ b T Q ) ≤ σ ( ∂ b T Q ) ≤ C σ ( Q ), where the cones and A Q are built from the dyadic decomposition of ∂ b T Q .What we want though is (5.29), with the cones and A Q built from the dyadic decomposition of E . To this end, it remains to pass from A Q in the latter statement (built with cones associated toWhitney decomposition of b T Q ) to A Q from (5.29) (built with truncated cones associated to Whitneydecomposition of Ω ). This is a fairly straightforward step, using, in particular, (5.10) to handle theWhitney cubes at distance roughly ℓ ( Q ) from Q . We leave the details to the interested reader. Thus,with possibly another renormalization, we are getting (5.29) and proceed to ( A ) on Ω as before. (cid:3) RANSFERENCE OF ESTIMATES 27 N < S bounds : from L ipschitz to NTA domains
Before the start, let us observe that for any bounded NTA domain D ⊂ R n + , with an ADRboundary E = ∂ D , there exists a cube Q ∈ D ( E ) such that Q = E and for any Q ∈ D ( E ) wehave Q ∈ D Q , and since the domain is NTA, there exists at least one interior corkscrew pointcorresponding to Q (or rather to a surface ball containing Q with the radius proportional to ℓ ( Q )– see (2.2)). We shall refer to this point as X + D .Also, recall the dyadic Hardy-Littlewood maximal function from Definition 5.3. In addition, wewill be using its continuous analogue. Definition 6.1.
Let E ⊂ R n + be an n -dimensional ADR set, as in (1.1). By M = M E we denote thecontinuous (non-centered) Hardy-Littlewood maximal function on E , that is, for f ∈ L ( E ) M f ( x ) = sup ∆ ∋ x ? ∆ | f ( y ) | d σ ( y ) , and we also write M p f = M ( | f | p ) p . Here, the supremum is taken over all ∆ , surface balls on E containing x .It is clear from (2.2) that M D f ( x ) . M f ( x ) for every x ∈ E . The converse might fail pointwise,but both maximal functions are bounded in L p ( E ), p > Theorem 6.2.
Given a bounded NTA domain D ⊂ R n + with an ADR boundary E = ∂ D let u ∈ W , ( D ) ∩ C ( D ) be such that for any c ∈ R , for any Q ∈ D ( E ) , (6.3) sup X ∈ U Q | u ( X ) − c | ≤ C ℓ ( Q ) − n − Z Z b U Q | u − c | dX ! / . Then the following holds.Suppose that the “ N < S ” estimates are valid on all bounded Lipschitz subdomains Ω ′ ⊂ D.That is, for any Ω ′ ⊂ D (6.4) (cid:13)(cid:13) N ∗ , Ω ′ ( u − u ( X +Ω ′ )) (cid:13)(cid:13) L ( Ω ′ ) ≤ C k S ∂ Ω ′ u k L ( ∂ Ω ′ ) . Here X +Ω ′ is any interior corkscrew point of Ω ′ at the scale of diam( Ω ′ ) (see the discussion above thestatement of the Theorem) and N ∗ , Ω ′ and S Ω ′ are defined on the boundaries of bounded Lipschitzdomains using the traditional non-tangential cones (1.16) on Ω ′ for κ > . The constant C in (6.4) depends on the Lipschitz character of Ω ′ , the dimension n, and the choice of κ only.Then there exists < c << depending on n, the NTA constants of D and the ADR constants of ∂ D only such that for every ε > , < γ < c ε and for all α > σ { x ∈ E : N ∗ ( u − u ( X + D ))( x ) > (1 + ε ) α, M ( b S u )( x ) ≤ γα }≤ C ∗ γ,ε σ { x ∈ E : N ∗ ( u − u ( X + D ))( x ) > α } . with the constant C ∗ γ,ε < . To be more precise, C ∗ γ,ε = − θ + C (cid:0) γε (cid:1) where C > is aconstant depending on n, the NTA constants of D and the ADR constants of ∂ D only and < θ < ,depending on the same parameters, is from Corollary 4.9.In particular, for every u ∈ L ∞ ( D ) satisfying the assumptions of the Theorem we have (6.6) k N ∗ ( u − u ( X + D )) k L p ( E ) ≤ C k b S u k L p ( E ) , for all p > . We remark that contrary to the previous sections, we do not consider general A G and N ∗ H anymore. This is a necessity as the argument of the area integral has to be the gradient of the argumentof the non-tangential maximal function in the course of this proof. Thus, we might as well work directly with S rather than A (cf. (5.1)). The assumption (6.3) is a standard interior regularityestimate for solutions of elliptic equations (also known as Moser estimate). In principle, we need aslightly weaker version,(6.7) | u ( Y Q ) − c | ≤ C ℓ ( Q ) − n − Z Z U Q | u − c | dX ! / , where Y Q is any point lying in U Q together with a ball centered at Y Q of radius proportional to ℓ ( Q ).Using (6.7) directly would permit us not to enlarge the “aperture of cones”, that is, in this context,not to pass from U Q to b U Q , but that is minor and (6.3) looks a bit more familiar and more in linewith (7.2).We also remark that we could be more careful, as in Theorem 5.7, to try to avoid the assumption u ∈ L ∞ ( D ) in (6.6), but in practice we will always work with bounded solutions. Proof.
For brevity, we shall write u D : = u ( X + D ). We can assume that the set on the right-hand sideof (6.5) is not empty (otherwise N ∗ ( u − u D )( x ) ≤ α for a.e. x ∈ E and the left-hand side of (6.5) hasmeasure zero, as desired). It is also finite as E is bounded by our assumptions.We also assume for the time being that the set on the right-hand side of (6.5) is not the entire E = Q . This case will be addressed in the end of the proof.Let { Q j } j ⊂ D ( E ) be a (disjoint) collection of maximal cubes such that(6.8) [ j Q j = { x ∈ E : ( N ∗ ( u − u D ))( x ) > α } . Indeed, one can subdivide Q = E into dyadic cubes stopping whenever for some Q ′ ⊂ Q thereexists Y ∈ U Q ′ such that u ( Y ) − u D > α. Since we assume for now that the right-hand side of(6.5) is not the entire E = Q , it follows that Q is not the stopping cube. The resulting collectionof stopping time cubes Q ′ is automatically maximal (for, the parent e Q ′ does not belong to thiscollection by construction) and disjoint (again, by construction). We will denote it by S j Q j . Thefact that the union of stopping time cubes coincides with the desired set, that is, (6.8) holds, can beseen as follows. Since there exists Y ∈ U Q j such that u ( Y ) − u D > α, by definition ( N ∗ ( u − u D ))( x ) > α for every x ∈ Q j . Thus, Q j ⊂ { x ∈ E : ( N ∗ ( u − u D ))( x ) > α } for every j . Conversely, if x is suchthat ( N ∗ ( u − u D ))( x ) > α then there exists Q ′ ⊂ Q containing x such that for some Y ∈ U Q ′ wehave u ( Y ) − u D > α. However, in that case either Q ′ or one of Q ∈ D ( E ) with Q ⊃ Q ′ must be thestopping time cube. Hence, Q ′ ⊂ S j Q j . This finishes the proof of (6.8).Let us denote by Q one of the maximal cubes from the collection { Q j } j constructed above. Wewill prove that for every such Q we have(6.9) σ { x ∈ Q : N ∗ ( u − u D )( x ) > (1 + ε ) α, M ( b S u )( x ) ≤ γα } ≤ (1 − θ + C ( γ, ε )) σ ( Q ) , with 0 < θ < C ( γ, ε ) ≈ C (cid:0) γε (cid:1) for all γ < c ε with a suitably small c . Here, C will be a constantform (6.4) for a collection of bounded Lipschitz subdomains of D as in Corollary 4.9. Hence, C will depend on n , the NTA constants of D and the ADR constants of ∂ D only. We write C ( γ, ε )as C (cid:0) γε (cid:1) in the statement of the theorem as both C and the implicit constant in the equivalence C ( γ, ε ) ≈ C (cid:0) γε (cid:1) depend on n , the NTA constants of D and the ADR constants of ∂ D only. Inparticular, given any ε > γ small enough, depending on n , the NTA constants of D and the ADR constants of ∂ D only, so that C ∗ γ,ε : = − θ + C ( γ, ε ) <
1. Then, possibly furthershrinking c , we have C ∗ γ,ε < γ < c ε , as desired.Let us now turn to (6.9). First, we claim that we can reduce matters to proving RANSFERENCE OF ESTIMATES 29 (6.10) σ { x ∈ Q : N Q ∗ ( u − u D )( x ) > (1 + ε ) α, M ( b S u )( x ) ≤ γα } ≤ (1 − θ + C ( γ, ε )) σ ( Q ) , where as before, N Q ∗ is defined by taking the supremum within a truncated cone Γ Q (see (2.23)).Indeed, by maximality, for any P ∈ D ( E ), P ⊃ Q , we have N ∗ ( u − u D ) ≤ α for some x ∈ P andhence, u − u D ≤ α on the entire U P for every such P . Hence, if N ∗ ( u − u D )( x ) > (1 + ε ) α for some x ∈ Q , then necessarily there is a point Y ∈ S Q ′ ∈ D Q , Q ′ ∋ x U Q ′ such that u ( Y ) − u D > α . And hence,the set on the left-hand side of (6.10) contains the set on the left-hand side of (6.9), as desired.Next we invoke Corollary 4.9 and take a bounded Lipschitz domain Ω Q ⊂ D satisfying properties(1)–(3) in the statement of the Corollary. In particular, we set F Q : = ∂ Ω Q ∩ Q ⊂ Q such that σ ( F Q ) ≥ θ σ ( Q ). Since σ ( Q \ F Q ) ≤ (1 − θ ) σ ( Q ) , it remains to prove that(6.11) σ { x ∈ F Q : N Q ∗ ( u − u D )( x ) > (1 + ε ) α, M ( b S u )( x ) ≤ γα } ≤ C ( γ, ε ) σ ( Q ) . Going further, let us denote by Y Q ∈ U Q the corkscrew point of Ω Q relative to Q (cf. property(2) in Corollary 4.9). Denoting by e Q the parent of Q , we observe that by maximality u − u D ≤ α in U e Q and, in particular,(6.12) u ( Y e Q ) − u D ≤ α (here Y e Q can retain the same significance as in Corollary 4.9 or just be any point lying in U e Q together with its corkscrew ball). On the other hand,(6.13) | u ( Y e Q ) − u ( Y Q ) | . γα. Indeed, since M D ( b S u )( x ) . M ( b S u )( x ) ≤ γα, we have, in particular,(6.14) Z Z b T e Q |∇ u ( Y ) | δ ( Y ) dY . ( γα ) σ ( e Q ) ≈ ( γα ) σ ( Q ) , and hence,(6.15) Z Z b U e Q S b U Q |∇ u ( Y ) | dY . ( γα ) ℓ ( Q ) n − . Then, denoting temporarily c Q : = | b U e Q S b U Q | Z Z b U e Q S b U Q u dX , we have(6.16) | u ( Y e Q ) − u ( Y Q ) | ≤ | u ( Y e Q ) − c Q | + | u ( Y Q ) − c Q | . ℓ ( Q ) − n − Z Z b U e Q S b U Q | u − c Q | dX ! / . ℓ ( Q ) − n + Z Z b U e Q S b U Q |∇ u | dX ! / , by (6.3). Now we can invoke standard Poincar´e inequality considerations to show (6.13) and thencombine this with (6.12) to reduce (6.11) to(6.17) σ { x ∈ F Q : N Q ∗ ( u − u ( Y Q ))( x ) > εα/ , M ( b S u )( x ) ≤ γα } ≤ C ( γ, ε ) σ ( Q ) , assuming that γ < c ε with a suitably small c depending on n , the NTA constants of D and theADR constants of ∂ D only.At this stage let us recall once again condition (2) of Corollary 4.9. By definition of a corkscrewpoint and given the fact that all implicit constants depend on n , the NTA constants of D and theADR constants of ∂ D only, we can assure that for a suitable κ , once again depending on n , theNTA constants of D and the ADR constants of ∂ D only, the non-tangential approach regions of Ω Q defined by Γ Ω Q ( x ) : = { Y ∈ Ω Q : | Y − x | ≤ (1 + κ ) dist( Y , ∂ Ω Q ) } , x ∈ ∂ Ω Q , contain arising corkscrew points. That is, in the notation of Corollary 4.9, Y Q ′ ∈ Γ Ω Q ( y ′ Q ) for all Q ′ ∈ D ( Q ).Now, if x ∈ F Q is such that N Q ∗ ( u − u ( Y Q ))( x ) > εα/
2, it follows that there exists Q ′ ∈ D ( Q ) with Q ′ ∋ x and there exists X ∈ U Q ′ such that u ( X ) − u ( Y Q ) > εα/
2. Much as above, using the fact that M D ( b S u )( x ) ≤ γα, and Poincar´e inequality considerations, we deduce that u ( Y Q ′ ) − u ( Y Q ) > εα/ γ < c ε with c small enough depending on n , the NTA constants of D and the ADRconstants of ∂ D only. Here Y Q ′ ∈ U Q ′ is a special point from the condition (2) of Corollary 4.9)corresponding to y Q ′ = x . Since by construction Y Q ′ ∈ Γ Ω Q ( x ), we have N ∗ , Ω Q ( u − u ( Y Q ))( x ) : = sup Γ Ω Q ( x ) ( u − u ( Y Q )) > εα/ , that is, we can change the cones in the definition of the non-tangential maximal function from thosewith respect to D to those with respect to ∂ Ω Q . Using (a simplified version of) the same argument,we also can switch from Y Q , which is a corkscrew point of ∂ Ω Q relative to some surface ball of theradius r ≈ ℓ ( Q ) to X +Ω Q in (6.4), and obtain N ∗ , Ω Q ( u − u ( X +Ω Q ))( x ) : = sup Γ Ω Q ( x ) ( u − u ( Y Q )) > εα/ . Now (6.17) further reduces to showing that(6.18) σ { x ∈ F Q : N ∗ , Ω Q ( u − u ( X +Ω Q ))( x ) > εα/ , M ( b S u )( x ) ≤ γα } ≤ C ( γ, ε ) σ ( Q ) . At this stage, using the Tchebyshev inequality and the assumption (6.4), we can write(6.19) σ { x ∈ F Q : N ∗ , Ω Q ( u − u ( X +Ω Q ))( x ) > εα/ , M ( b S u )( x ) ≤ γα }≤ σ { x ∈ F Q : N ∗ , Ω Q ( u − u ( X +Ω Q ))( x ) > εα/ }≤ (cid:18) εα (cid:19) Z ∂ Ω Q (cid:16) N ∗ , Ω Q ( u − u ( X +Ω Q ))( x ) (cid:17) d σ . C (cid:18) εα (cid:19) Z ∂ Ω Q (cid:0) S Ω Q u ( x ) (cid:1) d σ ≈ C (cid:18) εα (cid:19) Z Z Ω Q |∇ u ( Y ) | dist( Y , ∂ Ω Q ) dY . C (cid:18) εα (cid:19) Z Z D ∩ B ( x Q , C ℓ ( Q )) |∇ u ( Y ) | dist( Y , E ) dY . The last inequality is due to the fact that Ω Q ⊂ D ∩ B ( x Q , C ℓ ( Q )) (see (3) in Corollary 4.9) and, inparticular, dist( Y , ∂ Ω Q ) ≤ dist( Y , E ) for every Y ∈ Ω Q . Note that all our choices depended on n , theNTA constants of D and the ADR constants of ∂ D only, and hence, so does C .Now, let us return to the set on the left-hand side of (6.19). At this stage, we have dropped thecondition M ( b S u )( x ) ≤ γα . However, if for every x ∈ F Q we have M ( b S u )( x ) > γα , then the set onthe left-hand side of (6.19) has measure zero and there is nothing to prove. Hence, we can proceed RANSFERENCE OF ESTIMATES 31 assuming that there is a point x ∈ F Q such that M ( b S u )( x ) ≤ γα . Therefore, for all surfaceballs ∆ ∋ x we have > ∆ | ( b S u )( y ) | dy ≤ ( γα ) . This gives an upper bound for the right-hand sideof (6.19). Indeed, recall, e.g., from the argument in [HMM] passing from the Carleson measureon tent regions (4.13), [HMM], loc.cit. , to Carleson measure on balls (4.12), [HMM], loc.cit. , thatfor any B ( x Q , C ℓ ( Q )) ∩ D on the right-hand side of (6.19) there exists a collection of dyadic cubes { P j } Mj = ⊂ D ( E ) of uniformly controlled cardinality M with ℓ ( P j ) ≈ ℓ ( Q ) such that S j P j covers B ( x Q , C ℓ ( Q )) ∩ E and S j T P j covers B ( x Q , C ℓ ( Q )) ∩ D . Now if we take ∆ to be a surface ball on E of radius r ≈ ℓ ( Q ) containing S j P j and x ∈ F Q , then Z Z D ∩ B ( x Q , C ℓ ( Q )) |∇ u ( Y ) | dist( Y , E ) dY . M X j = Z Z T Pj |∇ u ( Y ) | dist( Y , E ) dY . Z ∆ | ( b S u )( y ) | dy ≤ ( γα ) σ ( ∆ ) . ( γα ) σ ( Q ) . Combining this with (6.19), we conclude that(6.20) σ { x ∈ F Q : N ∗ , Ω Q ( u − u ( X +Ω Q )))( x ) > εα/ , M ( b S u )( x ) ≤ γα } . C (cid:16) γε (cid:17) σ ( Q ) = : C ( γ, ε ) σ ( Q ) . Now it only remains to treat a special case when E = Q = Q is itself the first stopping cube. Inthis case, N ∗ = N Q ∗ by definition and hence, the reduction to (6.11) is automatic. Using (6.13) wecan swipe u D for u ( Y Q ) (as they are both corkscrew points of E at the scale ℓ ( Q ) ≈ diam( E )) andfurther reduce to (6.17). From that point on, the argument is the same as before.Finally, having at hand (6.5), an argument analogous to (5.25) yields (6.6). To be specific, weshow that taking ε > D and ADRconstants of E and then taking γ > ε , theestimate (6.5) yields (6.6). It is here that we use a possibility to pick ε > ffi ciently small.Indeed, fix any q >
2. The assumption that u ∈ L ∞ ( D ) and boundedness of D guarantee that k N ∗ ( u − u ( X + D )) k L q ( E ) is a priori finite. Then much as in (5.25), k N ∗ ( u − u ( X + D )) k qL q ( E ) = (1 + ε ) q Z ∞ q α q σ { x ∈ E : N ∗ ( u − u ( X + D ))( x ) > (1 + ε ) α } d αα (6.21) ≤ (1 + ε ) q Z ∞ q α q σ { x ∈ E : N ∗ ( u − u ( X + D ))( x ) > (1 + ε ) α, M ( b S u )( x ) ≤ γα } d αα + (cid:18) + εγ (cid:19) q k M ( b S u ) k qL q ( E ) ≤ C ∗ γ,ε (1 + ε ) q Z ∞ q α q σ { x ∈ E : N ∗ ( u − u ( X + D ))( x ) > α } d αα + (cid:18) + εγ (cid:19) q k M ( b S u ) k qL q ( E ) . At this point we note that we can choose ε > < γ < c ε depending on the NTA constantsof D and ADR constants of E such that C ∗ γ,ε (1 + ε ) q = (cid:16) − θ + C (cid:0) γε (cid:1) (cid:17) (1 + ε ) q < . Then theright-hand side of (6.21) is bounded by k N ∗ ( u − u ( X + D )) k qL q ( E ) + C k M ( b S u ) k qL q ( E ) (6.22) . k N ∗ ( u − u ( X + D )) k qL q ( E ) + C k b S u k qL q ( E ) , and a priori finiteness of k N ∗ ( u − u ( X + D )) k L q ( E ) allows us to hide the corresponding term on theleft-hand side. (cid:3) Remark . Working on a fixed Q ∈ D ( E ) rather than the entire E , and following the proof ofTheorem 6.2 we can show that for every u ∈ L ∞ ( D ) satisfying the assumptions of Theorem 6.2(6.24) k N Q ∗ ( u − u ( Y Q )) k L p ( Q ) ≤ C k b S CQ u k L p ( Q ) , for all Q ∈ D ( E ) , for all p > , for some C > E only. Here, as before, Y Q is anycorkscrew point of D relative to Q . Theorem 6.25.
Given a bounded NTA domain D ⊂ R n + with an ADR boundary E = ∂ D, letu ∈ W , ( D ) , continuous and bounded on D, be such that for any c ∈ R , for any Q ∈ D ( E ) (6.3) isvalid and (6.26) ℓ ( Q ) − n − Z Z U Q |∇ u | p dX ! / p . ℓ ( Q ) − n − Z Z b U Q |∇ u | dX ! / , for some p > . Then the following holds.If the local “ N < S ” estimates are valid on all bounded NTA subdomains Ω ′ ⊂ D for the samep > as above, that is, for every Q ∈ D ( ∂ Ω ′ ) and any Y Q , a corkscrew point of Ω ′ relative to Q, (6.27) k N Q ∗ ( u − u ( Y Q )) k L q ( Q ) ≤ C k b S C ℓ ( Q ) u k L q ( Q ) , then (6.6) holds in D for all < q < ∞ .Proof. Much as in the proof of Theorem 6.2, matters can be reduced to showing that there exists0 < c << n , the NTA constants of D and the ADR constants of ∂ D only such thatfor every ε >
0, 0 < γ < c ε and for all α > σ { x ∈ E : N ∗ ( u − u ( X + D ))( x ) > (1 + ε ) α, b S u ( x ) ≤ γα }≤ C ∗ γ,ε σ { x ∈ E : N ∗ ( u − u ( X + D ))( x ) > α } . with the constant C ∗ γ,ε <
1. We will show that in this case C ∗ γ,ε → γ → ε . In fact,at this stage we do not have to be as careful keeping ε > ε = Q beone of such cubes, and reduce (6.28) to(6.29) σ { x ∈ Q : N Q ∗ ( u − u ( Y Q ))( x ) > εα/ , b S u ( x ) ≤ γα } ≤ C ∗ γ,ε σ ( Q ) , assuming that γ < c ε with a suitably small c depending on n , the NTA constants of D and theADR constants of ∂ D only (and the constant C ∗ γ,ε satisfies the same constraints as before, althoughpossibly the actual value is di ff erent). The notation here is the same as in (6.17) and the reductionargument is, in fact, even simpler, because one does need to pass to F Q and because the use of b S u ( x ) ≤ γα for some x ∈ Q directly yields (6.15) avoiding (6.14).Now let us denote(6.30) E Q : = { x ∈ Q : N Q ∗ ( u − u ( Y Q ))( x ) > εα/ , b S u ( x ) ≤ γα } . By inner regularity, we can choose E ′ Q , a closed subset of E Q , with the size arbitrarily close to that of E Q . Now we denote by F the decomposition of an open set ( E ′ Q ) c into dyadic maximal cubes fromthe family D ( E ) and let b Ω F be a global sawtooth region corresponding to such a decomposition.The “fatness” of the Whitney regions defining b Ω F is bigger than that for N Q ∗ and smaller that for b S , RANSFERENCE OF ESTIMATES 33 as will become clear soon. For now, just take them slightly larger than the Whitney regions of N Q ∗ .We start with(6.31) σ ( E ′ Q ) ≤ (cid:18) εα (cid:19) p Z E ′ Q | N Q ∗ ( u − u ( Y Q ))( x ) | p dx , and now change the cones from those used in N Q ∗ (dyadic, with respect to D ) to the traditional ones(1.16) with respect to b Ω F . This is possible because every dyadic cone with respect to D , Γ ( x ), x ∈ E ′ Q , is comprised of U Q with Q ∋ x , and b Ω F contains all the corresponding b U Q , so that for Y ∈ U Q we have | Y − x | ≈ ℓ ( Q ) and dist( Y , ∂ b Ω F ) ≈ ℓ ( Q ).To lighten the notation, we will write Ω F in place of b Ω F from now on. The non-tangentialmaximal function defined with the traditional cones with respect to Ω F will be denoted by N ∗ , Ω F ,and we have then the right-hand side of (6.31) bounded by(6.32) (cid:18) εα (cid:19) p Z E ′ Q | N C ℓ ( Q ) ∗ , Ω F ( u − u ( Y Q ))( x ) | p dx . We remark that by construction Y Q , which is a corkscrew point for Q with respect to D is also acorkscrew point for some B ( x Q , C ℓ ( Q )) ∩ ∂ Ω F ⊃ Q with respect to Ω F , since int ( b U Q ) ⊂ Ω F . Thisis assuming that Q , F , but otherwise E Q = Ø and there is nothing to prove.Now, with the constant C changing value from line to line but still depending on the NTA / ADRconstants of D only, we have(6.33) (cid:18) εα (cid:19) p Z E ′ Q | N C ℓ ( Q ) ∗ , Ω F ( u − u ( Y Q ))( x ) | p dx ≤ (cid:18) εα (cid:19) p Z ∂ Ω F ∩ B ( x Q , C ℓ ( Q )) | N C ℓ ( Q ) ∗ , Ω F ( u − u ( Y Q ))( x ) | p dx . (cid:18) εα (cid:19) p Z ∂ Ω F ∩ B ( x Q , C ℓ ( Q )) | b S C ℓ ( Q ) Ω F u ( x ) | p dx . The first inequality here is just integration on a larger set. For the second one, we first recall that asawtooth domain with respect to an NTA domain with an ADR boundary is itself an NTA domainwith an ADR boundary (for the fact that the ADR property is preserved, see [HMM], and for NTAfeatures see [HM]). With this at hand, we pass from traditional cones with respect to Ω F to dyadiccones with respect to Ω F by Remark 2.35, cover ∂ Ω F ∩ B ( x Q , C ℓ ( Q )) by dyadic cubes of ∂ Ω F atthe scale C ℓ ( Q ) with a suitable C , use (6.27) on Ω F and then pass back to the traditional coneswith respect to Ω F , enlarging aperture and enlarging C in (6.33) in the process, but still keepingdependence on the NTA / ADR constants only.At this point, (6.31)–(6.33) can be summarized as(6.34) σ ( E ′ Q ) . (cid:18) εα (cid:19) p Z ∂ Ω F ∩ B ( x Q , C ℓ ( Q )) | b S C ℓ ( Q ) Ω F u ( x ) | p dx ≤ (cid:18) εα (cid:19) p Z E ′ Q | b S C ℓ ( Q ) Ω F u ( x ) | p dx + (cid:18) εα (cid:19) p Z ( ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q | b S C ℓ ( Q ) Ω F u ( x ) | p dx = : I + II . The estimate on part I is now quite straightforward. For points on the common boundary of ∂ D and ∂ Ω F , that is, for x ∈ E ′ Q , the traditional cones Γ Ω F ( x ) are trivially contained in traditional cones withrespect to Ω , Γ Ω ( x ), which are in turn contained in dyadic cones with respect to D , Γ ( x ), provided that η and K in their definition are su ffi ciently small and large, respectively. This means that b S C ℓ ( Q ) Ω F u ( x ) ≤ b S C ℓ ( Q ) u ( x ) ≤ γα, for x ∈ E ′ Q , provided that b S C ℓ ( Q ) in the definition of E Q uses su ffi ciently wide “cones”. Hence,(6.35) I . (cid:16) γε (cid:17) p σ ( Q ) . Turning to II , we start with the following Claim . For any x ∈ ∂ Ω F there exists x ∈ E ′ Q such that Γ Ω F ( x ) ⊂ Γ ( x ) , where Γ ( x ), x ∈ ∂ D , is a family of dyadic cones with respect to D , with suitably large aperture(that is, suitable η and K ). Proof.
We have already discussed that the Claim is straightforward with x = x when x ∈ E ′ Q , sowe concentrate on x ∈ ∂ Ω F \ E ′ Q . Since x ∈ ∂ Ω F \ E ′ Q , we know that x belongs to the closure of some b U Q , Q ∈ D ( E ), suchthat Q ∩ E ′ Q , Ø . Indeed, if Q ∩ E ′ Q = Ø then Q is either Q j or one of its subcubes, whichcontradicts the definition of the sawtooth region Ω F . Therefore, there exists a point x ∈ Q ∩ E ′ Q .By Remark 2.35, it is su ffi cient to show that a dyadic cone with respect to Ω F with a vertex at x , Γ D , Ω F ( x ), is contained in a dyadic cone with respect to D with a vertex at x , Γ ( x ).Now, if Y ∈ Γ D , Ω F ( x ) then Y ∈ U P , Ω F for some P ∈ D ( ∂ Ω F ), P ∋ x , where U P , Ω F is a Whitneyregion of Ω F corresponding to P ∈ D ( ∂ Ω F ). If ℓ ( P ) ≤ c ℓ ( Q ), then | Y − x | ≤ c ′ ℓ ( Q ) and hence, Y ∈ b U Q together with x , provided that c and hence c ′ are su ffi ciently small depending on the usualgeometric parameters only. Therefore, Y ∈ Γ ( x ) in this case.If ℓ ( P ) ≥ c ℓ ( Q ) then let Q P ∈ D ( ∂ D ) denote any cube containing Q at the scale c ℓ ( P ) . Weclaim that Y ∈ U Q P , provided that the parameters η and K in the definition of Whitney regions havebeen suitably adjusted. Indeed,dist( Y , Q P ) ≤ dist( Y , P ) + dist( P , Q P ) . ℓ ( P ) + | x − x | . ℓ ( P ) + ℓ ( Q ) . ℓ ( P ) . In particular, dist( Y , ∂ D ) . ℓ ( P ) . On the other hand,dist( Y , ∂ D ) ≥ dist( Y , ∂ Ω F ) ≥ ℓ ( P )since Y ∈ U P . This is su ffi cient to show that Y ∈ U Q P with suitable η and K , finishing the proof ofClaim 6.36. (cid:3) We observe that Γ ( x ) in the statement of Claim 6.36 can of course exceed the limits of Ω F ,which is not a problem.Let us now get back to the proof of the Theorem, specifically, to the estimate for II in (6.34). Tothis end, we split it further into II and II , as follows. The square function in the integrand of II is b S C ℓ ( Q ) Ω F u ( x ) = Z Z Γ C ℓ ( Q ) Ω F ( x ) |∇ u ( Y ) | dY dist( Y , ∂ Ω F ) n − ! / . We divide the domain of integration into Y ∈ Γ C ℓ ( Q ) Ω F ( x ) such that dist( Y , ∂ Ω F ) ≤ c dist( Y , ∂ D ) and Y ∈ Γ C ℓ ( Q ) Ω F ( x ) such that dist( Y , ∂ Ω F ) ≥ c dist( Y , ∂ D ), with small c to be determined below. Thecorresponding parts of II will be referred to as II and II , respectively.The estimate on II is easier. Using Claim 6.36, we have for every ( ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q RANSFERENCE OF ESTIMATES 35 (6.37)
Z Z Y ∈ Γ C ℓ ( Q ) Ω F ( x ):dist( Y ,∂ Ω F ) ≥ c dist( Y ,∂ D ) |∇ u ( Y ) | dY dist( Y , ∂ Ω F ) n − ! / . Z Z Y ∈ Γ C ℓ ( Q ) Ω F ( x ): dist( Y ,∂ Ω F ) ≥ c dist( Y ,∂ D ) |∇ u ( Y ) | dY dist( Y , ∂ D ) n − ! / . sup x ∈ E ′ Q (cid:18)Z Z Y ∈ Γ C ℓ ( Q ) ( x ) |∇ u ( Y ) | dY dist( Y , ∂ D ) n − (cid:19) / . sup x ∈ E ′ Q b S C ℓ ( Q ) u ( x ) ≤ γα. Thus,(6.38) II . (cid:16) γε (cid:17) p σ (( ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q ) ≤ (cid:16) γε (cid:17) p σ ( Q ) . It remains to handle II . To start,(6.39) Z ( ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z Y ∈ Γ C ℓ ( Q ) Ω F ( x ); dist( Y ,∂ Ω F ) ≤ c dist( Y ,∂ D ) |∇ u ( Y ) | dY dist( Y , ∂ Ω F ) n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p / dx . Z ( ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q Z Z Y ∈ Γ C ℓ ( Q ) Ω F ( x ); dist( Y ,∂ Ω F ) ≤ c dist( Y ,∂ D ) |∇ u ( Y ) | p dY dist( Y , ∂ Ω F ) n − p ×× Z Z Y ∈ Γ C ℓ ( Q ) Ω F ( x ); dist( Y ,∂ Ω F ) ≤ c dist( Y ,∂ D ) dY dist( Y , ∂ Ω F ) n ! p − dx . We claim that(6.40)
Z Z Y ∈ Γ C ℓ ( Q ) Ω F ( x ); dist( Y ,∂ Ω F ) ≤ c dist( Y ,∂ D ) dY dist( Y , ∂ Ω F ) n . dist( x , E ′ Q ) . It is here that the smallness of c is used. To show this, let us first observe that x ∈ ( ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q ⊂ D belongs to some U P , a Whitney region of D corresponding to P ∈ D ( E ),with dist( P , E ′ Q ) ≈ ℓ ( P ). Indeed, the sawtooth domain Ω F was formed based on cubes Q j suchthat ℓ ( Q j ) ≈ dist( Q j , E ′ Q ) and their subcubes. If ℓ ( P ) ≥ C dist( P , E ′ Q ) for su ffi ciently large C , thiscube cannot belong to F and thus, int U P ⊂ Ω F . If ℓ ( P ) ≤ c dist( P , E ′ Q ) with su ffi ciently small c then first of all, P ⊂ ∪ j Q j (since dist( P , E ′ Q ) >
0) and secondly, it cannot contain any Q j bymaximality. Hence, P ∈ F and x < Ω F . Thus, indeed, dist( P , E ′ Q ) ≈ ℓ ( P ) if U P ∋ x , or in otherwords, ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q is covered by such U P . Notice that dist( x , E ′ Q ) ≈ ℓ ( P ) in thisnotation: ℓ ( P ) ≈ dist( x , ∂ D ) ≤ dist( x , E ′ Q ) ≤ dist( x , P ) + dist( P , E ′ Q ) . ℓ ( P ) . If Y ∈ Γ C ℓ ( Q ) Ω F ( x ) is such that dist( Y , ∂ Ω F ) ≤ c dist( Y , ∂ D ), then | x − Y | ≤ (1 + κ ) dist( Y , ∂ Ω F ) ≤ c (1 + κ ) dist( Y , ∂ D ). Hence, if c is su ffi ciently small,(6.41) dist( Y , ∂ Ω F ) ≤ dist( Y , ∂ D ) ≈ dist( x , ∂ D ) ≈ ℓ ( P ) ≈ dist( x , E ′ Q ) . In fact, and it will be useful soon, we can choose c so small that Y belongs to b U P for the same U P that contains x . Having this at hand, (6.40) is established simply splitting the integral into the slices | Y − x | ≤ (1 + κ ) dist( Y , ∂ Ω F ) ≈ − j (1 + κ ) dist( x , E ′ Q ) , j ∈ N , and then summing up back.Now, using (6.40) and (6.41) again, we have (6.42) Z ( ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z Y ∈ Γ C ℓ ( Q ) Ω F ( x ); dist( Y ,∂ Ω F ) ≤ c dist( Y ,∂ D ) |∇ u ( Y ) | dY dist( Y , ∂ Ω F ) n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p / d σ ( x ) . Z ( ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q Z Z Y ∈ Γ C ℓ ( Q ) Ω F ( x ); dist( Y ,∂ Ω F ) ≤ c dist( Y ,∂ D ) |∇ u ( Y ) | p dY dist( Y , ∂ Ω F ) n ×× dist( x , E ′ Q ) p − d σ ( x ) . By Fubini’s theorem (keeping in mind our choice of small c ), the integral above is bounded by(6.43) X P ∈ D ( E ): U P ∩ ( ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q , Ø ℓ ( P ) p − Z Z Y ∈ b U P |∇ u ( Y ) | p dY dist( Y , ∂ Ω F ) n ×× Z x ∈ ( ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q : | x − Y | . (1 + κ ) dist( Y ,∂ Ω F ) d σ ( x ) dY . X P ∈ D ( E ): U P ∩ ( ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q , Ø ℓ ( P ) p − Z Z Y ∈ b U P |∇ u ( Y ) | p dY . Now, using the reverse H ¨older property for the gradients (6.26) and slightly enlarging b U Q (but wewill not write extra hats), the expression above is bounded by(6.44) X P ∈ D ( E ): U P ∩ ( ∂ Ω F ∩ B ( x Q , C ℓ ( Q ))) \ E ′ Q , Ø ℓ ( P ) n (cid:18) ℓ ( P ) − n − Z Z Y ∈ b U P | ℓ ( P ) ∇ u ( Y ) | dY (cid:19) p / . Finally, we recall that dist( P , E ′ Q ) ≈ ℓ ( P ) and hence, enlarging the Whitney regions that define thesquare function in (6.30) yet again, we can show that Y ∈ b Γ ( x ) for some x ∈ E ′ Q and hence, for Y as above, (cid:18) ℓ ( P ) − n − Z Z Y ∈ b U P | ℓ ( P ) ∇ u ( Y ) | dY (cid:19) p / . sup E ′ Q b S u ( x ) p ≤ ( γα ) p . Summing up ℓ ( P ) n over all P in (6.44) yields a multiple of σ ( Q ) and finishes the argument. (cid:3)
7. F rom N < S bounds on NTA domains to ε - approximability in a complement of a UR set Recall the definition of ε -approximability (Definition 1.11). The second main result in [HMM],stated there for harmonic functions but proved in full generality, can be formulated as follows. Theorem 7.1.
Let E ⊂ R n + be an n-dimensional UR set, Ω = R n + \ E, and suppose that u ∈ W , ( Ω ) ∩ C ( Ω ) is such that for any Q ∈ D ( E ) , for every component U iQ of U Q we have (7.2) sup X , Y ∈ U iQ | u ( X ) − u ( Y ) | ≤ C ℓ ( Q ) − n − Z Z b U iQ | u | dX ! / . Assume, in addition, that u ∈ L ∞ ( Ω ) and F : = |∇ u | / k u k L ∞ ( Ω ) satisfies the Carleson measure estimate (1.9) .Finally, suppose that for every Ω ± S defined by (2.33) (with S ′ = S ) we have (7.3) (cid:13)(cid:13)(cid:13) N ∗ ( u − u ( X Ω ± S )) (cid:13)(cid:13)(cid:13) L ( ∂ Ω ± S ) ≤ C k S u k L ( ∂ Ω ± S ) . Here X Ω ± S is any interior corkscrew point of Ω ± S at the scale of diam( Ω ± S ) (see the discussion abovethe statement of Theorem 6.2). Then u is ε -approximable on Ω , with the implicit constants depend-ing only on n, the ADR / UR constants of E, and the choice of η, K , τ, κ only. RANSFERENCE OF ESTIMATES 37
In particular, if one substitutes (7.3) by a more general assumption that the estimate (7.4) k N ∗ ( u − u ( X Ω ′ ) k L ( ∂ Ω ′ ) ≤ C k S u k L ( ∂ Ω ′ ) on every bounded NTA domain Ω ′ ⊂ Ω with an ADR boundary, with the constant C depending onthe NTA constants of Ω ′ and ADR constants of ∂ Ω ′ only, then the same conclusion follows. Strictly speaking, the Theorem above was proved in [HMM] departing from the bound (7.3) forthe non-tangential maximal function defined with traditional non-tangential cones (1.16) rather thanthe dyadic ones, but that is easy to change by Remark 2.35.8. A pplications : solutions of divergence form elliptic equations with bounded measurablecoefficients Second order divergence form elliptic operators with coe ffi cients satisfying a Carlesonmeasure condition. Let E ⊂ R n + be an n -dimensional ADR set and let Ω = R n + \ E . Consider adivergence form elliptic operator L : = − div A ( X ) ∇ , defined in Ω , where A is an ( n + × ( n +
1) matrix with real bounded measurable coe ffi cients,possibly non-symmetric, satisfying the ellipticity condition(8.1) λ | ξ | ≤ h A ( X ) ξ, ξ i : = n + X i , j = A i j ( X ) ξ j ξ i , k A k L ∞ ( R n ) ≤ λ − , for some λ >
0, and for all ξ ∈ R n + , X ∈ Ω . As usual, the divergence form equation is interpretedin the weak sense, i.e., we say that Lu = Ω if u ∈ W , ( Ω ) and(8.2) Z Ω A ( X ) ∇ u ( X ) · ∇ Ψ ( X ) dX = , for all Ψ ∈ C ∞ ( Ω ).Assume furthermore that the distributional derivatives of the coe ffi cients of A satisfy the follow-ing Carleson measure condition:(8.3) F ( X ) = ǫ ( X ) : = sup i , j = ,..., n + sup (cid:8) |∇ A ( Z ) | , Z ∈ B ( X , δ ( X ) / (cid:9) satisfies (1.9) . It has been demonstrated in [KP] that the condition (8.3) implies that solutions to the correspondingelliptic equation satisfy square function / non-tangential maximal function estimates on Lipschitzdomains. The results of the present paper allow us to establish analogous facts in the full generalityof uniformly rectifiable sets. To show this, we start with the following auxiliary fact (cf. Lemma 3.1in [KP]). Lemma 8.4.
Let E ⊂ R n + be an n-dimensional ADR set and let Ω = R n + \ E, and let A be an ( n + × ( n + matrix defined on Ω with real bounded measurable coe ffi cients, possibly non-symmetric, satisfying the ellipticity condition (8.1) .If the Carleson measure condition (8.3) is satisfied on Ω = R n + \ E then it is also satisfied onany subset D ⊂ Ω with an ADR boundary, that is, (8.5) ǫ D ( X ) : = sup (cid:8) |∇ A ( Z ) | , Z ∈ B ( X , dist( X , ∂ D ) / (cid:9) satisfies (8.6) sup x ∈ ∂ D , < r < ∞ r n Z Z B ( x , r ) ∩ D | ǫ D ( X ) | dist( X , ∂ D ) dX ≤ C , with the constant C depending on the constant in (8.3) and ADR constants of E only. Proof.
Fix any B ( x , r ) from (8.6). We shall consider two cases. First, if δ ( x ) ≤ r then thereexists B ( z , r ) such that z ∈ E and B ( x , r ) ⊂ B ( z , r ). In addition, for every X ∈ D we have B ( X , dist( X , ∂ D ) / ⊂ B ( X , δ ( X ) / ǫ D ≤ ǫ on D . Hence, in this case,(8.7) 1 r n Z Z B ( x , r ) ∩ D | ǫ D ( X ) | dist( X , ∂ D ) dX ≤ r n Z Z B ( z , r ) | ǫ ( X ) | δ ( X ) dX , so that (8.3) gives the desired bound.In the second case, δ ( x ) > r , we have δ ( x ) / < δ ( Y ) < δ ( x ) / Y ∈ B ( x , r ) ∩ D .Furthermore, by the ADR property of E , for any B ( y , r ) ∩ E , y ∈ E , there exists a corkscrew point Y ∈ E corresponding to B ( y , r ) ∩ E with the property that B ( Y , cr ) ⊂ Ω for some c depending onthe ADR constants of E only. Evidently, |∇ A ( Y ) | ≤ ǫ ( Z ) for all Z ∈ B ( Y , cr ). Now, for every y ∈ E , r > r n Z Z B ( y , r ) | ǫ ( X ) | δ ( X ) dX ≥ r n Z Z B ( Y , cr ) | ǫ ( X ) | δ ( X ) dX ≥ |∇ A ( Y ) | δ ( Y ) . Hence, (8.3) implies that |∇ A ( Y ) | δ ( Y ) ≤ C for all Y ∈ Ω , with C depending on the constant in (8.3)and ADR constants of E only.Returning to D , and specifically, to B ( x , r ), x ∈ ∂ D , δ ( x ) > r , we then have(8.9) 1 r n Z Z B ( x , r ) ∩ D | ǫ D ( X ) | dist( X , ∂ D ) dX ≤ r n Z Z B ( x , r ) δ ( X ) dist( X , ∂ D ) dX . r δ ( X ) ≤ C , as desired. (cid:3) Theorem 8.10.
Let E ⊂ R n + be an n-dimensional UR set and let Ω = R n + \ E. Let A be an ( n + × ( n + matrix defined on Ω with real bounded measurable coe ffi cients, possibly non-symmetric, satisfying the ellipticity condition (8.1) and the Carleson measure condition (8.3) on Ω = R n + \ E. Then any weak solution u to Lu = satisfies square function estimate (8.11) k S u k L p ( E ) ≤ C k b N ∗ u k L p ( E ) , < p < ∞ , as well as its local analogue (8.12) k S Q u k L p ( Q ) ≤ C k b N Q ∗ u k L p ( Q ) , Q ∈ D ( E ) , < p < ∞ . If u is, in addition, bounded, then the Carleson measure estimate (8.13) sup x ∈ E , < r < ∞ r n Z Z B ( x , r ) |∇ u ( Y ) | δ ( Y ) dY ≤ C k u k L ∞ ( Ω ) , holds and u is ε -approximable in the sense of Definition 1.11. All constants depend on the URcharacter of E only.Proof. First of all, all auxiliary estimates (3.2) for F : = |∇ u | / k u k L ∞ ( Ω ) , (7.2), (6.3), and (6.26) for u , (5.10) for G = ∇ u and H = u , hold by the usual interior estimates for solutions of elliptic PDEs(see, e.g., [K]).The square function bounds (8.11)–(8.12) and thus Carleson measure estimates (1.10) (as a par-ticular case p = N < S direction (and only for p > Ω as well. Then, by Theorem 3.3, the CME estimates hold on Ω and any RANSFERENCE OF ESTIMATES 39
UR subdomain of Ω , with the appropriate control of constants. This proves (8.13). Recalling thatall sawtooth subdomains of Ω are UR as well by Proposition A.10, we now use Theorem 5.7, toconclude (8.11)–(8.12).Passing to the question of ε -approximability, we point out again that N < S estimates (6.4) onall Lipschitz subdomains of Ω hold by [KP]. Once again, only the case p > p > p ’s, in particular, (6.4), by Theorem 6.25. By Theorem 6.2 we now get (6.6) and even (6.24) forall bounded NTA subdomains of Ω . This yields (6.6) for all 0 < p < ∞ by Theorem 6.25 and putsus in the context of Theorem 7.1. Then u is ε -approximable on Ω due to Theorem 7.1. (cid:3) Higher order elliptic equations and systems with constant coe ffi cients. In [DKPV] theauthors obtained square function / non-tangential maximal function estimates for higher order ellipticequations and systems on bounded Lipschitz domains. These results have never been extended,even to NTA domains, and here we present a generalization of Carleson measure estimates to thecomplements of UR sets. It is not clear if ε -approximability of (derivatives of) solutions to (higherorder) systems has any consequences: indeed, the traditional connection with elliptic measure is notavailable in this context, and for that reason we do not pursue N < S bounds and ε -approximabilityin this section.Let L kl = P | α | = | β | = m D α a kl αβ D β , where m , k , l ∈ N , α = ( α , ..., α n ) and β = ( β , ..., β n ) are mul-tiindices, and D α , D β are the corresponding vectors of partial derivatives. The coe ffi cients a kl αβ arereal constants. We say that Lu = u = ( u , ..., u K ), K ∈ N , u i ∈ W m , ( Ω ), if K X l = L kl u l = , k = , ..., K , as usual, in the weak sense, similarly to (8.2). Here, W m , ( Ω ) is the space of functions with allderivatives of orders 0 , ..., m in L ( Ω ) and W m , ( Ω ) is the space of functions locally in W m , ( Ω ).We assume, in addition, that L is symmetric: L kl = L lk for 1 ≤ k , l ≤ K , and that the Legendre-Hadamard ellipticity condition holds: there exists λ > K X k , l = X | α | = | β | = m a kl αβ ξ α ξ β ζ k ¯ ζ l ≥ λ | ξ | m | ζ | , for all ζ = ( ζ , ..., ζ n ) ∈ C n , ξ ∈ R n . Theorem 8.14.
Let E ⊂ R n + be an n-dimensional UR set and let Ω = R n + \ E. Let L be asymmetric constant coe ffi cient elliptic system on Ω , satisfying the Legendre-Hadamard ellipticitycondition, as above. Then any weak solution u to Lu = in Ω satisfies square function estimate (8.15) k S ( ∇ m − u ) k L p ( E ) ≤ C k b N ∗ ( |∇ m − u | ) k L p ( E ) , < p < ∞ , as well as its local analogue (8.16) k S Q ( ∇ m − u ) k L p ( Q ) ≤ C k b N Q ∗ ( |∇ m − u | ) k L p ( Q ) , Q ∈ D ( E ) , < p < ∞ . If u is, in addition, such that ∇ m − u is bounded, then the Carleson measure estimate (8.17) sup x ∈ E , < r < ∞ r n Z Z B ( x , r ) |∇ m u ( Y ) | δ ( Y ) dY ≤ C k∇ m − u k L ∞ ( Ω ) , holds. All constants depend on the UR character of E only. Here ∇ k , k ∈ N , stands for the vector ofall partial derivatives of order k.Proof. As mentioned above, the square function / non-tangential maximal function estimates onbounded Lipschitz domains in the present context have been proved in [DKPV]. In particular,the p = F = ∇ m u / k∇ m − u k L ∞ is Theorem 2, p. 1455, of [DKPV].Much as before, using Theorem 4.10, one concludes that the Carleson measure estimates (1.10) hold in all NTA subdomains of Ω as well. Then, by Theorem 3.3, the CME estimates hold on Ω andany UR subdomain of Ω , with the appropriate control of constants. This proves (8.17). Recallingthat all sawtooth subdomains of Ω are UR as well by Proposition A.10, we now use Theorem 5.7,to conclude (8.15)–(8.16). Throughout, ∇ m − u is used in place of u and auxiliary estimates (3.2)for F : = |∇ m u | / k∇ m − u k L ∞ ( Ω ) and (5.10) for G = ∇ m u and H = ∇ m − u , hold by the usual interiorestimates for solutions of elliptic PDEs – see, e.g., [PV]. (cid:3) A ppendix A. S awtooths have UR boundaries To start, recall from [HMM] the fact that the sawtooth regions and Carleson boxes inherit theADR property. In [HMM], we treated simultaneously the case that the set E is ADR, but notnecessarily UR, and also the case that E is UR. The point was that the Whitney regions in thetwo cases (and thus also the corresponding sawtooth regions and Carleson boxes) were somewhatdi ff erent. To make this more precise, we need to recall notational conventions set in Section 2.If the set E under consideration is merely ADR, but not UR, then we set W Q = W Q as definedin (2.8) (see “Case ADR” in Section 2). If in addition, the set E is UR, then we define W Q as in(2.29) (see “Case UR” in Section 2). In the first case, the constants involved in the constructionof W Q depend only on the ADR constant η and K , and in the UR case, on dimension and theADR / UR constants (compare (2.8) and (2.11)). Therefore there are numbers m ∈ Z + , C ∈ R + ,with the same dependence, such that(A.1) 2 − m ℓ ( Q ) ≤ ℓ ( I ) ≤ m ℓ ( Q ) , and dist( I , Q ) ≤ C ℓ ( Q ) , ∀ I ∈ W Q . This dichotomy in the choice of W Q is convenient for the results we have in mind, in the sense thatthe results that we quote from [HMM] may be applied in the purely ADR case, as well as in the URcase, under the conventions described above.For any I ∈ W such that ℓ ( I ) < diam( E ), we write Q ∗ I for the nearest dyadic cube to I with ℓ ( I ) = ℓ ( Q ∗ I ) so that I ∈ W Q ∗ I . Notice that there can be more than one choice of Q ∗ I , but at this pointwe fix one so that in what follows Q ∗ I is unambiguously defined.Let us now recall some results from [HMM] that we shall use in the sequel. Proposition A.2. [HMM, Proposition A.2]
Let E ⊂ R n + be an n-dimensional ADR set. Then alldyadic local sawtooths Ω F , Q and all Carleson boxes T Q have n-dimensional ADR boundaries. Inall cases, the implicit constants are uniform and depend only on dimension, the ADR constant of Eand the parameters m and C . Given a cube Q ∈ D and a family F of disjoint cubes F = { Q j } ⊂ D Q (for the case F = Ø thechanges are straightforward and we leave them to the reader, also the case F = { Q } is disregardedsince in that case Ω F , Q is the null set). We write Ω ⋆ = Ω F , Q and Σ = ∂ Ω ⋆ \ E . Given Q ∈ D weset R Q : = [ Q ′ ∈ D Q W Q ′ , and Σ Q = Σ \ (cid:16) [ I ∈R Q I (cid:17) . Let C be a su ffi ciently large constant, to be chosen below, depending on n , the ADR constant of E , m and C . Let us introduce some new collections: F || : = (cid:8) Q ∈ D \ { Q } : ℓ ( Q ) = ℓ ( Q ) , dist( Q , Q ) ≤ C ℓ ( Q ) (cid:9) , F ⊤ : = (cid:8) Q ′ ∈ D : dist( Q ′ , Q ) ≤ C ℓ ( Q ) , ℓ ( Q ) < ℓ ( Q ′ ) ≤ C ℓ ( Q ) (cid:9) , F ∗|| : = (cid:8) Q ∈ F || : Σ Q , Ø (cid:9) = (cid:8) Q ∈ F || : ∃ I ∈ R Q such that Σ ∩ I , Ø (cid:9) , F ∗ : = (cid:8) Q ∈ F : Σ Q , Ø (cid:9) = (cid:8) Q ∈ F : ∃ I ∈ R Q such that Σ ∩ I , Ø (cid:9) , RANSFERENCE OF ESTIMATES 41
We also set R ⊥ = [ Q ∈F ∗ R Q , R || = [ Q ∈F ∗|| R Q , R ⊤ = [ Q ∈F ⊤ W Q . Lemma A.3. [HMM, Lemma A.3]
Set W Σ = { I ∈ W : I ∩ Σ , Ø } and define W ⊥ Σ = [ Q ∈F ∗ W Σ , Q , W || Σ = [ Q ∈F ∗|| W Σ , Q , W ⊤ Σ = (cid:8) I ∈ W Σ : Q ∗ I ∈ F ⊤ (cid:9) . where for every Q ∈ F ∗ ∪ F ∗|| we set W Σ , Q = (cid:8) I ∈ W Σ : Q ∗ I ∈ D Q } ; and where we recall that Q ∗ I is the nearest dyadic cube to I with ℓ ( I ) = ℓ ( Q ∗ I ) as defined above.Then (A.4) W Σ = W ⊥ Σ ∪ W || Σ ∪ W ⊤ Σ , where (A.5) W ⊥ Σ ⊂ R ⊥ , W || Σ ⊂ R || , W ⊤ Σ ⊂ R ⊤ . As a consequence, (A.6)
Σ = Σ ⊥ ∪ Σ || ∪ Σ ⊤ : = (cid:16) [ I ∈W ⊥ Σ Σ ∩ I (cid:17) [ (cid:16) [ I ∈W || Σ Σ ∩ I (cid:17) [ (cid:16) [ I ∈W ⊤ Σ Σ ∩ I (cid:17) . Lemma A.7. [HMM, Lemma A.7]
Given I ∈ W Σ , we can find Q I ∈ D , with Q I ⊂ Q ∗ I , such that ℓ ( I ) ≈ ℓ ( Q I ) , dist( Q I , I ) ≈ ℓ ( I ) , and in addition, (A.8) X I ∈W Σ , Q Q I . Q , for any Q ∈ F ∗ ∪ F ∗|| , and (A.9) X I ∈W ⊤ Σ Q I . B ∗ Q ∩ E , where the implicit constants depend on n, the ADR constant of E, m and C , and where B ∗ Q = B ( x Q , C ℓ ( Q )) with C large enough depending on the same parameters. With the preceding results in hand, we turn to the main purpose of this appendix: to prove thatuniform rectifiability is also inherited by the sawtooth domains and Carleson boxes.
Proposition A.10.
Let E ⊂ R n + be an n-dimensional UR set. Then all dyadic local sawtooths Ω F , Q and all Carleson boxes T Q have n-dimensional UR boundaries. In all cases, the implicitconstants are uniform and depend only on dimension, the ADR and UR constants of E and theparameters m and C . The proof of this result follows the ideas from [HM, Appendix C] which in turn uses some ideasfrom Guy David, and uses the following singular integral characterization of UR sets, establishedin [DS1]. Suppose that E ⊂ R n + is n -dimensional ADR. The singular integral operators that weshall consider are those of the form T E ,ε f ( x ) = T ε f ( x ) : = Z E K ε ( x − y ) f ( y ) dH n ( y ) , where K ε ( x ) : = K ( x ) Φ ( | x | /ε ), with 0 ≤ Φ ≤ Φ ( ρ ) ≡ ρ ≥ , Φ ( ρ ) ≡ ρ ≤
1, and Φ ∈ C ∞ ( R ), and where the singular kernel K is an odd function, smooth on R n + \{ } , and satisfying | K ( x ) | ≤ C | x | − n (A.11) |∇ m K ( x ) | ≤ C m | x | − n − m , ∀ m = , , , . . . ... . (A.12) Then E is UR if and only if for every such kernel K , we have that(A.13) sup ε> Z E | T ε f | dH n ≤ C K Z E | f | dH n . We refer the reader to [DS1] for the proof. For K as above, set(A.14) T E f ( X ) : = Z E K ( X − y ) f ( y ) dH n ( y ) , X ∈ R n + \ E . We define (possibly disconnected) non-tangential approach regions Υ α ( x ) as follows. Set W α ( x ) : = { I ∈ W : dist( I , x ) < αℓ ( I ) } . Then we define Υ α ( x ) : = [ I ∈W α ( x ) I ∗ (thus, roughly speaking, α is the “aperture” of Υ α ( x )). For F ∈ C ( R n + \ E ) we may then also definethe non-tangential maximal function N ∗ ,α ( F )( x ) : = sup Y ∈ Υ α ( x ) | F ( Y ) | . We shall sometimes write simply N ∗ when there is no chance of confusion in leaving implicit thedependence on the aperture α . The following lemma is a standard consequence of the usual Cotlarinequality for maximal singular integrals, and we omit the proof. Lemma A.15.
Suppose that E ⊂ R n + is n-dimensional UR, and let T E be defined as in (A.14) .Then for each < p < ∞ and α ∈ (0 , ∞ ) , there is a constant C p ,α, K depending only on p , n , α, Kand the UR constants such that (A.16) Z E (cid:0) N ∗ ,α ( T E f ) (cid:1) p dH n ≤ C α, K Z E | f | p dH n . Proof of Proposition A.10.
We now fix Q ∈ D and a family F of disjoint cubes F = { Q j } ⊂ D Q (for the case F = Ø the changes are straightforward and we leave them to the reader, also the case F = { Q } is disregarded since Ω F , Q is the null set). We write Ω ⋆ = Ω F , Q and also E ⋆ = ∂ Ω ⋆ .We fix 0 ≤ Φ ≤ Φ ( ρ ) ≡ ρ ≥ , Φ ( ρ ) ≡ ρ ≤
1, and Φ ∈ C ∞ ( R ). According to previousconsideration we fix ǫ > T E ⋆ ,ǫ is bounded in L ( E ⋆ ) with boundsthat are independent of ǫ . To simplify the notation we write K = K ǫ and set for every X ∈ R n + T E , f ( X ) = Z E K ( x − y ) f ( y ) d σ ( y ) , T E ⋆ , g ( X ) = Z E ⋆ K ( x − y ) g ( y ) d σ ⋆ ( y ) . We first observe that K is not singular and therefore, for any p , 1 < p < ∞ , and for every f ∈ L p ( E ), respectively g ∈ L p ( E ⋆ ), the previous operators are well-defined (by means of an absolutelyconvergent integral) for every X ∈ R n + . Also for such functions it is easy to see that the dominatedconvergence theorem implies that T E , f , T E ⋆ , g ∈ C ( R n + ). Remark
A.17 . We notice that K is an odd smooth function which satisfies (A.11) and (A.12) withuniform constants (i.e. with no dependence on ǫ ) and therefore the fact that E is UR implies that(A.13) and (A.16) hold with constants that do not depend on ǫ .We are going to see that T E , : L p ( E ) −→ L p ( E ⋆ ) for every 1 < p < ∞ . To do that we take f ∈ L p ( E ) and write Z E ⋆ |T E , f ( x ) | p d σ ⋆ ( x ) = Z E ⋆ ∩ E |T E , f ( x ) | p d σ ⋆ ( x ) + Z E ⋆ \ E |T E , f ( x ) | p d σ ⋆ ( x ) = : A + S . RANSFERENCE OF ESTIMATES 43
The estimate for A follows from the fact that E is UR A ≤ Z E |T E , f ( x ) | p d σ ( x ) = Z E | T E ,ǫ f ( x ) | p d σ ( x ) ≤ C K Z E | f ( x ) | p d σ ( x )where we have used (A.13) and the standard Calder´on-Zygmund theory (taking place in the ADRset E ) and C K does not depend on ǫ . For S we use that Σ = E ⋆ \ E = ∂ Ω ⋆ \ E and invoke LemmasA.3 and A.7; let Q I be the cube constructed in the latter, so that S = X I ∈W Σ Z I ∩ Σ |T E , f ( x ) | p d σ ⋆ ( x ) = X I ∈W Σ ? Q I Z I ∩ Σ |T E , f ( x ) | p d σ ⋆ ( x ) d σ ( y ) . Notice that if y ∈ Q I and x ∈ Σ ∩ I then dist( I , y ) . ℓ ( Q I ). Then taking α > I ⊂ W α ( y ). Write e F = F ∗ ∪ F ∗|| , and observe that by construction the cubes in e F arepairwise disjoint. Then by the ADR property of E ⋆ , along with Lemmas A.3 and A.7, S ≤ X I ∈W Σ σ ⋆ ( Σ ∩ I ) ? Q I | N ∗ ,α ( T E , f )( y ) | p d σ ( y ) . X Q ∈ e F X I ∈W Σ , Q Z Q I | N ∗ ,α ( T E , f )( y ) | p d σ ( y ) + X I ∈W ⊤ Σ Z Q I | N ∗ ,α ( T E , f )( y ) | p d σ ( y ) . X Q ∈ e F Z Q | N ∗ ,α ( T E , f )( y ) | p d σ ( y ) + Z B ∗ Q ∩ E | N ∗ ,α ( T E , f )( y ) | p d σ ( y ) . Z E | N ∗ ,α ( T E , f )( y ) | p d σ ( y ) . Z E | f ( y ) | p d σ ( y ) , where in the last estimate we have employed Lemma A.15 and Remark A.17, and the implicitconstants do not depend on ǫ .We have thus established that T E , : L p ( E ) −→ L p ( E ⋆ ) for every 1 < p < ∞ . Since K is odd, sois K , and by duality we therefore obtain that(A.18) T E ⋆ , : L p ( E ⋆ ) −→ L p ( E ) , < p < ∞ . Our goal is to show that T E ⋆ , : L ( E ⋆ ) −→ L ( E ⋆ ) with bounds that do not depend on ǫ . Note that T E ⋆ , f is a continuous function for every f ∈ L ( E ⋆ ) and therefore T E ⋆ , f (cid:12)(cid:12) E ⋆ = T E ⋆ ,ǫ f everywhereon E ⋆ .We take f ∈ L ( E ⋆ ) and write as before(A.19) Z E ⋆ |T E ⋆ , f ( x ) | d σ ⋆ ( x ) = Z E ⋆ ∩ E |T E ⋆ , f ( x ) | d σ ⋆ ( x ) + X I ∈W Σ Z I ∩ Σ |T E ⋆ , f ( x ) | d σ ⋆ ( x ) = : A + X I ∈W Σ S I = A + S . For A we use (A.18) with p = A ≤ Z E ⋆ ∩ E |T E ⋆ , f ( x ) | d σ ⋆ ( x ) ≤ Z E |T E ⋆ , f ( x ) | d σ ( x ) ≤ Z E ⋆ | f ( x ) | d σ ⋆ ( x ) . We next fix I ∈ W Σ and estimate each S I . Let M > ζ I = ℓ ( I ) / M , ξ I = M ℓ ( I ). Write (A.21) K ( x ) = K ( x ) Φ (cid:16) | x | ξ I (cid:17) + K ( x ) (cid:16) Φ (cid:16) | x | ζ I (cid:17) − Φ (cid:16) | x | ξ I (cid:17)(cid:17) + K ( x ) (cid:16) − Φ (cid:16) | x | ζ I (cid:17)(cid:17) = : K ,ξ I ( x ) + K ,ζ I ,ξ I ( x ) + K ζ I ( x ) . Corresponding to any of these kernels we respectively set the operators T E ⋆ , ,ξ I , T E ⋆ , ,ζ I ,ξ I and T ζ I E ⋆ , .We start with T E ⋆ , ,ξ I . Fix x ∈ Σ ∩ I . Write ∆ ⋆, I = B ( x , ξ I ) ∩ E ⋆ and split f = f + f : = f ∆ ⋆, I + f E ⋆ \ ∆ ⋆, I . Then we use Remark A.17, the fact supp Φ ⊂ [1 , ∞ ) and that E ⋆ is ADR toeasily obtain that for every y ∈ Q I , with Q I as in Lemma A.7,(A.22) |T E ⋆ , ,ξ I f ( x ) | + |T E ⋆ , ,ξ I f ( y ) |≤ Z ∆ ⋆, I (cid:16) | K ( x − z ) | Φ (cid:16) | x − z | ξ I (cid:17) + | K ( y − z ) | Φ (cid:16) | y − z | ξ I (cid:17)(cid:17) | f ( z ) | d σ ⋆ ( z ) . ξ nI Z ∆ ⋆, I | f ( y ) | d σ ⋆ ( z ) ≈ ? ∆ ⋆, I | f ( y ) | d σ ⋆ ( z ) ≤ M E ⋆ f ( x ) , where M E ⋆ is the Hardy-Littlewood maximal function on E ⋆ , and the constants are independent of ǫ and I .On the other hand, very much as before we have that K ,ξ I is a Calder´on-Zygmund kernel withconstants that are uniform in ǫ and ξ I . Also, if M is taken large enough we have that 2 | x − y | < M ℓ ( I ) ≤ | x − z | for every z ∈ E ⋆ \ ∆ ⋆, I , x ∈ Σ ∩ I and y ∈ Q I . Therefore using standard Calder´on-Zygmund estimates and the fact that E ⋆ is ADR we obtain that for every and y ∈ Q I (A.23) |T E ⋆ , ,ξ I f ( x ) − T E ⋆ , ,ξ I f ( y ) |≤ Z E ⋆ \ ∆ ⋆, I (cid:12)(cid:12) K ,ξ I ( x − z ) − K ,ξ I ( y − z ) (cid:12)(cid:12) | f ( z ) | d σ ⋆ ( z ) . Z E ⋆ \ ∆ ⋆, I | x − y || x − z | n + | f ( z ) | d σ ⋆ ( z ) ≤ C M M E ⋆ f ( x ) . We next use (A.22) and (A.23) to conclude that (cid:12)(cid:12)(cid:12) T E ⋆ , ,ξ I f ( x ) − ? Q I T E ⋆ , ,ξ I f ( y ) d σ ( y ) (cid:12)(cid:12)(cid:12) . |T E ⋆ , ,ξ I f ( x ) | + ? Q I |T E ⋆ , ,ξ I f ( y ) | d σ ( y ) + ? Q I |T E ⋆ , ,ξ I f ( x ) − T E ⋆ , ,ξ I f ( y ) | d σ ( y ) . M E ⋆ f ( x ) , which in turn yields(A.24) Z Σ ∩ I (cid:12)(cid:12)(cid:12) T E ⋆ , ,ξ I f ( x ) − ? Q I T E ⋆ , ,ξ I f ( y ) d σ ( y ) (cid:12)(cid:12)(cid:12) d σ ⋆ ( x ) . Z Σ ∩ I M E ⋆ f ( x ) d σ ⋆ ( x ) . We next introduce another operator T E ⋆ , ,ξ I f ( y ) = Z z ∈ E ⋆ : | y − z |≥ ξ I K ( y − z ) f ( z ) d σ ⋆ ( z ) , y ∈ E . We fix x ∈ Σ ∩ I and y ∈ Q I . We first observe that, for M large enough, Remark A.17 and the ADRproperty for E ⋆ imply that (cid:12)(cid:12) T E ⋆ , ,ξ I f ( y ) − T E ⋆ , ,ξ I f ( y ) (cid:12)(cid:12) ≤ Z E ⋆ | K ( y − z ) | (cid:12)(cid:12)(cid:12)(cid:12) Φ (cid:16) | y − z | ξ I (cid:17) − [1 , ∞ ) (cid:16) | y − z | ξ I (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) | f ( z ) | d σ ⋆ ( z ) RANSFERENCE OF ESTIMATES 45 . ξ nI Z z ∈ E ⋆ : | y − z |≤ ξ I | f ( z ) | d σ ⋆ ( z ) . ξ nI Z z ∈ E ⋆ : | x − z |≤ ξ I | f ( z ) | d σ ⋆ ( z ) . M E ⋆ f ( x ) . On the other hand, we can introduce another decomposition f = f + f : = f B ( y ,ξ I ) ∩ E ⋆ + f E ⋆ \ B ( y ,ξ I ) , and then for every ¯ y ∈ Q I (A.25) | T E ⋆ , ,ξ I f ( y ) | = |T E ⋆ , f ( y ) | ≤ |T E ⋆ , f ( y ) − T E ⋆ , f (¯ y ) | + |T E ⋆ , f (¯ y ) |≤ |T E ⋆ , f ( y ) − T E ⋆ , f (¯ y ) | + |T E ⋆ , f (¯ y ) | + |T E ⋆ , f (¯ y ) | . We estimate each term in turn. We first observe that, for M large enough, 2 | y − ¯ y | < M ℓ ( I ) ≤ | y − z | for every z ∈ E ⋆ \ B ( y , ξ I ) and ¯ y ∈ Q I . Therefore, using standard Calder´on-Zygmund estimates andthe fact that E ⋆ is ADR, we obtain that for every and ¯ y ∈ Q I (A.26) |T E ⋆ , f ( y ) − T E ⋆ , f (¯ y ) | ≤ Z E ⋆ \ B ( y ,ξ I ) | K ( y − z ) − K (¯ y − z ) | | f ( z ) | d σ ⋆ ( z ) . Z E ⋆ \ B ( y ,ξ I ) | y − ¯ y || y − z | n + | f ( z ) | d σ ⋆ ( z ) . M E ⋆ f ( x ) , where we have used that, for M large enough, x ∈ B ( y , ξ I / < p <
2. We next average(A.25) on ¯ y ∈ Q I and use (A.26) and (A.18) to obtain | T E ⋆ , ,ξ I f ( y ) | (A.27) ≤ ? Q I (cid:0) |T E ⋆ , f ( y ) − T E ⋆ , f (¯ y ) | + |T E ⋆ , f (¯ y ) | + |T E ⋆ , f (¯ y ) | (cid:1) d σ (¯ y ) . M E ⋆ f ( x ) + M E ( T E ⋆ , f )( y ) + σ ( Q I ) − p kT E ⋆ , f k L p ( E ) . M E ⋆ f ( x ) + M E ( T E ⋆ , f )( y ) + σ ( Q I ) − p k f k L p ( E ⋆ ) . M E ⋆ f ( x ) + M E ( T E ⋆ , f )( y ) + (cid:16) ℓ ( I ) n Z B ( y ,ξ I ) ∩ E ⋆ | f ( z ) | p d σ ⋆ ( z ) (cid:17) p . M E ⋆ , p f ( x ) + M E ( T E ⋆ , f )( y ) , where M E is the Hardy-Littlewood maximal function on E and we also write M E ⋆ , p f = M E ⋆ ( | f | p ) p .Note that this estimate holds for every x ∈ Σ ∩ I and for every y ∈ Q I . Hence,(A.28) Z Σ ∩ I (cid:12)(cid:12)(cid:12) ? Q I T E ⋆ , ,ξ I f ( y ) d σ ( y ) (cid:12)(cid:12)(cid:12) d σ ⋆ ( x ) . Z Σ ∩ I M E ⋆ , p f ( x ) d σ ⋆ ( x ) + Z Q I M E ( T E ⋆ , f )( y ) d σ ( y ) , where we have used that σ ⋆ ( Σ ∩ I ) . ℓ ( I ) n . We now gather (A.24) and (A.28) to obtain that forevery I ∈ W Σ Z Σ ∩ I (cid:12)(cid:12) T E ⋆ , ,ξ I f ( x ) (cid:12)(cid:12) d σ ⋆ ( x )(A.29) . Z Σ ∩ I (cid:12)(cid:12)(cid:12) T E ⋆ , ,ξ I f ( x ) − ? Q I T E ⋆ , ,ξ I f ( y ) d σ ( y ) (cid:12)(cid:12)(cid:12) d σ ⋆ ( x ) + Z Σ ∩ I (cid:12)(cid:12)(cid:12) ? Q I T E ⋆ , ,ξ I f ( y ) d σ ( y ) (cid:12)(cid:12)(cid:12) d σ ⋆ ( x ) . Z Σ ∩ I M E ⋆ , p f ( x ) d σ ⋆ ( x ) + Z Q I M E ( T E ⋆ , f )( y ) d σ ( y ) . We next consider T E ⋆ , ,ζ I ,ξ I . Notice that for every x ∈ Σ ∩ I and z ∈ E ⋆ we have | K ,ζ I ,ξ I ( z − x ) | = | K ( z − x ) | (cid:12)(cid:12)(cid:12) Φ (cid:16) | z − x | ζ I (cid:17) − Φ (cid:16) | z − x | ξ I (cid:17)(cid:12)(cid:12)(cid:12) . | z − x | n ζ I ≤| z − x |≤ ξ I . ζ nI | z − x |≤ ξ I , and therefore(A.30) Z Σ ∩ I | T E ⋆ , ,ζ I ,ξ I f ( x ) | d σ ⋆ ( x ) . Z Σ ∩ I (cid:18) ζ nI Z B ( x , ξ I ) ∩ E ⋆ | f ( z ) | d σ ⋆ ( z ) (cid:19) d σ ⋆ ( x ) ≤ C M Z Σ ∩ I M E ⋆ f ( x ) d σ ⋆ ( x ) . Let us finally address T ζ I E ⋆ , . Observe first that K ζ I ( · ) = K ( · ) Φ (cid:16) | · | ǫ (cid:17) (cid:16) − Φ (cid:16) | · | ζ I (cid:17)(cid:17) . We consider di ff erent cases. Case 1: ζ I ≤ ǫ . We have that K ζ I ≡ T ζ I E ⋆ , ≡ Case 2: ǫ < ζ I ≤ ǫ . In this case for every x ∈ Σ ∩ I and z ∈ E ⋆ | K ζ I ( x − z ) | . | x − z | n ǫ ≤| z − x |≤ ζ I . ǫ n | z − x |≤ ǫ , and therefore(A.31) Z Σ ∩ I |T ζ I E ⋆ , f ( x ) | d σ ⋆ ( x ) . Z Σ ∩ I (cid:18) ǫ n Z B ( x , ǫ ) ∩ E ⋆ | f ( z ) | d σ ⋆ ( z ) (cid:19) d σ ⋆ ( x ) . Z Σ ∩ I M E ⋆ f ( x ) d σ ⋆ ( x )where the implicit constants are independent of ǫ and ζ I . Case 3: ζ I > ǫ . In this case T ζ I E ⋆ , f is a double truncated integral whose smooth Calder´on-Zygmund kernel K ζ I is odd, smooth in R n + and satisfies the estimates (A.11), (A.12). with uniformbounds (i.e., independent of ǫ and ζ I ). Fix z I ∈ Σ ∩ I and notice that if x ∈ Σ ∩ I and z ∈ B ( x , ζ I ) ∩ E ⋆ then, taking M large enough, we have | z − z I | ≤ | z − x | + | x − z I | ≤ ζ I + diam( I ) = ℓ ( I )2 M + diam( I ) <
32 diam( I )and therefore the fact that supp K ζ I ⊂ B (0 , ζ I ) immediately gives T ζ I E ⋆ , f ( x ) = T ζ I E ⋆ , ( f e ∆ ⋆, I )( x )where e ∆ ⋆, I : = e B ⋆, I ∩ E ⋆ : = B ( z I , I )) ∩ E ⋆ . Notice that (2.7) yields4 diam( I ) ≤ dist(4 I , E ) ≤ dist( z I , E ) ≤ dist( e B ⋆, I , E ) + I )and therefore dist( e B ⋆, I , E ) ≥ I ). This implies that e B ⋆, I ⊂ R n + \ E . Also if J ∈ W satisfiesthat J ∗ ∩ e B ⋆, I , Ø we can easily check that ℓ ( I ) ≈ ℓ ( J ) and dist( I , J ) . ℓ ( I ). This implies that only RANSFERENCE OF ESTIMATES 47 a bounded number of J ’s have the property that J ∗ intersects e B ⋆, I . We recall that Σ = E ⋆ \ E is aunion of portion of faces of fattened Whitney cubes J ∗ . Thus we have e ∆ ⋆, I ⊂ M [ m = F m , I , where M is a uniform constant and each F m , I is either a portion of a face of some J ∗ , or else F m , I = Ø (since M is not necessarily equal to the number of faces, but is rather an upper bound forthe number of faces.) Note also that I ⊂ e B ⋆, I and therefore we also have that Σ ∩ I ⊂ M [ m = F m , I . Thus Z Σ ∩ I |T ζ I E ⋆ , f ( x ) | d σ ⋆ ( x ) = Z Σ ∩ I |T ζ I E ⋆ , ( f e ∆ ⋆, I )( x ) | d σ ⋆ ( x ) . X ≤ m , m ′ ≤ M Z F m , I |T ζ I E ⋆ , ( f F m ′ , I )( x ) | d σ ⋆ ( x ) . In the case m = m ′ we take the hyperplane H m , I with F m , I ⊂ H m , I and then Z F m , I |T ζ I E ⋆ , ( f F m , I )( x ) | d σ ⋆ ( x ) ≤ Z H m , I |T ζ I H m , I , ( f F m , I )( x ) | dH n ( x ) . Z F m , I | f ( x ) | dH n ( x ) = Z F m , I | f ( x ) | d σ ⋆ ( x ) , where, after a rotation, we have used the L bounds of Calder´on-Zygmund operators with nice ker-nels on R n . For m , m ′ we consider two cases: either dist( F m , I , F m ′ , I ) ≈ ℓ ( I ) or dist( F m , I , F m ′ , I ) ≪ ℓ ( I ). In the first scenario, using that K ζ I satisfies (A.11) uniformly we obtain that Z F m , I |T ζ I E ⋆ , ( f F m ′ , I )( x ) | d σ ⋆ ( x ) . Z F m , I (cid:16) Z F m ′ , I | x − z | n | f ( z ) | d σ ⋆ ( z ) (cid:17) d σ ⋆ ( x ) . Z F m , I (cid:16) ℓ ( I ) n Z B ( x , C ℓ ( I )) ∩ E ⋆ | f ( z ) | d σ ⋆ ( z ) (cid:17) d σ ⋆ ( x ) . Z F m , I M E ⋆ f ( x ) d σ ⋆ ( x )Finally if dist( F m , I , F m ′ , I ) ≪ ℓ ( I ), we have that F m , I and F m ′ , I are contained in respective faceswhich either lie in the same hyperplane, or else meet at an angle of π/
2. In the first case we mayproceed as in the case m = m ′ . In the second case, after a possible rotation of co-ordinates, we mayview F jm ∪ F jm ′ as lying in a Lipschitz graph with Lipschitz constant 1, so that we may estimate T ζ I E ⋆ , using an extension of the Coifman-McIntosh-Meyer theorem: Z F m , I |T ζ I E ⋆ , ( f F m ′ , I )( x ) | d σ ⋆ ( x ) . Z F m ′ , I | f ( x ) | d σ ⋆ ( x ) . Gathering all the possible cases we may conclude that(A.32) Z Σ ∩ I |T ζ I E ⋆ , f ( x ) | d σ ⋆ ( x ) . X ≤ m ≤ M Z F m , I M E ⋆ f ( x ) d σ ⋆ ( x ) . X I ′ ∈W Σ : I ′ ∩ e ∆ ⋆, I , Ø Z I ′ ∩ Σ M E ⋆ f ( x ) d σ ⋆ ( x ) . We now gather (A.29), (A.30) and (A.32) to get the following estimate for S I after using (A.21): S I = Z Σ ∩ I |T E ⋆ , f ( x ) | d σ ⋆ ( x )(A.33) . Z Σ ∩ I |T E ⋆ , ,ξ I f ( x ) | d σ ⋆ ( x ) + Z Σ ∩ I |T E ⋆ , ,ζ I ,ξ I f ( x ) | d σ ⋆ ( x ) + Z Σ ∩ I |T ζ I E ⋆ , | d σ ⋆ ( x ) . Z Σ ∩ I M E ⋆ , p f ( x ) d σ ⋆ ( x ) + Z Q I M E ( T E ⋆ , f )( y ) d σ ( y ) + X I ′ ∈W Σ : I ′ ∩ e ∆ ⋆, I , Ø Z I ′ ∩ Σ M E ⋆ f ( x ) d σ ⋆ ( x ) . Notice that since 1 < p < X I ∈W Σ Z Σ ∩ I M E ⋆ , p f ( x ) d σ ⋆ ( x ) ≤ Z E ⋆ M E ⋆ , p f ( x ) d σ ⋆ ( x ) . Z E ⋆ | f ( x ) | d σ ⋆ ( x ) . On the other hand, we set e F = F ∗ ∪ F ∗|| and observe that, by construction, the cubes in e F arepairwise disjoint. Lemmas A.3 and A.7 then imply that X I ∈W Σ Z Q I M E ( T E ⋆ , f )( y ) d σ ( y )(A.35) = X Q ∈ e F X I ∈W Σ , Q Z Q I M E ( T E ⋆ , f )( y ) d σ ( y ) + X I ∈W ⊤ Σ Z Q I M E ( T E ⋆ , f )( y ) d σ ( y ) . X Q ∈ e F Z Q M E ( T E ⋆ , f )( y ) d σ ( y ) + Z B ∗ Q ∩ E M E ( T E ⋆ , f )( y ) d σ ( y ) . Z E M E ( T E ⋆ , f )( y ) d σ ( y ) . Z E |T E ⋆ , f ( y ) | d σ ( y ) . Z E ⋆ | f ( y ) | d σ ⋆ ( y ) , where in the last estimate we have used (A.18) with p = { I } I ∈W has thebounded overlap property and therefore X I ∈W Σ X I ′ ∈W Σ : I ′ ∩ e ∆ ⋆, I , Ø Σ ∩ I ′ . sup I ′ ∈W Σ (cid:8) I ∈ W Σ : I ′ ∩ ∆ ⋆, I , Ø (cid:9) which we claim that is uniformly bounded. Indeed, fix I ′ ∈ W Σ and let I , I ∈ W Σ with I ′ ∩ e ∆ ⋆, I , Ø and I ′ ∩ e ∆ ⋆, I , Ø. Recall that dist( e B ⋆, I , E ) ≥ I ) with e B ⋆, I = B ( z I , I )) and z I ∈ I ∩ Σ . This implies that ℓ ( I ) ≈ ℓ ( I ′ ) ≈ ℓ ( I ) and also dist( I , I ) . ℓ ( I ). This easily gives ourclaim. Using this we conclude that(A.36) X I ∈W Σ X I ′ ∈W Σ : I ′ ∩ e ∆ ⋆, I , Ø Z I ′ ∩ Σ M E ⋆ f ( x ) d σ ⋆ ( x ) . Z E ⋆ M E ⋆ f ( x ) d σ ⋆ ( x ) . Z E ⋆ | f ( x ) | d σ ⋆ ( x ) . RANSFERENCE OF ESTIMATES 49
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