On the A_\infty condition for elliptic operators in 1-sided NTA domains satisfying the capacity density condition
Mingming Cao, ?scar Domínguez, José María Martell, Pedro Tradacete
aa r X i v : . [ m a t h . C A ] J a n ON THE A ∞ CONDITION FOR ELLIPTIC OPERATORS IN 1-SIDED NTA DOMAINSSATISFYING THE CAPACITY DENSITY CONDITION
MINGMING CAO, ´OSCAR DOM´INGUEZ, JOS ´E MAR´IA MARTELL, AND PEDRO TRADACETEA bstract . Let Ω ⊂ R n + , n ≥
2, be a 1-sided non-tangentially accessible domain, that is, a set which isquantitatively open and path-connected. Assume also that Ω satisfies the capacity density condition. Let L u = − div( A ∇ u ), Lu = − div( A ∇ u ) be two real (not necessarily symmetric) uniformly elliptic operatorsin Ω , and write ω L , ω L for the respective associated elliptic measures. We establish the equivalencebetween the following properties: (i) ω L ∈ A ∞ ( ω L ), (ii) L is L p ( ω L )-solvable for some p ∈ (1 , ∞ ), (iii)bounded null solutions of L satisfy Carleson measure estimates with respect to ω L , (iv) S < N (i.e., theconical square function is controlled by the non-tangential maximal function) in L q ( ω L ) for some (orfor all) q ∈ (0 , ∞ ) for any null solution of L , and (v) L is BMO( ω L )-solvable. Moreover, in each of theproperties (ii)-(v) it is enough to consider the class of solutions given by characteristic functions of Borelsets (i.e, u ( X ) = ω XL ( S ) for an arbitrary Borel set S ⊂ ∂ Ω ).Also, we obtain a qualitative analog of the previous equivalences. Namely, we characterize the ab-solute continuity of ω L with respect to ω L in terms of some qualitative local L ( ω L ) estimates for thetruncated conical square function for any bounded null solution of L . This is also equivalent to the finite-ness ω L -almost everywhere of the truncated conical square function for any bounded null solution of L .As applications, we show that ω L is absolutely continuous with respect to ω L if the disagreement of thecoe ffi cients satisfies some qualitative quadratic estimate in truncated cones for ω L -almost everywherevertex. Finally, when L is either the transpose of L or its symmetric part, we obtain the correspondingabsolute continuity upon assuming that the antisymmetric part of the coe ffi cients has some controlledoscillation in truncated cones for ω L -almost every vertex. C ontents
1. Introduction 22. Preliminaries 73. Uniformly elliptic operators, elliptic measure and the Green function 144. Proof of Theorem 1.2 19
Date : January 14, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Uniformly elliptic operators, elliptic measure, Green function, 1-sided non-tangentially acces-sible domains, 1-sided chord-arc domains, capacity density condition, A ∞ Muckenhoupt weights, Reverse H¨older, Car-leson measures, square function estimates, non-tangential maximal function estimates, dyadic analysis, sawtooth domains,perturbation.The first author is supported by Spanish Ministry of Science and Innovation, through the Juan de la Cierva-Formaci´on2018, FJC2018-038526-I. The first, third, and last authors acknowledge financial support from the Spanish Ministry of Sci-ence and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S) andfrom the Spanish National Research Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa”(20205CEX001). The second author was partially supported by Spanish Ministry of Science and Innovation through grantMTM2017-84058-P (AEI / FEDER,UE). The third author also acknowledges that the research leading to these results has re-ceived funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7 / / ERC agreement no. 615112 HAPDEGMT. The third author was partially supported by the Spanish Ministry of Scienceand Innovation, MTM PID2019-107914GB-I00. The last author was partially supported by Spanish Ministry of Science andInnovation through grant MTM2016-75196-P (AEI / FEDER,UE).
5. Proof of Theorem 1.9 396. Proof of Theorems 1.10 and 1.12 43References 521. I ntroduction
The solvability of the Dirichlet problem (1.1) on rough domains has been of great interest in the lastfifty years. Given a domain Ω ⊂ R n + and a uniformly elliptic operator L on Ω , it consists on findinga solution u (satisfying natural conditions in accordance to what is known for the boundary data f ) tothe boundary value problem(1.1) ( Lu = Ω , u = f on ∂ Ω . To address this question, one typically investigates the properties of the corresponding elliptic measure,since it is the fundamental tool that enables us to construct solutions of (1.1). The techniques fromharmonic analysis and geometric measure theory have allowed us to study the regularity of ellipticmeasures and hence understand this subject well. Conversely, the good properties of elliptic measuresallow us to e ff ectively use the machinery from these fields to obtain information about the topologyand the regularity of the domains. These ideas have led to a quite active research at the intersection ofharmonic analysis, partial di ff erential equations, and geometric measure theory.The connection between the geometry of a domain and the absolute continuity properties of itsharmonic measure goes back to the classical result of F. and M. Riesz [47], which showed that for asimply connected domain in the plane, the rectifiability of its boundary implies that harmonic mea-sure is mutually absolutely continuous with respect to the surface measure. After that, considerableattention has focused on establishing higher dimensional analogues and the converse of the F. and M.Riesz theorem. For a planar domain, Bishop and Jones [5] proved that if only a portion of the boundaryis rectifiable, harmonic measure is absolutely continuous with respect to arclength on that portion. Acounter-example was also constructed to show that the result of [47] may fail in the absence of somestrong connectivity property (like simple connectivity). In dimensions greater than 2, Dahlberg [12] es-tablished a quantitative version of the absolute continuity of harmonic measures with respect to surfacemeasure on the boundary of a Lipschitz domain. This result was extended to BMO domains by Jerisonand Kenig [38], and to chord-arc domains by David and Jerison [16] (see also [4, 28, 33] for the caseof 1-sided chord-arc domains). In this direction, this was culminated in the recent results of [3] undersome optimal background hypothesis (an open set in R n + satisfying an interior corkscrew conditionwith an n -dimensional Ahlfors-David regular boundary). Indeed, [3] gives a complete picture of therelationship between the quantitative absolute continuity of harmonic measure with respect to surfacemeasure (or, equivalently, the solvability of (1.1) for singular data, see [26]) and the rectifiability ofthe boundary plus some weak local John condition (that is, local accessibility by non-tangential pathsto some pieces of the boundary). Another significant extension of the F. and M. Riesz theorem wasobtained in [2], where it was proved that, in any dimension and in the absence of any connectivity con-dition, every piece of the boundary with finite surface measure is rectifiable, provided surface measureis absolutely continuous with respect to harmonic measure on that piece. It is worth pointing out thatall the aforementioned results are restricted to the n -dimensional boundaries of domains in R n + . Someanalogues have been obtained in [14, 15, 17, 44] on lower-dimensional sets.On the other hand, the solvability of the Dirichlet problem (1.1) is closely linked with the abso-lute continuity properties of elliptic measures. The importance of the quantitative absolute continuity LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 3 of the elliptic measure with respect to the surface measure comes from the fact that ω L ∈ RH q ( σ )(short for the Reverse H ¨older class with respect to σ , being σ the surface measure) is equivalent to the L q ′ ( σ )-solvability of the Dirichlet problem (see, e.g. [26]). In 1984, Dahlberg formulated a conjectureconcerning the optimal conditions on a matrix of coe ffi cients guaranteeing that the Dirichlet problem(1.1) with L p data for some p ∈ (1 , ∞ ) is solvable. Kenig and Pipher [41] made the first attempt onbounded Lipschitz domains and gave an a ffi rmative answer to Dahlberg’s conjecture. More precisely,they showed that elliptic measure is quantitatively absolutely continuous with respect to surface mea-sure whenever the gradient of the coe ffi cients satisfies a Carleson measure condition. This was donein Lipschitz domains but can be naturally extended to chord-arc domains. In some sense, some recentresults have shown that this class of domains is optimal. First, [28, 33, 4] show that in the case ofthe Laplacian and for 1-sided chord-arc domains, the fact that the harmonic measure is quantitativelyabsolutely continuous with respect to surface measure (equivalently, the L p ( σ )-Dirichlet problem issolvable for some finite p ) implies that the domains must have exterior corkscrews, hence they arechord-arc domains. Indeed, in a first attempt to generalize this to the class of Kenig-Pipher operators,Hofmann, the third author of the present paper, and Toro [31] were able to consider variable coe ffi -cients whose gradient satisfies some L -Carleson condition (in turn, stronger than the one in [41]). Thegeneral case, on which the operators are in the optimal Kenig-Pipher–class (that is, the gradient of thecoe ffi cients satisfies an L -Carleson condition) has been recently solved by Hofmann et al. [30].In another direction, one can consider perturbations of elliptic operators in rough domains. Thatis, one seeks for conditions on the disagreement of two coe ffi cient matrices so that the solvability ofthe Dirichlet problem or the quantitative absolute continuity with respect to the surface measure of theelliptic measure for one elliptic operator could be transferred to the other operator. This problem wasinitiated by Fabes, Jerison and Kenig [18] in the case of continuous and symmetric coe ffi cients, andextended by Dahlberg [13] to a more general setting under a vanishing Carleson measure condition.Soon after, working again in the domain Ω = B (0 ,
1) and with symmetric operators, Fe ff erman [19]improved Dahlberg’s result by formulating the boundedness of a conical square function, which allowsone to preserve the A ∞ property of elliptic measures, but without preserving the reverse H ¨older expo-nent (see [20, Theorem 2.24]). A major step forward was made by Fe ff erman, Kenig and Pipher [20]by giving an optimal Carleson measure perturbation on Lipschitz domains. Additionally, they estab-lished another kind of perturbation to study the quantitative absolute continuity between two ellipticmeasures. Beyond the Lipschitz setting, these results were extended to chord-arc domains [45, 46], 1-sided chord-arc domains [7, 8], and 1-sided non-tangentially accessible domains satisfying the capacitydensity condition [1]. It is worth mentioning that the so-called extrapolation of Carleson measure wasutilized in [1, 7]. Nevertheless, a simpler and novel argument was presented in [8] to get the largeconstant perturbation. More specifically, A ∞ property of elliptic measures can be characterized by thefact that every bounded weak solution of L satisfies Carleson measure estimates. Also, it is worthmentioning that [1] considers for the first time perturbation results on sets with bad surface measures.The goal of this paper is to continue with the line of research initiated in [1]. We work with Ω ⊂ R n + , n ≥
2, a 1-sided non-tangentially accessible domain satisfying the capacity density condition. Weconsider two real (not necessarily symmetric) uniformly elliptic operators L u = − div( A ∇ u ) and Lu = − div( A ∇ u ) in Ω , and denote by ω L , ω L the respective associated elliptic measures. The paper[1] considered the perturbation theory in this context providing natural conditions on the disagreementof the coe ffi cients so that ω L is quantitatively absolutely continuous with respect to ω L (see also [20]).In our first main result we single out the latter property and characterize it in terms of the solvabilityof the Dirichlet problem or some other properties that certain solutions satisfy. In a nutshell, we showthat such condition is equivalent to the fact that null solutions of L have a good behavior with respectto ω L . The precise statement is as follows: MINGMING CAO, ´OSCAR DOM´INGUEZ, JOS ´E MAR´IA MARTELL, AND PEDRO TRADACETE
Theorem 1.2.
Let Ω ⊂ R n + , n ≥ , be a 1-sided NTA domain (cf. Definition 2.5) satisfying the capacitydensity condition (cf. Definition 2.10), and let Lu = − div( A ∇ u ) and L u = − div( A ∇ u ) be real (non-necessarily symmetric) elliptic operators. Bearing in mind the notions introduced in Definition 3.6, thefollowing statements are equivalent: (a) ω L ∈ A ∞ ( ∂ Ω , ω L ) (cf. Definition 3.3). (b) L is L p ( ω L ) -solvable for some p ∈ (1 , ∞ ) . (b) ′ L is L p ( ω L ) -solvable for characteristic functions for some p ∈ (1 , ∞ ) . (c) L satisfies
CME( ω L ) . (c) ′ L satisfies
CME( ω L ) for characteristic functions. (d) L satisfies S < N in L q ( ω L ) for some (or all) q ∈ (0 , ∞ ) . (d) ′ L satisfies S < N in L q ( ω L ) for characteristic functions for some (or all) q ∈ (0 , ∞ ) . (e) L is
BMO( ω L ) -solvable. (e) ′ L is
BMO( ω L ) -solvable for characteristic functions. (f) L is
BMO( ω L ) -solvable in the sense of [26] . (f) ′ L is
BMO( ω L ) -solvable in the sense of [26] for characteristic functions.Furthermore, for any p ∈ (1 , ∞ ) there hold (a) p ′ ω L ∈ RH p ′ ( ∂ Ω , ω L ) ⇐⇒ (b) p L is L p ( ω L ) -solvable , (b) p L is L p ( ω L ) -solvable = ⇒ (b) ′ p L is L p ( ω L ) -solvable for characteristic functions , and (b) p L is L p ( ω L ) -solvable = ⇒ (b) q L is L q ( ω L ) -solvable for all q ≥ p.Remark . Note that in Definition 3.6 the L p ( ω L )-solvability depends on some fixed α and N . How-ever, in the previous result what we prove is that if (a) holds then (b) is valid for all α and N . For theconverse we see that if (b) holds for some α and N then we get (a). This eventually says that if (b) holdsfor some α and N , then it also holds for every α and N . The same occurs with (d) where now there isonly α .As an immediate consequence of Theorem 1.2, if we take L = L , in which case we clearly have ω L ∈ A ∞ ( ∂ Ω , ω L ) (indeed, ω L ∈ RH p ( ∂ Ω , ω L ) for any 1 < p < ∞ ), then we obtain the followingestimates for the null solutions of L (note that (ii) and (iii) coincide with [1, Theorems 5.1 and 5.3]respectively): Corollary 1.4.
Let Ω ⊂ R n + , n ≥ , be a 1-sided NTA domain (cf. Definition 2.5) satisfying thecapacity density condition (cf. Definition 2.10), and let Lu = − div( A ∇ u ) be a real (non-necessarilysymmetric) elliptic operator. Bearing in mind the notions introduced in Definition 3.6, the followingstatements hold: (i) L is L p ( ω L ) -solvable, and also L p ( ω L ) -solvable for characteristic functions, for all p ∈ (1 , ∞ ) . (ii) L satisfies
CME( ω L ) . (iii) L satisfies S < N in L q ( ω L ) for all q ∈ (0 , ∞ ) . (iv) L is
BMO( ω L ) -solvable, and also BMO( ω L ) -solvable for characteristic functions. LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 5 (v)
L is
BMO( ω L ) -solvable, and also BMO( ω L ) -solvable for characteristic functions, in the senseof [26] .Remark . We would like to emphasize that in (i) the L p ( ω L )-solvability holds for all α and N , thesame occurs with (iii) which holds for all α , see Definition 3.6.Our second application is a direct consequence of [1, Theorems 1.5, 1.10] and Theorem 1.2: Corollary 1.6.
Let Ω ⊂ R n + , n ≥ , be a 1-sided NTA domain (cf. Definition 2.5) satisfying thecapacity density condition (cf. Definition 2.10), and let Lu = − div( A ∇ u ) and L u = − div( A ∇ u ) bereal (non-necessarily symmetric) elliptic operators. Define (1.7) ̺ ( A , A )( X ) : = sup Y ∈ B ( X ,δ ( X ) / | A ( Y ) − A ( Y ) | , X ∈ Ω , and ||| ̺ ( A , A ) ||| : = sup B sup B ′ ω X ∆ L ( ∆ ′ ) ¨ B ′ ∩ Ω ̺ ( A , A )( X ) G L ( X ∆ , X ) δ ( X ) dX , where ∆ = B ∩ Ω , ∆ ′ = B ′ ∩ Ω , and the sup is taken respectively over all balls B = B ( x , r ) with x ∈ ∂ Ω and < r < diam( ∂ Ω ) , and B ′ = B ( x ′ , r ) with x ′ ∈ ∆ and < r ′ < c r / , and c is the Corkscrewconstant. We also define A α ( ̺ ( A , A ))( x ) : = (cid:18) ¨ Γ α ( x ) ̺ ( A , A )( X ) δ ( X ) n + dX (cid:19) , x ∈ ∂ Ω , where Γ α ( x ) : = { X ∈ Ω : | X − x | ≤ (1 + α ) δ ( X ) } .If (1.8) ||| ̺ ( A , A ) ||| < ∞ or A α ( ̺ ( A , A )) ∈ L ∞ ( ∂ Ω , ω L ) , then all the properties (a) – (f) ′ in Theorem 1.2 are satisfied.Moreover, if given < p < ∞ , there exists ε p > (depending only on dimension, the 1-sided NTAand CDC constants, the ellipticity constants of L and L, and p) such that if ||| ̺ ( A , A ) ||| ≤ ε p or k A α ( ̺ ( A , A )) k L ∞ ( ω L ) ≤ ε p , then ω L ∈ RH p ′ ( ∂ Ω , ω L ) and hence L is L q ( ω L ) -solvable for q ≥ p. Our next goal is to state a qualitative version of Theorem 1.2 in line with [6]. The A ∞ conditionwill turn into absolute continuity. The qualitative analog of S < N is going to be that the conicalsquare function satisfies L q estimates in some pieces of the boundary. On the other hand, as seen fromthe proof of Theorem 1.2 (see Lemma 4.7 and (4.34)), the CME condition, more precisely, the left-hand side term of (3.11) is connected with the local L -norm of the conical square function. Thus, the L -estimates for the conical square function are the qualitative version of CME. In turn, all these areequivalent to the simple fact that the truncated conical square function is finite almost everywhere withrespect to the elliptic measure ω L . Theorem 1.9.
Let Ω ⊂ R n + , n ≥ , be a 1-sided NTA domain (cf. Definition 2.5) satisfying thecapacity density condition (cf. Definition 2.10). There exists α > (depending only on the 1-sidedNTA and CDC constants) such that for each fixed α ≥ α and for every real (not necessarily symmetric)elliptic operators L u = − div( A ∇ u ) and Lu = − div( A ∇ u ) the following statements are equivalent: (a) ω L ≪ ω L on ∂ Ω . MINGMING CAO, ´OSCAR DOM´INGUEZ, JOS ´E MAR´IA MARTELL, AND PEDRO TRADACETE (b) ∂ Ω = S N ≥ F N , where ω L ( F ) = , for each N ≥ , F N = ∂ Ω ∩ ∂ Ω N for some bounded 1-sidedNTA domain Ω N ⊂ Ω satisfying the capacity density condition, and S α r u ∈ L q ( F N , ω L ) for everyweak solution u ∈ W , ( Ω ) ∩ L ∞ ( Ω ) of Lu = in Ω , for all (or for some) r > , and for all (orfor some) q ∈ (0 , ∞ ) . (b) ′ ∂ Ω = S N ≥ F N , where ω L ( F ) = , for each N ≥ , F N = ∂ Ω ∩ ∂ Ω N for some bounded 1-sidedNTA domain Ω N ⊂ Ω satisfying the capacity density condition, and S α r u ∈ L q ( F N , ω L ) whereu ( X ) = ω XL ( S ) , X ∈ Ω , for any arbitrary Borel set S ⊂ ∂ Ω , for all (or for some) r > , and forall (or for some) q ∈ (0 , ∞ ) . (c) S α r u ( x ) < ∞ for ω L -a.e. x ∈ ∂ Ω , for every weak solution u ∈ W , ( Ω ) ∩ L ∞ ( Ω ) of Lu = in Ω and for all (or for some) r > . (c) ′ S α r u ( x ) < ∞ for ω L -a.e. x ∈ ∂ Ω where u ( X ) = ω XL ( S ) , X ∈ Ω , for any arbitrary Borel setS ⊂ ∂ Ω , and for all (or for some) r > . (d) For every weak solution u ∈ W , ( Ω ) ∩ L ∞ ( Ω ) of Lu = in Ω and for ω L -a.e. x ∈ ∂ Ω thereexists r x > such that S α r x u ( x ) < ∞ . (d) ′ For every Borel set S ⊂ ∂ Ω and for ω L -a.e. x ∈ ∂ Ω there exists r x > such that S α r x u ( x ) < ∞ ,where u ( X ) = ω XL ( S ) , X ∈ Ω . Our first application of the previous result is a qualitative version of [1, Theorem 1.10]:
Theorem 1.10.
Let Ω ⊂ R n + , n ≥ , be a 1-sided NTA domain (cf. Definition 2.5) satisfying thecapacity density condition (cf. Definition 2.10). There exists α > (depending only on the 1-sidedNTA and CDC constants) such that, if the real (not necessarily symmetric) elliptic operators L u = − div( A ∇ u ) and Lu = − div( A ∇ u ) satisfy for some α ≥ α and for some r > ¨ Γ α r ( x ) ̺ ( A , A )( X ) δ ( X ) n + dX < ∞ , for ω L -a.e. x ∈ ∂ Ω , (1.11) where ̺ ( A , A ) is as in (1.7) , then ω L ≪ ω L . To present another application of Theorem 1.9, we introduce some notation. For any real (not nec-essarily symmetric) elliptic operator Lu = − div( A ∇ u ), we let L ⊤ denote the transpose of L , and let L sym = L + L ⊤ be the symmetric part of L . These are respectively the divergence form elliptic operatorswith associated matrices A ⊤ (the transpose of A ) and A sym = A + A ⊤ . Theorem 1.12.
Let Ω ⊂ R n + , n ≥ , be a 1-sided NTA domain (cf. Definition 2.5) satisfying thecapacity density condition (cf. Definition 2.10). There exists α > (depending only on the 1-sidedNTA and CDC constants) such that, if Lu = − div( A ∇ u ) is a real (not necessarily symmetric) ellipticoperator, and we assume that ( A − A ⊤ ) ∈ Lip loc ( Ω ) and that for some α ≥ α and for some r > onehas (1.13) F α r ( x ; A ) : = ¨ Γ α r ( x ) (cid:12)(cid:12) div C A − A ⊤ )( X ) (cid:12)(cid:12) δ ( X ) − n dX < ∞ , for ω L -a.e. x ∈ ∂ Ω , where div C ( A − A ⊤ )( X ) : = (cid:18) n + X i = ∂ i ( a i , j − a j , i )( X ) (cid:19) ≤ j ≤ n + , X ∈ Ω , then ω L ≪ ω L ⊤ and ω L ≪ ω L sym . LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 7
Moreover, if (1.14) F α r ( x ; A ) < ∞ , for ω L -a.e. and ω L ⊤ -a.e. x ∈ ∂ Ω , then ω L ≪ ω L ⊤ ≪ ω L ≪ ω L sym . The structure of this paper is as follows. Section 2 contains some preliminaries, definitions, and toolsthat will be used throughout. Also, for convenience of the reader, we gather in Section 3 several factsconcerning elliptic measures and Green functions which can be found in the upcoming [32]. The proofof Theorem 1.2 is in Section 4. Section 5 is devoted to proving Theorem 1.9. In Section 6, we willpresent the proofs of Theorems 1.10 and 1.12 which follow easily from a more general perturbationresult which is interesting in its own right.We note that some interesting related work has been carried out while this manuscript was in prepa-ration due to Feneuil and Poggi [21]. This work can be particularized to our setting and contains someresults which overlap with ours. First, [21, Theorem 1.22] corresponds to (c) ′ = ⇒ (a) in Theorem 1.2.It should be mentioned that both arguments use the ideas originated in [39] (see also [40]) which presentsome problems when extended to the 1-sided NTA setting. Namely, elliptic measure may not always bea probability and also it could happen that for a uniformly bounded number of generations the dyadicchildren of a given cube may agree with that cube. These two issues have been carefully addressedin [8, Lemma 3.10] (see Lemma 4.4 with β >
0) and although such a result is stated in the setting of1-sided CAD it is straightforward to see that it readily adapts to our case. Our proof of (c) ′ = ⇒ (a)in Theorem 1.2 follows easily from that lemma. Second, [21, Theorem 1.27] (see also [21, Corol-lary 1.33]) shows (d) in Theorem 1.2 with q = L . In our setting, weare showing that (d) follows if (a) holds for any given operator L (whether or not it is a generalizedperturbation of L .) 2. P reliminaries Notation and conventions. • We use the letters c , C to denote harmless positive constants, not necessarily the same at each oc-currence, which depend only on dimension and the constants appearing in the hypotheses of thetheorems (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 areas above, unless explicitly noted to the contrary. Unless otherwise specified upper case constants aregreater than 1 and lower case constants are smaller than 1. In some occasions it is important to keeptrack of the dependence on a given parameter γ , in that case we write a . γ b or a ≈ γ b to emphasizethat the implicit constants in the inequalities depend on γ . • Our ambient space is R n + , n ≥ • Given E ⊂ R n + we write diam( E ) = sup x , y ∈ E | x − y | to denote its diameter. • Given an open set Ω ⊂ R n + , we shall use lower case letters x , y , z , etc., to denote points on ∂ Ω , andcapital letters X , Y , Z , etc., to denote generic points in R n + (especially those in R n + \ ∂ Ω ). • The open ( n + r will be denoted B ( x , r ) when the center x lieson ∂ Ω , or B ( X , r ) when the center X ∈ R n + \ ∂ Ω . A surface ball is denoted ∆ ( x , r ) : = B ( x , r ) ∩ ∂ Ω ,and unless otherwise specified it is implicitly assumed that x ∈ ∂ Ω . • If ∂ Ω is bounded, it is always understood (unless otherwise specified) that all surface balls have radiicontrolled by the diameter of ∂ Ω , that is, if ∆ = ∆ ( x , r ) then r . diam( ∂ Ω ). Note that in this way ∆ = ∂ Ω if diam( ∂ Ω ) < r . diam( ∂ Ω ). MINGMING CAO, ´OSCAR DOM´INGUEZ, JOS ´E MAR´IA MARTELL, AND PEDRO TRADACETE • For X ∈ R n + , we set δ ( X ) : = dist( X , ∂ Ω ). • We let H n denote the n -dimensional Hausdor ff measure. • For a Borel set A ⊂ R n + , we let A denote the usual indicator function of A , i.e. A ( X ) = X ∈ A ,and A ( X ) = X < A . • We shall use the letter I (and sometimes J ) to denote a closed ( n + ℓ ( I ) denote the side length of I . We use Q todenote dyadic “cubes” on E or ∂ Ω . The latter exist as a consequence of Lemma 2.13 below.2.2. Some definitions.Definition 2.1 ( Corkscrew condition).
Following [38], we say that a domain Ω ⊂ R n + satisfies the Corkscrew condition if for some uniform constant 0 < c < x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ), if we write ∆ : = ∆ ( x , r ), there is a ball B ( X ∆ , c r ) ⊂ B ( x , r ) ∩ Ω . The point X ∆ ⊂ Ω is calleda Corkscrew point relative to ∆ (or, relative to B ). We note that we may allow r < C diam( ∂ Ω ) for anyfixed C , simply by adjusting the constant c . Definition 2.2 ( Harnack Chain condition).
Again following [38], we say that Ω satisfies the HarnackChain condition if there are uniform constants C , C > X , X ′ ∈ Ω there is a chain of balls B , B , . . . , B N ⊂ Ω with N ≤ C (2 + log + Π ) where(2.3) Π : = | X − X ′ | min { δ ( X ) , δ ( X ′ ) } such that X ∈ B , X ′ ∈ B N , B k ∩ B k + , Ø and for every 1 ≤ k ≤ N (2.4) C − diam( B k ) ≤ dist( B k , ∂ Ω ) ≤ C diam( B k ) . The chain of balls is called a
Harnack Chain .We note that in the context of the previous definition if Π ≤ B = B ( X , δ ( X ) /
5) and B = B ( X ′ , δ ( X ′ ) /
5) where (2.4) holds with C =
3. Hence the Harnackchain condition is non-trivial only when Π > Definition 2.5 ( We say that a domain Ω is a (1-sided NTA) if it satisfies both the Corkscrew and Harnack Chain conditions. Fur-thermore, we say that Ω is a non-tangentially accessible domain (NTA domain) if it is a 1-sided NTAdomain and if, in addition, Ω ext : = R n + \ Ω also satisfies the Corkscrew condition. Remark . In the literature, 1-sided NTA domains are also called uniform domains . We remark thatthe 1-sided NTA condition is a quantitative form of openness and path connectedness.
Definition 2.7 ( Ahlfors regular).
We say that a closed set E ⊂ R n + is n-dimensional Ahlfors regular (AR for short) if there is some uniform constant C > C − r n ≤ H n ( E ∩ B ( x , r )) ≤ C r n , x ∈ E , < r < diam( E ) . Definition 2.9 ( A (1-sided CAD) is a 1-sidedNTA domain with AR boundary. A chord-arc domain (CAD) is an NTA domain with AR boundary.We next recall the definition of the capacity of a set. Given an open set D ⊂ R n + (where we recallthat we always assume that n ≥
2) and a compact set K ⊂ D we define the capacity of K relative to D as Cap ( K , D ) = inf (cid:26) ¨ D |∇ v ( X ) | dX : v ∈ C ∞ c ( D ) , v ( x ) ≥ K (cid:27) . LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 9
Definition 2.10 ( Capacity density condition ) . An open set Ω is said to satisfy the capacity densitycondition (CDC for short) if there exists a uniform constant c > ( B ( x , r ) \ Ω , B ( x , r ))Cap ( B ( x , r ) , B ( x , r )) ≥ c for all x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ).The CDC is also known as the uniform 2-fatness as studied by Lewis in [42]. Using [25, Example2.12] one has that(2.12) Cap ( B ( x , r ) , B ( x , r )) ≈ r n − , for all x ∈ R n + and r > , and hence the CDC is a quantitative version of the Wiener regularity, in particular every x ∈ ∂ Ω isWiener regular. It is easy to see that the exterior Corkscrew condition implies CDC. Also, it wasproved in [48, Section 3] and [27, Lemma 3.27] that a set with Ahlfors regular boundary satisfies thecapacity density condition with constant c depending only on n and the Ahlfors regular constant.2.3. Dyadic grids and sawtooths.
In this section we introduce a dyadic grid from [1, Lemma 2.33]along the lines of that obtained in [9] but using the dyadic structure from [36, 37, 34]:
Lemma 2.13 ( Existence and properties of the “dyadic grid” , [1, Lemma 2.33]) . Let E ⊂ R n + be aclosed set. Then there exists a constant C ≥ depending just on n such that for each k ∈ Z there is acollection of Borel sets (called “cubes”) D k : = (cid:8) Q kj ⊂ E : j ∈ J k (cid:9) , where J k denotes some (possibly finite) index set depending on k satisfying: ( a ) E = S j ∈ J k Q kj for each k ∈ Z . ( b ) If m ≤ k then either Q kj ⊂ Q mi or Q mi ∩ Q kj = Ø . ( c ) For each k ∈ Z , j ∈ J k , and m < k, there is a unique i ∈ J m such that Q kj ⊂ Q mi . ( d ) For each k ∈ Z , j ∈ J k there is x kj ∈ E such thatB ( x kj , C − − k ) ∩ E ⊂ Q kj ⊂ B ( x kj , C − k ) ∩ E . In what follows given B = B ( x , r ) with x ∈ E we will denote ∆ = ∆ ( x , r ) = B ∩ E . A few remarksare in order concerning this lemma. Note that within the same generation (that is, within each D k ) thecubes are pairwise disjoint (hence, there are no repetitions). On the other hand, we allow repetitions inthe di ff erent generations, that is, we could have that Q ∈ D k and Q ′ ∈ D k − agree. Then, although Q and Q ′ are the same set, as cubes we understand that they are di ff erent. In short, it is then understoodthat D is an indexed collection of sets where repetitions of sets are allowed in the di ff erent generationsbut not within the same generation. With this in mind, we can give a proper definition of the “length” ofa cube (this concept has no geometric meaning in this context). For every Q ∈ D k , we set ℓ ( Q ) = − k ,which is called the “length” of Q . Note that the “length” is well defined when considered on D , but it isnot well-defined on the family of sets induced by D . It is important to observe that the “length” refersto the way the cubes are organized in the dyadic grid. It is clear from ( d ) that diam( Q ) . ℓ ( Q ). When E = ∂ Ω , with Ω being a 1-sided NTA domain satisfying the CDC condition, the converse holds, hencediam( Q ) ≈ ℓ ( Q ), see [1, Remark 2.73]. This means that the “length” is related to the diameter of thecube. Let us observe that if E is bounded and k ∈ Z is such that diam( E ) < C − − k , then there cannot betwo distinct cubes in D k . Thus, D k = { Q k } with Q k = E . Therefore, we are going to ignore those k ∈ Z such that 2 − k & diam( E ). Hence, we shall denote by D ( E ) the collection of all relevant Q kj , i.e., D ( E ) : = [ k D k , where, if diam( E ) is finite, the union runs over those k ∈ Z such that 2 − k . diam( E ). We write Ξ = C , with C being the constant in Lemma 2.13, which is purely dimensional. For Q ∈ D ( E ) wewill set k ( Q ) = k if Q ∈ D k . Property ( d ) implies that for each cube Q ∈ D ( E ), there exist x Q ∈ E and r Q , with Ξ − ℓ ( Q ) ≤ r Q ≤ ℓ ( Q ) (indeed r Q = (2 C ) − ℓ ( Q )), such that(2.14) ∆ ( x Q , r Q ) ⊂ Q ⊂ ∆ ( x Q , Ξ r Q ) . We shall denote these balls and surface balls by(2.15) B Q : = B ( x Q , r Q ) , ∆ Q : = ∆ ( x Q , r Q ) , (2.16) e B Q : = B ( x Q , Ξ r Q ) , e ∆ Q : = ∆ ( x Q , Ξ r Q ) , and we shall refer to the point x Q as the “center” of Q .Let Q ∈ D k and consider the family of its dyadic children { Q ′ ∈ D k + : Q ′ ⊂ Q } . Note that for anytwo distinct children Q ′ , Q ′′ , one has | x Q ′ − x Q ′′ | ≥ r Q ′ = r Q ′′ = r Q /
2, otherwise x Q ′′ ∈ Q ′′ ∩ ∆ Q ′ ⊂ Q ′′ ∩ Q ′ , contradicting the fact that Q ′ and Q ′′ are disjoint. Also x Q ′ , x Q ′′ ∈ Q ⊂ ∆ ( x Q , Ξ r Q ), hence bythe geometric doubling property we have a purely dimensional bound for the number of such x Q ′ andhence the number of dyadic children of a given dyadic cube is uniformly bounded.We next introduce the “discretized Carleson region” relative to Q ∈ D ( E ), D Q = { Q ′ ∈ D : Q ′ ⊂ Q } .Let F = { Q i } ⊂ D ( E ) be a family of pairwise disjoint cubes. The “global discretized sawtooth” relativeto F is the collection of cubes Q ∈ D ( E ) that are not contained in any Q i ∈ F , that is, D F : = D \ [ Q i ∈F D Q i . For a given Q ∈ D ( E ), the “local discretized sawtooth” relative to F is the collection of cubes in D Q that are not contained in any Q i ∈ F or, equivalently, D F , Q : = D Q \ [ Q i ∈F D Q i = D F ∩ D Q . We also allow F to be the null set in which case D Ø = D ( E ) and D Ø , Q = D Q .In the sequel, Ω ⊂ R n + , n ≥
2, will be a 1-sided NTA domain satisfying the CDC. Write D = D ( ∂ Ω )for the dyadic grid obtained from Lemma 2.13 with E = ∂ Ω . In [1, Remark 2.73] it is shown that underthe present assumptions one has that diam( ∆ ) ≈ r ∆ for every surface ball ∆ and diam( Q ) ≈ ℓ ( Q ) forevery Q ∈ D . Given Q ∈ D we define the “Corkscrew point relative to Q ” as X Q : = X ∆ Q . We note that δ ( X Q ) ≈ dist( X Q , Q ) ≈ diam( Q ) . We also introduce the “geometric” Carleson regions and sawtooths. Given Q ∈ D we want to definesome associated regions which inherit the good properties of Ω . Let W = W ( Ω ) denote a collectionof (closed) dyadic Whitney cubes of Ω ⊂ R n + , so that the cubes in W form a covering of Ω withnon-overlapping interiors, and satisfy(2.17) 4 diam( I ) ≤ dist(4 I , ∂ Ω ) ≤ dist( I , ∂ Ω ) ≤
40 diam( I ) , ∀ I ∈ W , and diam( I ) ≈ diam( I ) , whenever I and I touch . LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 11
Let X ( I ) denote the center of I , let ℓ ( I ) denote the side length of I , and write k = k I if ℓ ( I ) = − k .Given 0 < λ < I ∈ W we write I ∗ = (1 + λ ) I for the “fattening” of I . By taking λ smallenough, we can arrange matters, so that, first, dist( I ∗ , J ∗ ) ≈ dist( I , J ) for every I , J ∈ W . Secondly, I ∗ meets J ∗ if and only if ∂ I meets ∂ J (the fattening thus ensures overlap of I ∗ and J ∗ for any pair I , J ∈ W whose boundaries touch, so that the Harnack Chain property then holds locally in I ∗ ∪ J ∗ ,with constants depending upon λ ). By picking λ su ffi ciently small, say 0 < λ < λ , we may alsosuppose that there is τ ∈ ( ,
1) such that for distinct I , J ∈ W , we have that τ J ∩ I ∗ = Ø. In whatfollows we will need to work with dilations I ∗∗ = (1 + λ ) I or I ∗∗∗ = (1 + λ ) I , and in order to ensurethat the same properties hold we further assume that 0 < λ < λ / ϑ ∈ N , for every cube Q ∈ D we set(2.18) W ϑ Q : = (cid:8) I ∈ W : 2 − ϑ ℓ ( Q ) ≤ ℓ ( I ) ≤ ϑ ℓ ( Q ) , and dist( I , Q ) ≤ ϑ ℓ ( Q ) (cid:9) . We will choose ϑ ≥ ϑ , with ϑ large enough depending on the constants of the Corkscrew condition(cf. Definition 2.1) and in the dyadic cube construction (cf. Lemma 2.13), so that X Q ∈ I for some I ∈ W ϑ Q , and for each dyadic child Q j of Q , the respective corkscrew points X Q j ∈ I j for some I j ∈ W ϑ Q . Moreover, we may always find an I ∈ W ϑ Q with the slightly more precise property that ℓ ( Q ) / ≤ ℓ ( I ) ≤ ℓ ( Q ) and W ϑ Q ∩ W ϑ Q , Ø , whenever 1 ≤ ℓ ( Q ) ℓ ( Q ) ≤ , and dist( Q , Q ) ≤ ℓ ( Q ) . For each I ∈ W ϑ Q , we form a Harnack chain from the center X ( I ) to the Corkscrew point X Q and callit H ( I ). We now let W ϑ, ∗ Q denote the collection of all Whitney cubes which meet at least one ball in theHarnack chain H ( I ) with I ∈ W ϑ Q , that is, W ϑ, ∗ Q : = { J ∈ W : there exists I ∈ W ϑ Q such that H ( I ) ∩ J , Ø } . We also define U ϑ Q : = [ I ∈W ϑ, ∗ Q (1 + λ ) I = : [ I ∈W ϑ, ∗ Q I ∗ . By construction, we then have that W ϑ Q ⊂ W ϑ, ∗ Q ⊂ W and X Q ∈ U ϑ Q , X Q j ∈ U ϑ Q , for each child Q j of Q . It is also clear that there is a uniform constant k ∗ (depending only on the 1-sidedCAD constants and ϑ ) such that2 − k ∗ ℓ ( Q ) ≤ ℓ ( I ) ≤ k ∗ ℓ ( Q ) , ∀ I ∈ W ϑ, ∗ Q , X ( I ) → U ϑ Q X Q , ∀ I ∈ W ϑ, ∗ Q , dist( I , Q ) ≤ k ∗ ℓ ( Q ) , ∀ I ∈ W ϑ, ∗ Q . Here, X ( I ) → U ϑ Q X Q means that the interior of U ϑ Q contains all balls in Harnack Chain (in Ω ) con-necting X ( I ) to X Q , and moreover, for any point Z contained in any ball in the Harnack Chain, wehave dist( Z , ∂ Ω ) ≈ dist( Z , Ω \ U ϑ Q ) with uniform control of implicit constants. The constant k ∗ and theimplicit constants in the condition X ( I ) → U ϑ Q X Q , depend at most on the allowable parameters, on λ ,and on ϑ . Moreover, given I ∈ W we have that I ∈ W ϑ, ∗ Q I , where Q I ∈ D satisfies ℓ ( Q I ) = ℓ ( I ), andcontains any fixed b y ∈ ∂ Ω such that dist( I , ∂ Ω ) = dist( I , b y ). The reader is referred to [28, 32] for fulldetails. We note however that in [28] the parameter ϑ is fixed. Here we need to allow ϑ to depend onthe aperture of the cones and hence it is convenient to include the superindex ϑ . For a given Q ∈ D , the “Carleson box” relative to Q is defined by T ϑ Q : = int (cid:18) [ Q ′ ∈ D Q U ϑ Q ′ (cid:19) . For a given family F = { Q i } ⊂ D of pairwise disjoint cubes and a given Q ∈ D , we define the “localsawtooth region” relative to F by(2.19) Ω ϑ F , Q : = int (cid:18) [ Q ′ ∈ D F , Q U ϑ Q ′ (cid:19) = int (cid:18) [ I ∈W ϑ F , Q I ∗ (cid:19) , where W ϑ F , Q : = S Q ′ ∈ D F , Q W ϑ, ∗ Q ′ . Note that in the previous definition we may allow F to be empty inwhich case clearly Ω ϑ Ø , Q = T ϑ Q . Similarly, the “global sawtooth region” relative to F is defined as(2.20) Ω ϑ F : = int (cid:18) [ Q ′ ∈ D F U ϑ Q ′ (cid:19) = int (cid:18) [ I ∈W ϑ F I ∗ (cid:19) , where W ϑ F : = S Q ′ ∈ D F W ϑ, ∗ Q ′ . If F is the empty set clearly Ω ϑ Ø = Ω . For a given Q ∈ D and x ∈ ∂ Ω letus introduce the “truncated dyadic cone” Γ ϑ Q ( x ) : = [ x ∈ Q ′ ∈ D Q U ϑ Q ′ , where it is understood that Γ ϑ Q ( x ) = Ø if x < Q . Analogously, we can slightly fatten the Whitney boxesand use I ∗∗ to define new fattened Whitney regions and sawtooth domains. More precisely, for every Q ∈ D , T ϑ, ∗ Q : = int (cid:18) [ Q ′ ∈ D Q U ϑ, ∗ Q ′ (cid:19) , Ω ϑ, ∗F , Q : = int (cid:18) [ Q ′ ∈ D F , Q U ϑ, ∗ Q ′ (cid:19) , Γ ϑ, ∗ Q ( x ) : = [ x ∈ Q ′ ∈ D Q U ϑ, ∗ Q ′ , where U ϑ, ∗ Q : = S I ∈W ϑ, ∗ Q I ∗∗ . Similarly, we can define T ϑ, ∗∗ Q , Ω ϑ, ∗∗F , Q , Γ ϑ, ∗∗ Q ( x ), and U ϑ, ∗∗ Q by using I ∗∗∗ inplace of I ∗∗ .To define the “Carleson box” T ϑ ∆ associated with a surface ball ∆ = ∆ ( x , r ), let k ( ∆ ) denote theunique k ∈ Z such that 2 − k − < r ≤ − k , and set(2.21) D ∆ : = (cid:8) Q ∈ D k ( ∆ ) : Q ∩ ∆ , Ø (cid:9) . We then define(2.22) T ϑ ∆ : = int (cid:18) [ Q ∈ D ∆ T ϑ Q (cid:19) . We can also consider fattened versions of T ϑ ∆ given by T ϑ, ∗ ∆ : = int (cid:18) [ Q ∈ D ∆ T ϑ, ∗ Q (cid:19) , T ϑ, ∗∗ ∆ : = int (cid:18) [ Q ∈ D ∆ T ϑ, ∗∗ Q (cid:19) . Following [28, 32], one can easily see that there exist constants 0 < κ < κ ≥ Ξ (with Ξ theconstant in (2.14)), depending only on the allowable parameters and on ϑ , so that κ B Q ∩ Ω ⊂ T ϑ Q ⊂ T ϑ, ∗ Q ⊂ T ϑ, ∗∗ Q ⊂ T ϑ, ∗∗ Q ⊂ κ B Q ∩ Ω = : B ∗ Q ∩ Ω , (2.23) B ∆ ∩ Ω ⊂ T ϑ ∆ ⊂ T ϑ, ∗ ∆ ⊂ T ϑ, ∗∗ ∆ ⊂ T ϑ, ∗∗ ∆ ⊂ κ B ∆ ∩ Ω = : B ∗ ∆ ∩ Ω , (2.24) LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 13 and also(2.25) Q ⊂ κ B ∆ ∩ ∂ Ω = B ∗ ∆ ∩ ∂ Ω = : ∆ ∗ , ∀ Q ∈ D ∆ , where B Q is defined as in (2.15), ∆ = ∆ ( x , r ) with x ∈ ∂ Ω , 0 < r < diam( ∂ Ω ), and B ∆ = B ( x , r ) is sothat ∆ = B ∆ ∩ ∂ Ω . From our choice of the parameters one also has that B ∗ Q ⊂ B ∗ Q ′ whenever Q ⊂ Q ′ . Lemma 2.26 ([1, Lemma 2.54] and [28, Appendices A.1-A.2]) . Let Ω ⊂ R n + , n ≥ , be a 1-sided NTA domain satisfying the CDC. For every ϑ ≥ ϑ all of its Carleson boxes T ϑ Q , T ϑ, ∗ Q , T ϑ, ∗∗ Q andT ϑ ∆ , T ϑ, ∗ ∆ , T ϑ, ∗∗ ∆ , and sawtooth regions Ω ϑ F , Ω ϑ, ∗F , Ω ϑ, ∗∗F , and Ω ϑ F , Q , Ω ϑ, ∗F , Q , Ω ϑ, ∗∗F , Q are 1-sided NTA domainsand satisfy the CDC with uniform implicit constants depending only on dimension, the correspondingconstants for Ω , and ϑ . Given Q we define the “localized dyadic conical square function”(2.27) S ϑ Q u ( x ) : = (cid:18) ¨ Γ ϑ Q ( x ) |∇ u ( Y ) | δ ( Y ) − n dY (cid:19) , x ∈ ∂ Ω , for every u ∈ W , ( T ϑ Q ). Note that S ϑ Q u ( x ) = x ∈ ∂ Ω \ Q since Γ ϑ Q ( x ) = Ø in such a case.The “localized dyadic non-tangential maximal function” is given by(2.28) N ϑ Q u ( x ) : = sup Y ∈ Γ ϑ, ∗ Q ( x ) | u ( Y ) | , x ∈ ∂ Ω , for every u ∈ C ( T ϑ, ∗ Q ), where it is understood that N ϑ Q u ( x ) = x ∈ ∂ Ω \ Q .Given α > x ∈ ∂ Ω we introduce the “cone with vertex at x and aperture α ” defined as Γ α ( x ) = { X ∈ Ω : | X − x | ≤ (1 + α ) δ ( X ) } . One can also introduce the “truncated cone”, for every x ∈ ∂ Ω and 0 < r < ∞ we set Γ α r ( x ) = B ( x , r ) ∩ Γ α ( x ).The “conical square function” and the “non-tangential maximal function” are defined respectivelyas(2.29) S α u ( x ) : = (cid:18) ¨ Γ α ( x ) |∇ u ( Y ) | δ ( Y ) − n dY (cid:19) , N α u ( x ) : = sup X ∈ Γ α ( x ) | u ( X ) | , x ∈ ∂ Ω , for every u ∈ W , ( Ω ) and u ∈ C ( Ω ) respectively. Analogously, the “truncated conical square function”and the “truncated non-tangential maximal function” are defined respectively as(2.30) S α r u ( x ) : = (cid:18) ¨ Γ α r ( x ) |∇ u ( Y ) | δ ( Y ) − n dY (cid:19) , N α r u ( x ) : = sup X ∈ Γ α r ( x ) | u ( X ) | , x ∈ ∂ Ω , < r < ∞ , for every u ∈ W , ( Ω ∩ B ( x , r )) and u ∈ C ( Ω ∩ B ( x , r )) respectively.We would like to note that truncated dyadic cones are never empty. Indeed, in our construction wehave made sure that X Q ∈ U ϑ Q for every Q ∈ D , hence for any Q ∈ D and x ∈ Q one has X Q ∈ Γ ϑ Q ( x ).Moreover, X Q ′ ∈ Γ ϑ Q ( x ) for every Q ′ ∈ D Q with Q ′ ∋ x . For the regular truncated cones it could happenthat Γ α r ( x ) = Ø unless α is su ffi ciently large. Suppose for instance that Ω = { X = ( x , . . . , x n + ) ∈ R n + : x , . . . , x n + > } is the first orthant, then Γ α r (0) = Ø for any 0 < r < ∞ if α < √ n + −
1. On theother hand, if α is su ffi ciently large, more precisely, if α ≥ c − −
1, where c is the corkscrew constant(cf. Definition 2.1), then(2.31) X ∆ ( x , r ) ∈ Γ α r ( x ) , ∀ x ∈ ∂ Ω , < r < diam( ∂ Ω ) .
3. U niformly elliptic operators , elliptic measure and the G reen function Next, we recall several facts concerning elliptic measures and Green functions. To set the stage let Ω ⊂ R n + be an open set. Throughout we consider elliptic operators L of the form Lu = − div( A ∇ u )with A ( X ) = ( a i , j ( X )) n + i , j = being a real (non-necessarily symmetric) matrix such that a i , j ∈ L ∞ ( Ω ) andthere exists Λ ≥ Λ − | ξ | ≤ A ( X ) ξ · ξ, | A ( X ) ξ · η | ≤ Λ | ξ | | η | (3.1)for all ξ, η ∈ R n + and for almost every X ∈ Ω . We write L ⊤ to denote the transpose of L , or, in otherwords, L ⊤ u = − div( A ⊤ ∇ u ) with A ⊤ being the transpose matrix of A .We say that u is a weak solution to Lu = Ω provided that u ∈ W , ( Ω ) satisfies ¨ Ω A ( X ) ∇ u ( X ) · ∇ φ ( X ) dX = φ ∈ C ∞ c ( Ω ) . Associated with L one can construct an elliptic measure { ω XL } X ∈ Ω and a Green function G L (see [32]for full details). If Ω satisfies the CDC then it follows that all boundary points are Wiener regular andhence for a given f ∈ C c ( ∂ Ω ) we can define u ( X ) : = ˆ ∂ Ω f ( z ) d ω XL ( z ) , whenever X ∈ Ω , and u : = f on ∂ Ω and obtain that u ∈ W , ( Ω ) ∩ C ( Ω ) and Lu = Ω . Moreover,if f ∈ Lip( ∂ Ω ) then u ∈ W , ( Ω ).We first define the reverse H ¨older class and the A ∞ classes with respect to a fixed elliptic measure in Ω . One reason we take this approach is that we do not know whether H n | ∂ Ω is well-defined since wedo not assume any Ahlfors regularity in Theorem 1.2. Hence we have to develop these notions in termsof elliptic measures. To this end, let Ω satisfy the CDC and let L and L be two real (non-necessarilysymmetric) elliptic operators associated with L u = − div( A ∇ u ) and Lu = − div( A ∇ u ) where A and A satisfy (3.1). Let ω XL and ω XL be the elliptic measures of Ω associated with the operators L and L respectively with pole at X ∈ Ω . Note that if we further assume that Ω is connected, then Harnack’sinequality readily implies that ω XL ≪ ω YL on ∂ Ω for every X , Y ∈ Ω . Hence if ω X L ≪ ω Y L on ∂ Ω forsome X , Y ∈ Ω then ω XL ≪ ω YL on ∂ Ω for every X , Y ∈ Ω and thus we can simply write ω L ≪ ω L on ∂ Ω . In the latter case we will use the notation(3.2) h ( · ; L , L , X ) = d ω XL d ω XL to denote the Radon-Nikodym derivative of ω XL with respect to ω XL , which is a well-defined function ω XL -almost everywhere on ∂ Ω . Definition 3.3 (Reverse H ¨older and A ∞ classes) . Fix ∆ = B ∩ ∂ Ω where B = B ( x , r ) with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ). Given 1 < p < ∞ , we say that ω L ∈ RH p ( ∆ , ω L ), provided that ω L ≪ ω L on ∆ , and there exists C ≥ (cid:18) ∆ h ( y ; L , L , X ∆ ) p d ω X ∆ L ( y ) (cid:19) p ≤ C ∆ h ( y ; L , L , X ∆ ) d ω X ∆ L ( y ) = C ω X ∆ L ( ∆ ) ω X ∆ L ( ∆ ) , (3.4)for every ∆ = B ∩ ∂ Ω where B ⊂ B ( x , r ), B = B ( x , r ) with x ∈ ∂ Ω , 0 < r < diam( ∂ Ω ). The infimumof the constants C as above is denoted by [ ω L ] RH p ( ∆ ,ω L ) . LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 15
Similarly, we say that ω L ∈ RH p ( ∂ Ω , ω L ) provided that for every ∆ = ∆ ( x , r ) with x ∈ ∂ Ω and0 < r < diam( ∂ Ω ) one has ω L ∈ RH p ( ∆ , ω L ) uniformly on ∆ , that is,[ ω L ] RH p ( ∂ Ω ,ω L ) : = sup ∆ [ ω L ] RH p ( ∆ ,ω L ) < ∞ . Finally, A ∞ ( ∆ , ω L ) : = [ p > RH p ( ∆ , ω L ) and A ∞ ( ∂ Ω , ω L ) : = [ p > RH p ( ∂ Ω , ω L ) . Definition 3.5 (BMO) . Fix ∆ = B ∩ ∂ Ω where B = B ( x , r ) with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ).We say that f ∈ BMO( ∆ , ω L ) provided f ∈ L ( ∆ , ω X ∆ L ) and k f k BMO( ∆ ,ω L ) : = sup ∆ inf c ∈ R ∆ | f ( x ) − c | d ω X ∆ L ( x ) < ∞ , where the sup is taken over all surface balls ∆ = B ∩ ∂ Ω where B ⊂ B ( x , r ), B = B ( x , r ) with x ∈ ∂ Ω ,0 < r < diam( ∂ Ω ).Similarly, we say that f ∈ BMO( ∂ Ω , ω L ) provided that for every ∆ = ∆ ( x , r ) with x ∈ ∂ Ω and0 < r < diam( ∂ Ω ) one has f ∈ BMO( ∆ , ω L ) uniformly on ∆ , that is, f ∈ L ( ∂ Ω , ω L ) (that is, k f ∆ k L ( ∂ Ω ,ω XL ) < ∞ for every surface ball ∆ ⊂ ∂ Ω and for every X ∈ Ω —albeit with a constant thatmay depend on ∆ and X ) and satisfies k f k BMO( ∂ Ω ,ω L ) = sup ∆ sup ∆ inf c ∈ R ∆ | f ( x ) − c | d ω X ∆ L ( x ) < ∞ , where the sups are taken respectively over all surface balls ∆ = B ( x , r ) ∩ ∂ Ω with x ∈ ∂ Ω and0 < r < diam( ∂ Ω ), and ∆ = B ∩ ∂ Ω , B = B ( x , r ) ⊂ B with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ). Definition 3.6 (Solvability, CME, S < N ) . Let Ω ⊂ R n + , n ≥
2, be a 1-sided NTA domain (cf.Definition 2.5) satisfying the capacity density condition (cf. Definition 2.10), and let Lu = − div( A ∇ u )and L u = − div( A ∇ u ) be real (non-necessarily symmetric) elliptic operators. • Given 1 < p < ∞ , we say that L is L p ( ω L ) -solvable if for a given α > N ≥ C α, N ≥ n , the 1-sided NTA constants, the CDC constant, the ellipticity of L and L , α , N , and p ) such that for every ∆ = ∆ ( x , r ) with x ∈ ∂ Ω , < r < diam( ∂ Ω ), and forevery f ∈ C ( ∂ Ω ) with supp f ⊂ N ∆ if one sets(3.7) u ( X ) : = ˆ ∂ Ω f ( y ) d ω XL ( y ) , X ∈ Ω , then(3.8) kN α r u k L p ( ∆ ,ω X ∆ L ) ≤ C α, N k f k L p ( N ∆ ,ω X ∆ L ) . • We say that
L is
BMO( ω L ) -solvable , if there exists C ≥ n , the 1-sidedNTA constants, the CDC constant, and the ellipticity of L and L ) such that for every f ∈ C ( ∂ Ω ) ∩ L ∞ ( ∂ Ω , ω L ) if one takes u as in (3.7) and we set u L , Ω ( X ) : = ω XL ( ∂ Ω ), X ∈ Ω , then(3.9) sup B sup B ′ ω X ∆ L ( ∆ ′ ) ¨ B ′ ∩ Ω |∇ ( u − f ∆ , L u L , Ω )( X ) | G L ( X ∆ , X ) dX ≤ C k f k ∂ Ω ,ω L ) , where ∆ = B ∩ ∂ Ω , ∆ ′ = B ′ ∩ ∂ Ω , f ∆ , L = ∆ f d ω X ∆ L , and the sups are taken respectively over all balls B = B ( x , r ) with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ), and B ′ = B ( x ′ , r ′ ) with x ′ ∈ ∆ and 0 < r ′ < rc / c is the Corkscrew constant. • We say that
L is
BMO( ω L ) -solvable in the sense of [26], that is, there exists C ≥ n , the 1-sided NTA constants, the CDC constant, and the ellipticity of L and L ) such thatfor every ε ∈ (0 ,
1] there exists ̺ ( ε ) ≥ ̺ ( ε ) −→ ε → + in such a way that for every f ∈ C ( ∂ Ω ) ∩ L ∞ ( ∂ Ω , ω L ) if one takes u as in (3.7), then(3.10) sup B ε sup B ′ ω X ∆ ε L ( ∆ ′ ) ¨ B ′ ∩ Ω |∇ u ( X ) | G L ( X ∆ ε , X ) dX ≤ C (cid:0) k f k ∂ Ω ,ω L ) + ̺ ( ε ) k f k L ∞ ( ∂ Ω ,ω L ) (cid:1) , where ∆ ε = B ε ∩ ∂ Ω , ∆ ′ = B ′ ∩ ∂ Ω , and the sups are taken respectively over all balls B ε = B ( x , ε r )with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ), and B ′ = B ( x ′ , r ′ ) with x ′ ∈ ∆ ε and 0 < r ′ < ε rc /
4, and c isthe Corkscrew constant. • We say that
L satisfies
CME( ω L ), if there exists C ≥ n , the 1-sided NTAconstants, the CDC constant, and the ellipticity of L and L ) such that for every u ∈ W , ( Ω ) ∩ L ∞ ( Ω )satisfying Lu = Ω the following estimate holds(3.11) sup B sup B ′ ω X ∆ L ( ∆ ′ ) ¨ B ′ ∩ Ω |∇ u ( X ) | G L ( X ∆ , X ) dX ≤ C k u k L ∞ ( Ω ) , where ∆ = B ∩ ∂ Ω , ∆ ′ = B ′ ∩ ∂ Ω , and the sups are taken respectively over all balls B = B ( x , r ) with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ), and B ′ = B ( x ′ , r ′ ) with x ′ ∈ ∆ and 0 < r ′ < rc /
4, and c is theCorkscrew constant. • Given 0 < q < ∞ , we say that L satisfies S < N in L q ( ω L ) if for some given α >
0, there exists C α ≥ n , the 1-sided NTA constants, the CDC constant, the ellipticity of L and L , α , and q ) such that for every ∆ = ∆ ( x , r ) with x ∈ ∂ Ω , < r < diam( ∂ Ω ), and for every u ∈ W , ( Ω ) satisfying Lu = Ω the following estimate holds(3.12) kS α r u k L q ( ∆ ,ω X ∆ L ) ≤ C α kN α r u k L q (5 ∆ ,ω X ∆ L ) . • We say that any of the previous properties holds for characteristic functions if the correspondingestimate is valid for all solutions of the form u ( X ) = ω XL ( S ), X ∈ Ω , with S ⊂ ∂ Ω being an arbitraryBorel set (with S ⊂ N ∆ in the case of L p ( ω L )-solvability). Remark . We would like to observe that when either Ω and ∂ Ω are both bounded or ∂ Ω is un-bounded, the elliptic measure is a probability (that is, u L , Ω ( X ) = ω XL ( ∂ Ω ) ≡ X ∈ Ω ). Hence,it has vanishing gradient and one can then remove the term f ∆ , L u L , Ω in (3.9). This means that the onlycase on which subtracting f ∆ , L u L , Ω is relevant is that where Ω is unbounded and ∂ Ω is bounded (e.g.,the complementary of a ball). As a matter of fact, one must subtract that term or a similar one for(3.9) to hold. To see this, take f ≡ ∈ BMO( ∂ Ω , ω L ) so that k f k BMO( ∂ Ω ,ω L ) = u = u L , Ω bethe associated elliptic measure solution. One can see (cf. [32]) that the function u L , Ω is non-constant(it decays at infinity), hence 0 < u L , Ω ( X ) < X ∈ Ω and |∇ u L , Ω | .
0. This means that theversion of (3.9) without the term f ∆ , L u L , Ω cannot hold. Moreover, note that in this case (3.9) is trivial: f ∆ , L u L , Ω = u L , Ω and the left-hand side of (3.9) vanishes. Remark . As just explained in the previous remark when either Ω and ∂ Ω are both bounded or ∂ Ω is unbounded, the left-hand sides of (3.9) and (3.10) are the same, as a result (e) clearly implies (f)—and (e) ′ implies (f) ′ — upon taking ̺ ( ε ) ≡ Ω is unbounded and ∂ Ω is bounded, (3.10) needs toincorporate the term ̺ ( ε ) k f k L ∞ ( ∂ Ω ,ω L ) , otherwise it would fail for u = u L , Ω . Remark . In (3.9) one can replace f ∆ , L by f ∆ ′ , L (see Remark 4.43 below). Also, when Ω isunbounded and ∂ Ω bounded one can subtract a constant that does not depend on ∆ nor ∆ ′ . Namely, let LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 17 X Ω ∈ Ω satisfy δ ( X Ω ) ≈ diam( ∂ Ω ) (say, X Ω = X ∆ ( x , r ) with x ∈ ∂ Ω and r ≈ diam( ∂ Ω )). Then in (3.9)one can replace f ∆ , L by f ∂ Ω , L = ∂ Ω f d ω X Ω L , see Remark 4.43.The following lemmas state some properties of Green functions and elliptic measures, proofs maybe found in [32]. Lemma 3.16.
Suppose that Ω ⊂ R n + , n ≥ , is an open set satisfying the CDC. Given a real (non-necessarily symmetric) elliptic operator L = − div( A ∇ ) , there exist C > (depending only on dimensionand on the ellipticity constant of L) and c θ > (depending on the above parameters and on θ ∈ (0 , )such that G L , the Green function associated with L, satisfiesG L ( X , Y ) ≤ C | X − Y | − n ;(3.17) c θ | X − Y | − n ≤ G L ( X , Y ) , if | X − Y | ≤ θδ ( X ) , θ ∈ (0 , G L ( · , Y ) ∈ C (cid:0) Ω \ { Y } (cid:1) and G L ( · , Y ) | ∂ Ω ≡ ∀ Y ∈ Ω ;(3.19) G L ( X , Y ) ≥ , ∀ X , Y ∈ Ω , X , Y ;(3.20) G L ( X , Y ) = G L ⊤ ( Y , X ) , ∀ X , Y ∈ Ω , X , Y . (3.21) Moreover, G L ( · , Y ) ∈ W , ( Ω \ { Y } ) for any Y ∈ Ω and satisfies LG L ( · , Y ) = δ Y in the sense ofdistributions, that is, (3.22) ¨ Ω A ( X ) ∇ X G L ( X , Y ) · ∇ ϕ ( X ) dX = ϕ ( Y ) , ∀ ϕ ∈ C ∞ c ( Ω ) . In particular, G L ( · , Y ) is a weak solution to LG L ( · , Y ) = in the open set Ω \ { Y } .Finally, the following Riesz formula holds: ¨ Ω A ⊤ ( X ) ∇ X G L ⊤ ( X , Y ) · ∇ ϕ ( X ) dX = ϕ ( Y ) − ˆ ∂ Ω ϕ d ω YL , for a.e. Y ∈ Ω , for every ϕ ∈ C ∞ c ( R n + ) .Remark . If we also assume that Ω is bounded, following [32] we know that the Green function G L coincides with the one constructed in [24]. Consequently, for each Y ∈ Ω and 0 < r < δ ( Y ), thereholds(3.24) G L ( · , Y ) ∈ W , ( Ω \ B ( Y , r )) ∩ W , ( Ω ) . Moreover, for every ϕ ∈ C ∞ c ( Ω ) such that 0 ≤ ϕ ≤ ϕ ≡ B ( Y , r ) with 0 < r < δ ( Y ), we havethat(3.25) (1 − ϕ ) G L ( · , Y ) ∈ W , ( Ω ) . The following result lists a number of properties which will be used throughout the paper:
Lemma 3.26.
Suppose that Ω ⊂ R n + , n ≥ , is a 1-sided NTA domain satisfying the CDC. LetL = − div( A ∇ ) and L = − div( A ∇ ) be two real (non-necessarily symmetric) elliptic operators, thereexist C ≥ , ρ ∈ (0 , (depending only on dimension, the 1-sided NTA constants, the CDC constant,and the ellipticity of L) and C ≥ (depending on the same parameters and on the ellipticity of L ),such that for every B = B ( x , r ) with x ∈ ∂ Ω , < r < diam( ∂ Ω ) , and ∆ = B ∩ ∂ Ω we have thefollowing properties: ( a ) ω YL ( ∆ ) ≥ C − for every Y ∈ C − B ∩ Ω and ω X ∆ L ( ∆ ) ≥ C − . ( b ) If B = B ( x , r ) with x ∈ ∂ Ω and ∆ = B ∩ ∂ Ω is such that B ⊂ B , then for all X ∈ Ω \ B we havethat C ω XL ( ∆ ) ≤ r n − G L ( X , X ∆ ) ≤ C ω XL ( ∆ ) . ( c ) If X ∈ Ω \ B , then ω XL (2 ∆ ) ≤ C ω XL ( ∆ ) . ( d ) If B = B ( x , r ) with x ∈ ∂ Ω and ∆ : = B ∩ ∂ Ω is such that B ⊂ B , then for every X ∈ Ω \ κ B with κ as in (2.24) , we have that C ω X ∆ L ( ∆ ) ≤ ω XL ( ∆ ) ω XL ( ∆ ) ≤ C ω X ∆ L ( ∆ ) . As a consequence, C ω XL ( ∆ ) ≤ d ω X ∆ L d ω XL ( y ) ≤ C ω XL ( ∆ ) , for ω XL -a.e. y ∈ ∆ . ( e ) For every X ∈ B ∩ Ω and for any j ≥ d ω XL d ω X j ∆ L ( y ) ≤ C (cid:18) δ ( X )2 j r (cid:19) ρ , for ω XL -a.e. y ∈ ∂ Ω \ j ∆ . ( f ) If ≤ u ∈ W , ( B ∩ Ω ) ∩ C ( B ∩ Ω ) satisfies Lu = in the weak-sense in B ∩ Ω and u ≡ in ∆ then u ( X ) ≤ C (cid:16) δ ( X ) r (cid:17) ρ u ( X ∆ ) , X ∈ B ∩ Ω . Remark . We note that from ( d ) in the previous result and Harnack’s inequality one can easily seethat given Q , Q ′ , Q ′′ ∈ D ( ∂ Ω )(3.28) ω X Q ′′ L ( Q ) ω X Q ′′ L ( Q ′ ) ≈ ω X Q ′ L ( Q ) , whenever Q ⊂ Q ′ ⊂ Q ′′ . Also, ( d ), Harnack’s inequality, and (2.14) give(3.29) d ω X Q ′ L d ω X Q ′′ L ( y ) ≈ ω X Q ′′ L ( Q ′ ) , for ω X Q ′′ L -a.e. y ∈ Q ′ , whenever Q ′ ⊂ Q ′′ . Observe that since ω X Q ′′ L ≪ ω X Q ′ L we can easily get an analogous inequality for the reciprocal of theRadon-Nikodym derivative. Remark . It is not hard to see that if ω L ≪ ω L then Lemma 3.26 gives the following:(3.31) ω L ∈ RH p ( ∂ Ω , ω L ) ⇐⇒ sup x ∈ ∂ Ω , < r < diam( ∂ Ω ) k h ( · ; L , L , X ∆ ( x , r ) ) k L p ( ∆ ( x , r ) ,ω X ∆ ( x , r ) L ) < ∞ . The left-to-right implication follows at once from (3.4) by taking B = B (hence ∆ = ∆ ) andLemma 3.26 part ( a ). For the converse, fix B = B ( x , r ) and B = B ( x , r ) with B ⊂ B , x , x ∈ ∂ Ω and0 < r , r < diam( ∂ Ω ). Write ∆ = B ∩ ∂ Ω and ∆ = B ∩ ∂ Ω . If r ≈ r we have, by Lemma 3.26 part( a ), (cid:18) ∆ h ( y ; L , L , X ∆ ) p d ω X ∆ L ( y ) (cid:19) p . k h ( · ; L , L , X ∆ ) k L p ( ∆ ,ω X ∆ L )LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 19 ≈ k h ( · ; L , L , X ∆ ) k L p ( ∆ ,ω X ∆ L ) ω X ∆ L ( ∆ ) ω X ∆ L ( ∆ ) . On the other hand, if r ≪ r , we have by Lemma 3.26 part ( d ) and the fact that ω L ≪ ω L that h ( · ; L , L , X ∆ ) = d ω X ∆ L d ω X ∆ L = d ω X ∆ L d ω X ∆ L d ω X ∆ L d ω X ∆ L d ω X ∆ L d ω X ∆ L ≈ h ( · ; L , L , X ∆ ) ω X ∆ L ( ∆ ) ω X ∆ L ( ∆ ) , ω L -a.e. in ∆ .This and Lemma 3.26 part ( d ) give (cid:18) ∆ h ( y ; L , L , X ∆ ) p d ω X ∆ L ( y ) (cid:19) p ≈ k h ( · ; L , L , X ∆ ) k L p ( ∆ ,ω X ∆ L ) ≈ k h ( · ; L , L , X ∆ ) k L p ( ∆ ,ω X ∆ L ) ω X ∆ L ( ∆ ) ω X ∆ L ( ∆ )Thus, (3.4) holds and the right-to-left implication holds. Remark . It is not di ffi cult to see that under the assumptions of Lemma 3.26 one has k f k BMO( ∂ Ω ,ω L ) ≈ sup ∆ ⊂ ∂ Ω inf c ∈ R ∆ | f ( x ) − c | d ω X ∆ L ( x ) , where the sup is taken over all surface balls ∆ = B ( x , r ) ∩ ∂ Ω with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ).Thus, we could have taken this as the definition of f ∈ BMO( ∂ Ω , ω L ). Remark . Under the assumptions of Lemma 3.26, for every ∆ as above if f ∈ BMO( ∆ , ω L ),then John-Nirenberg’s inequality holds locally in ∆ and the implicit constants depend on the doublingproperty of ω X ∆ L in 2 ∆ . Thus, if one further assumes that f ∈ BMO( ∂ Ω , ω L ), then for every 1 < q < ∞ there holds(3.34) k f k BMO( ∂ Ω ,ω L ) ≈ sup ∆ sup ∆ inf c ∈ R (cid:16) ∆ | f ( x ) − c | q d ω X ∆ L ( x ) (cid:17) q < ∞ , where the sups are taken respectively over all surface balls ∆ = B ( x , r ) ∩ ∂ Ω with x ∈ ∂ Ω and0 < r < diam( ∂ Ω ), and ∆ = B ∩ ∂ Ω , B = B ( x , r ) ⊂ B with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ). Notethat the implicit constants depend only on dimension, the 1-sided NTA constants, the CDC constant,the ellipticity of L , and q . 4. P roof of T heorem p ∈ (1 , ∞ ) the equivalence (a) p ′ ⇐⇒ (b) p easily implies (a) ⇐⇒ (b).Also, since Jensen’s inequality readily gives that ω L ∈ RH p ′ ( ∂ Ω , ω L ) implies ω L ∈ RH q ′ ( ∂ Ω , ω L ) forall q ≥ p , the equivalence (a) p ′ ⇐⇒ (b) p yields (b) p = ⇒ (b) q for all q ≥ p . Finally, (b) p = ⇒ (b) ′ p clearly implies (b) = ⇒ (b) ′ . With all these in mind, we will follow the scheme(a) p ′ ⇐⇒ (b) p = ⇒ (b) ′ p , (b) ′ = ⇒ (a) , (a) = ⇒ (d) = ⇒ (d) ′ = ⇒ (a) , (c) = ⇒ (c) ′ , (e) = ⇒ (f) = ⇒ (c) ′ , (e) ′ = ⇒ (f) ′ = ⇒ (c) ′ = ⇒ (a) , (a) = ⇒ (c) , (a) = ⇒ (e) , (a) = ⇒ (e) ′ . Before proving all these implications we present some auxiliary results:
Lemma 4.1.
Let Ω ⊂ R n + , n ≥ , be a 1-sided NTA domain (cf. Definition 2.5) satisfying the capacitydensity condition (cf. Definition 2.10), and let Lu = − div( A ∇ u ) and L u = − div( A ∇ u ) be real(non-necessarily symmetric) elliptic operators. There exists ρ ∈ (0 , (depending only on dimension,the 1-sided NTA constants, the CDC constant, and the ellipticity of L) and C ≥ (depending onthe same parameters and on the ellipticity of L ) such that the following holds: If ∆ = B ∩ ∂ Ω and ∆ ′ = B ′ ∩ ∂ Ω , where B = B ( x , r ) with x ∈ ∂ Ω and < r < diam( ∂ Ω ) , and B ′ = B ( x ′ , r ′ ) with x ′ ∈ ∆ and < r ′ < rc / , where c is the Corkscrew constant, and u L , Ω ( X ) : = ω XL ( ∂ Ω ) , X ∈ Ω , then (4.2) 1 ω X ∆ L ( ∆ ′ ) ¨ B ′ ∩ Ω |∇ u L , Ω ( X ) | G L ( X ∆ , X ) dX ≤ C (cid:16) r ′ diam( ∂ Ω ) (cid:17) ρ . Proof.
Fix B = B ( x , r ) with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ) and B ′ = B ( x ′ , r ′ ) with x ′ ∈ ∆ and0 < r ′ < rc /
4. Let ∆ = B ∩ ∂ Ω , ∆ ′ = B ′ ∩ ∂ Ω .We note that when either ∂ Ω is unbounded or ∂ Ω and Ω are both bounded then elliptic measureis a probability, hence u L , Ω ≡ ∂ Ω is bounded and Ω is unbounded (e.g, the complement of a closed ball). In that scenario, u L , Ω decays at ∞ , 0 < u L , Ω < Ω , and u L , Ω | ∂ Ω ≡
1. Define v : = − u L , Ω and note that our assumptionsguarantee that v ∈ W , ( Ω ) ∩ C ( Ω ) with 0 ≤ v ≤ v | ∂ Ω ≡
0. By Lemma 3.26 part ( f ) applied in B ( x ′ , diam( ∂ Ω ) /
2) we have0 ≤ v ( X ) . (cid:16) δ ( X )diam( ∂ Ω ) (cid:17) ρ v ( X ∆ ( x ′ , diam( ∂ Ω ) / ) ≤ (cid:16) δ ( X )diam( ∂ Ω ) (cid:17) ρ , X ∈ B ′ ∩ Ω . Set W B ′ : = { I ∈ W : I ∩ B ′ , Ø } and we pick Z I , B ′ ∈ I ∩ B ′ for I ∈ W B ′ . Caccioppoli’s and Harnack’sinequalities, and the previous estimate yield ¨ I |∇ v ( X ) | dX . ℓ ( I ) − ¨ I ∗ v ( X ) dX . ℓ ( I ) n − v ( Z I , B ′ ) . ℓ ( I ) n − (cid:16) ℓ ( I )diam( ∂ Ω ) (cid:17) ρ . Thus, Lemma 3.26 gives ¨ B ′ ∩ Ω |∇ v ( X ) | G L ( X ∆ , X ) dX . X I ∈W B ′ ω X ∆ L ( Q I ) ℓ ( I ) − n ¨ I |∇ v ( X ) | dX . X I ∈W B ′ ω X ∆ L ( Q I ) (cid:16) ℓ ( I )diam( ∂ Ω ) (cid:17) ρ . X k :2 − k . r ′ (cid:16) − k diam( ∂ Ω ) (cid:17) ρ X I ∈W B ′ : ℓ ( I ) = − k ω X ∆ L ( Q I ) , where Q I ∈ D ( ∂ Ω ) is so that ℓ ( Q I ) = ℓ ( I ) and contains b y I ∈ ∂ Ω such that dist( I , ∂ Ω ) = dist( b y I , I ). It iseasy to see that if 2 − k . r , then the family { Q I } I ∈W B ′ ,ℓ ( I ) = − k has bounded overlap uniformly on k , andalso that Q I ⊂ C ∆ ′ for every I ∈ W B ′ , where C is some harmless dimensional constant. Hence, ¨ B ′ ∩ Ω |∇ v ( X ) | G L ( X ∆ , X ) dX . X k :2 − k . r ′ (cid:16) − k diam( ∂ Ω ) (cid:17) ρ ω X ∆ L ( C ∆ ′ ) . (cid:16) r ′ diam( ∂ Ω ) (cid:17) ρ ω X ∆ L ( ∆ ′ ) . This gives the desired estimate. (cid:3)
Given Q ∈ D ( ∂ Ω ) , ϑ ∈ N , and for every η ∈ (0 ,
1) we define the modified non-tangential cone(4.3) Γ ϑ Q ,η ( x ) : = [ Q ∈ D Q Q ∋ x U ϑ Q ,η , U ϑ Q ,η : = [ Q ′ ∈ D Q ℓ ( Q ′ ) >η ℓ ( Q ) U ϑ Q ′ . It is not hard to see that the sets { U ϑ Q ,η } Q ∈ D Q have bounded overlap with constant depending on η .The following result was obtained in [8, Lemma 3.10] (for β >
0) and in [6, Lemma 3.40] (for β = LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 21
Lemma 4.4.
Let Ω ⊂ R n + , n ≥ , be a 1-sided NTA domain (cf. Definition 2.5) satisfying the capacitydensity condition (cf. Definition 2.10), and let Lu = − div( A ∇ u ) be a real (non-necessarily symmetric)elliptic operator. There exist < η ≪ (depending only on dimension, the 1-sided NTA constants, theCDC constant, and the ellipticity of L), and β ∈ (0 , , C η ≥ both depending on the same parametersand additionally on η , such that for every Q ∈ D ( ∂ Ω ) , for every < β < β , and for every Borel setF ⊂ Q satisfying ω X Q L ( F ) ≤ βω X Q L ( Q ) , there exists a Borel set S ⊂ Q such that the bounded weaksolution u ( X ) = ω XL ( S ) , X ∈ Ω , satisfies (4.5) S ϑ Q ,η u ( x ) : = (cid:18) ¨ Γ ϑ Q ,η ( x ) |∇ u ( Y ) | δ ( Y ) − n dY (cid:19) ≥ C − η (cid:0) log( β − ) (cid:1) , ∀ x ∈ F . Furthermore, in the case β = , that is, when ω X Q L ( F ) = , there exists a Borel set S ⊂ Q such thatthe bounded weak solution u ( X ) = ω XL ( S ) , X ∈ Ω , satisfies (4.6) S ϑ Q ,η u ( x ) = ∞ , ∀ x ∈ F . Lemma 4.7.
Let Ω ⊂ R n + , n ≥ , be a 1-sided NTA domain (cf. Definition 2.5) satisfying the capacitydensity condition (cf. Definition 2.10), and let Lu = − div( A ∇ u ) and L u = − div( A ∇ u ) be real (non-necessarily symmetric) elliptic operators. There exists C ≥ (depending only on dimension, the 1-sidedNTA constants, the CDC constant, and the ellipticity of L and L ) such that the following holds. GivenB = B ( x , r ) with x ∈ ∂ Ω and < r < diam( ∂ Ω ) , and B ′ = B ( x ′ , r ′ ) with x ′ ∈ ∆ and < r ′ < rc / , let ∆ = B ∩ ∂ Ω , ∆ ′ = B ′ ∩ ∂ Ω , for every u ∈ W , ( Ω ) ∩ L ∞ ( Ω ) satisfying Lu = in the weak sense in Ω there holds ω X ∆ L ( ∆ ′ ) ¨ B ′ ∩ Ω |∇ u ( X ) | G L ( X ∆ , X ) dX ≤ C ˆ ∆ ′ S C α r ′ u ( y ) d ω X ∆ ′ L ( y ) + C sup {| u ( Y ) | : Y ∈ B ′ , δ ( Y ) ≥ r ′ / C } . Proof.
Fix B , B ′ , ∆ , ∆ ′ , and u as in the statement. Define W B ′ : = { I ∈ W : I ∩ B ′ , Ø } and W MB ′ : = { I ∈ W B ′ : ℓ ( I ) < r ′ / M } for M ≥ I ∈ W B ′ pick Z I ∈ I ∩ B ′ and Q I ∈ D ( ∂ Ω ) so that ℓ ( Q I ) = ℓ ( I ) and contains b y I ∈ ∂ Ω such that dist( I , ∂ Ω ) = dist( b y I , I ).If z ∈ Q I and I ∈ W MB ′ , then | z − x ′ | ≤ | z − b y I | + dist( b y I , I ) + diam( I ) + | Z I − x ′ | ≤ C n ℓ ( I ) + r ′ < (1 + C n / M ) r ′ < r ′ , provided M > C n . Hence, Q I ⊂ ∆ ′ for every I ∈ W MB ′ . Write F for the collection of maximal cubesin { Q I } I ∈W MB ′ , with respect to the inclusion (maximal cubes exist since Q I ⊂ ∆ ′ for every I ∈ W MB ′ ).Hence Q I ⊂ Q for some Q ∈ F . Let ϑ = ϑ and by construction I ∈ W ϑ Q I ⊂ W ϑ, ∗ Q I (see Section 2.3).Hence, for every y ∈ Q ∈ F [ I ∈W MB ′ : y ∈ Q I ∈ D Q I ⊂ [ I ∈W MB ′ : y ∈ Q I ∈ D Q U ϑ Q I ⊂ [ y ∈ Q ′ ∈ D Q U ϑ Q ′ = Γ ϑ Q ( y ) . This gives Σ : = X I ∈W MB ′ ω X ∆ L ( Q I ) ¨ I |∇ u ( X ) | δ ( X ) − n dX = X Q ∈F X I ∈W MB ′ : Q I ∈ D Q ω X ∆ L ( Q I ) ¨ I |∇ u ( X ) | δ ( X ) − n dX = X Q ∈F ˆ Q X I ∈W MB ′ : y ∈ Q I ∈ D Q ¨ I |∇ u ( X ) | δ ( X ) − n dX d ω X ∆ L ( y ) ≤ X Q ∈F ˆ Q ¨ Γ ϑ Q ( y ) |∇ u ( X ) | δ ( X ) − n dXd ω X ∆ L ( y ) = X Q ∈F ˆ Q S ϑ Q u ( y ) d ω X ∆ L ( y ) . To continue let y ∈ Q ∈ F and X ∈ Γ ϑ Q ( y ). Then X ∈ I ∗ with I ∈ W ϑ, ∗ Q ′ and y ∈ Q ′ ∈ D Q . Thus, | X − y | ≤ diam( I ∗ ) + dist( I , Q ′ ) + diam( Q ′ ) . ϑ ℓ ( I ) ≈ δ ( X ) . r ′ / M . where we have used (2.23) and the last estimate holds since ℓ ( I ) < r ′ / M for every I ∈ W MB ′ . This showsthat taking M large enough X ∈ Γ α ′ r ′ ( y ) for some α ′ = α ′ ( ϑ ). Note also that 2 r ′ < rc / < diam( ∂ Ω ),and we can now conclude that Σ . X Q ∈F ˆ Q S α ′ r ′ u ( y ) d ω X ∆ L ( y ) . ˆ ∆ ′ S α ′ r ′ u ( y ) d ω X ∆ L ( y ) ≈ ω X ∆ L ( ∆ ′ ) ˆ ∆ ′ S α ′ r ′ u ( y ) d ω X ∆ ′ L ( y ) , where we have used Lemma 3.26.Now, we note that for each I ∈ W B ′ \ W MB ′ we have ℓ ( Q I ) = ℓ ( I ) ≈ M r ′ , hence for every Y ∈ I ∗ wehave r ′ . M δ ( Y ) ≤ | Y − Z I | + δ ( Z I ) ≤ diam( I ∗ ) + δ ( Z I ) < dist( I , ∂ Ω ) + δ ( Z I ) ≤ δ ( Z I ) ≤ | Z I − x ′ | < r ′ . Also, | b y I − x ′ | + dist( b y I , I ) + diam( I ) + | Z I − x ′ | . dist( I , ∂ Ω ) + | Z I − x ′ | ≤ | Z I − x ′ | < r ′ . Thus, Lemma 3.26 implies that ω X ∆ L ( Q I ) ≈ M ω X ∆ L ( ∆ ′ ). As a consequence of this, we get Σ : = X I ∈W B ′ \W MB ′ ω X ∆ L ( Q I ) ¨ I |∇ u ( X ) | δ ( X ) − n dX . ω X ∆ L ( ∆ ′ ) X I ∈W B ′ \W MB ′ ℓ ( I ) − n ¨ I |∇ u ( X ) | dX . ω X ∆ L ( ∆ ′ ) X I ∈W B ′ \W MB ′ ℓ ( I ) − n − ¨ I ∗ | u ( X ) | dX . ω X ∆ L ( ∆ ′ ) W B ′ \ W MB ′ ) sup {| u ( Y ) | : Y ∈ B ′ , δ ( Y ) ≥ r ′ / C } . M ω X ∆ L ( ∆ ′ ) sup {| u ( Y ) | : Y ∈ B ′ , δ ( Y ) ≥ r ′ / C } , where we have used that W B ′ \ W MB ′ has bounded cardinality depending on n and M .To complete the proof we use Lemma 3.26 and the estimates proved for Σ and Σ : ¨ B ′ ∩ Ω |∇ u ( X ) | G L ( X ∆ , X ) dX ≤ X I ∈W B ′ ¨ I |∇ u ( X ) | G L ( X ∆ , X ) dX ≈ X I ∈W B ′ ω X ∆ L ( Q I ) ¨ I |∇ u ( X ) | δ ( X ) − n dX = Σ + Σ . ω X ∆ L ( ∆ ′ ) (cid:16) ˆ ∆ ′ S α ′ r ′ u ( y ) d ω X ∆ ′ L ( y ) + sup {| u ( Y ) | : Y ∈ B ′ , δ ( Y ) ≥ r ′ / C } (cid:17) . This completes the proof. (cid:3)
For the following result we need to introduce some notation: A α r F ( x ) : = (cid:18) ¨ Γ α r ( x ) | F ( Y ) | dY (cid:19) , x ∈ ∂ Ω , < r < ∞ , α > , for any F ∈ L ( Ω ∩ B ( x , r )). Lemma 4.8.
Let Ω ⊂ R n + , n ≥ , be a 1-sided NTA domain (cf. Definition 2.5) satisfying the capacitydensity condition (cf. Definition 2.10), and let L u = − div( A ∇ u ) be a real (non-necessarily symmetric)elliptic operator. Given < q < ∞ , < α, α ′ < ∞ there exists C ≥ (depending only on dimension,the 1-sided NTA constants, the CDC constant, the ellipticity of L , q, α , and α ′ ) such that the followingholds. Given B = B ( x , r ) with x ∈ ∂ Ω and < r < diam( ∂ Ω ) , let ∆ = B ∩ ∂ Ω , for every F ∈ L ( Ω ) there holds (4.9) kA α r F k L q ( ∆ ,ω X ∆ L ) ≤ C kA α ′ r F k L q (3 ∆ ,ω X ∆ L ) , F ∈ L ( Ω ∩ B ) , and (4.10) kN α r F k L q ( ∆ ,ω X ∆ L ) ≤ C kN α ′ r F k L q (4 ∆ ,ω X ∆ L ) , F ∈ C ( Ω ∩ B ) . Proof.
We start with (4.9) and borrow some ideas from [43, Proposition 3.2]. We may assume that α > α ′ , otherwise the desired estimate follows trivially. Let v ∈ A ∞ ( ∂ Ω , ω L ). By the classical theoryof weights (cf. [10, 22]), we can find p ∈ (1 , ∞ ) such for every ∆ as in the statement we have C : = sup ∆ [ v ] A p ( ∆ ,ω L ) : = sup ∆ sup ∆ ′ (cid:16) ∆ ′ v ( x ) d ω X ∆ L ( x ) (cid:17)(cid:16) ∆ ′ v ( x ) − p ′ d ω X ∆ L ( x ) (cid:17) p − < ∞ , where the sups are taken over all ∆ ′ = B ′ ∩ ∂ Ω with B ′ ⊂ B , B ′ = B ( x ′ , r ′ ), x ′ ∈ ∂ Ω , 0 < r ′ < diam( ∂ Ω ),and where C depends on [ v ] A ∞ ( ∂ Ω ,ω L ) . Note that for any such ∆ ′ and for any Borel set F ⊂ ∆ ′ wehave, by H ¨older’s inequality,(4.11) (cid:16) ω X ∆ L ( F ) ω X ∆ L ( ∆ ′ ) (cid:17) p = (cid:16) ∆ ′ F d ω X ∆ L (cid:17) p = (cid:16) ∆ ′ F v p v − p d ω X ∆ L (cid:17) p ≤ (cid:16) ∆ ′ F v d ω X ∆ L (cid:17)(cid:16) ∆ ′ v − p ′ d ω X ∆ L (cid:17) p − ≤ C (cid:16) ∆ ′ F v d ω X ∆ L (cid:17)(cid:16) ∆ ′ v d ω X ∆ L (cid:17) − = C ´ F v d ω X ∆ L ´ ∆ ′ v d ω X ∆ L . Let y ∈ ∆ and X ∈ Γ α r ( y ) and pick b x so that | X − b x | = δ ( X ). Then one can easily see that X ∈ B , δ ( X ) < r , y ∈ ∆ (cid:0)b x , min { (3 + α ) δ ( X ) , r } (cid:1) = : e ∆ , e B : = B (cid:0)b x , min { (3 + α ) δ ( X ) , r } (cid:1) ⊂ B . Then, by (4.11) and Lemma 3.26 we get ˆ e ∆ v d ω X ∆ L ≤ C (cid:16) ω X ∆ L ( e ∆ ) ω X ∆ L ( b ∆ ) (cid:17) p ˆ b ∆ v d ω X ∆ L . α,α ′ , p C ˆ b ∆ v d ω X ∆ L , where b ∆ : = ∆ ( b x , min { α ′ , } δ ( X )). Moreover, if X ∈ B with δ ( X ) < r and y ∈ b ∆ one can easily showthat | y − x | < r , | X − y | ≤ min { + α ′ , } δ ( X ) . If we now combine the previous estimates, then we conclude that kA α r F k L ( ∆ , v d ω X ∆ L ) = ˆ ∆ ¨ Γ α r ( y ) | F ( X ) | dX v ( y ) d ω X ∆ L ( y ) ≤ ¨ B ∩{ δ ( X ) < r } | F ( X ) | (cid:16) ˆ e ∆ v ( y ) d ω X ∆ L ( y ) (cid:17) dX . α,α ′ , p C ¨ B ∩{ δ ( X ) < r } | F ( X ) | (cid:16) ˆ b ∆ v ( y ) d ω X ∆ L ( y ) (cid:17) dX ≤ C ˆ ∆ ¨ Γ α ′ r ( y ) | F ( X ) | dX v ( y ) d ω X ∆ L ( y ) = C kA α ′ r F k L (3 ∆ , v d ω X ∆ L ) . We can now extrapolate (locally in 3 ∆ ) as in [11, Corollary 3.15] to conclude that kA α r F k L q ( ∆ , v d ω X ∆ L ) . α,α ′ , q kA α ′ r F k L (3 ∆ , v d ω X ∆ L ) . The desired estimate follows at once by taking v ≡ A ∞ ( ∂ Ω , ω L ).Let us next consider (4.10). First, introduce M ∆ ω L h ( z ) : = sup < s ≤ r ∆ ( z , s ) | h | d ω X ∆ L = sup < s ≤ r ∆ ( z , s ) | h | ∆ d ω X ∆ L , z ∈ ∆ . We proceed as in [35, Proposition 2.2] and write for any λ > β > E ( β, r , λ ) : = { y ∈ ∂ Ω : N β r F ( y ) > λ } . Let y ∈ E ( α, r , λ ) ∩ ∆ . Hence, there is X ∈ Γ α r ( y ) with | F ( X ) | > λ . Pick b x ∈ ∂ Ω so that | X − b x | = δ ( X ).Note that e ∆ : = ∆ ( b x , min { , α ′ } δ ( X )) ⊂ b ∆ : = ∆ ( y , min { (2 + α + α ′ ) δ ( X ) , r } ) and e ∆ ⊂ ∆ . One can easily see that if z ∈ e ∆ then X ∈ Γ α ′ r ( z ). Hence, e ∆ ⊂ E ( α ′ , r , λ ) ∩ b ∆ and M ∆ ω L E ( α ′ , r ,λ ) ( y ) ≥ ω X ∆ L ( E ( α ′ , r , λ ) ∩ b ∆ ) ω X ∆ L ( b ∆ ) ≥ ω X ∆ L ( e ∆ ) ω X ∆ L ( b ∆ ) > γ = γ α,α ′ , where in the last estimate we have used that ω X ∆ L ( b ∆ ) ≤ ω X ∆ L ( ∆ ( b x , min { (4 + α + α ′ ) δ ( X ) , r } )) . α,α ′ ω X ∆ L ( e ∆ ) . We have then shown that E ( α, r , λ ) ∩ ∆ ⊂ { y ∈ ∆ : M ∆ ω L E ( α ′ , r ,λ ) ( y ) > γ } , and by the Hardy-Littlewood maximal inequality we get ω X ∆ L ( E ( α, r , λ ) ∩ ∆ ) ≤ ω X ∆ L ( { y ∈ ∆ : M ∆ ω L E ( α ′ , r ,λ ) ( y ) > γ } ) . ω X ∆ L ( E ( α ′ , r , λ ) ∩ ∆ ) . ω X ∆ L ( E ( α ′ , r , λ ) ∩ ∆ ) . This readily implies (4.10). (cid:3)
LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 25
Proof of (a) p ′ = ⇒ (b) p . Fix α > N ≥
1. Take ∆ = ∆ ( x , r ) with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ) and fix f ∈ C ( ∂ Ω ) with supp f ⊂ N ∆ . We may assume that Nr < ∂ Ω ), otherwise ∂ Ω is bounded and 4 diam( ∂ Ω ) / N ≤ r < diam( ∂ Ω ) and we can work with N ′ = ∂ Ω ) / r ∈ (2 , N /
2] and N ′ ∆ = ∂ Ω .Let u be the associated elliptic measure L -solution as in (3.7). Assume ω L ∈ RH p ′ ( ∂ Ω , ω L ) and ourgoal is to obtain that (3.8) holds. By Gehring’s lemma [23] (see also [10]) there exists s > ω L ∈ RH p ′ s ( ∂ Ω , ω L ).Introduce the family of pairwise disjoint cubes F ∆ : = { Q ∈ D ( ∂ Ω ) : ( N + Ξ ) r < ℓ ( Q ) ≤ N + Ξ ) r , Q ∩ Ξ∆ , Ø } . Take x ∈ ∆ and X ∈ Γ α r ( x ). Let I X ∈ W be such that X ∈ I X . Take y X ∈ ∂ Ω such that dist( I X , ∂ Ω ) = dist( I X , y X ) and let Q X ∈ D be the unique dyadic cube satisfying ℓ ( Q X ) = ℓ ( I X ) and y X ∈ Q X . Byconstruction (see Section 2.3), I X ∈ W ϑ, ∗ Q X and thus I ∗ ⊂ Γ Q X ( y X ). Thus, by the properties of theWhitney cubes δ ( X ) ≤ | X − y X | ≤ diam( I X ) + dist( I X , y X ) ≤
54 dist( I X , ∂ Ω ) ≤ δ ( X )and 4 ℓ ( Q X ) = ℓ ( I X ) ≤ dist( I X , ∂ Ω ) ≤ δ ( X ) ≤
54 dist( I X , ∂ Ω ) ≤ √ n + ℓ ( I X ) = √ n + ℓ ( Q X ) . These and the fact that X ∈ Γ α r ( x ) give ℓ ( Q X ) < δ ( X ) ≤ | X − x | < r . Also, for every z ∈ Q X | z − x | ≤ | z − y X | + | y X − X | + | X − x | + | x − x | < Ξ ℓ ( Q X ) + | X − x | + r < ( Ξ + r ≤ Ξ r , since Ξ ≥
2, and | z − x | ≤ | z − y X | + | y X − X | + | X − x | < Ξ ℓ ( Q X ) + (3 + α ) δ ( X ) < (2 Ξ + α ) δ ( X ) = : C α δ ( X ) , since X ∈ Γ α r ( x ). Thus, Q X ⊂ Ξ∆ ∩ ∆ ( x , C α δ ( X )) and there exists a unique e Q X ∈ F ∆ such that Q X ( e Q X . In particular, X ∈ I X ⊂ U Q X ⊂ Γ e Q X ( y ) for all y ∈ Q X and | u ( X ) | ≤ N e Q X u ( y ) , for all y ∈ Q X . Taking the average over Q X with respect to ω X ∆ L we arrive at | u ( X ) | ≤ Q X N e Q X u ( y ) d ω X ∆ L ( y ) ≤ Q X sup Q ∈F ∆ N Q u ( y ) d ω X ∆ L ( y ) . α ∆ ( x , C α δ ( X )) sup Q ∈F ∆ N Q u ( y ) d ω X ∆ L ( y ) ≤ sup < r ≤ C α r ∆ ( x , r ) sup Q ∈F ∆ N Q u ( y ) d ω X ∆ L ( y ) , where in the last inequality we have used that δ ( X ) ≤ | X − x | < r since Γ α r ( x ) ⊂ B ( x , r ). Taking nowthe supremum over all X ∈ Γ α r ( x ), we arrive at N α r u ( x ) . α sup < r ≤ C α r ∆ ( x , r ) sup Q ∈F ∆ N Q u ( y ) d ω X ∆ L ( y ) , for all x ∈ ∆ . Applying the Hardy-Littlewood maximal inequality and the fact that the set F ∆ has bounded cardinal-ity, we have (4.12) kN α r u k L p ( ∆ ,ω X ∆ L ) . α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup < r ≤ C α r ∆ ( · , r ) sup Q ∈F ∆ N Q u ( y ) d ω X ∆ L ( y ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( ∆ ,ω X ∆ L ) . (cid:13)(cid:13)(cid:13) sup Q ∈F ∆ N Q u (cid:13)(cid:13)(cid:13) L p ( ∆ ,ω X ∆ L ) . sup Q ∈F ∆ kN Q u k L p ( ∆ ,ω X ∆ L ) ≈ N sup Q ∈F ∆ kN Q u k L p ( Q ,ω X ∆ L ) , where we have used that for every Q ∈ F ∆ we have supp( N Q u ) ⊂ Q .Let us also observe that for every Q ∈ F ∆ we can pick y Q ∈ Q ∩ Ξ∆ so that if z ∈ N ∆ there holds | z − x Q | ≤ | z − x | + | x − y Q | + | y Q − x Q | ≤ ( N + Ξ ) r + Ξ r Q < Ξ r Q . That is, N ∆ ⊂ e ∆ Q and we are now ready to invoke [1, Lemma 3.20] to see that(4.13) N Q u ( x ) . sup ∆ ∋ x < r ∆ < Ξ r Q ∆ | f ( y ) | d ω X Q L ( y ) , x ∈ Q . To continue let x ∈ Q ∈ F ∆ and let ∆ be a surface ball such that x ∈ ∆ and 0 < r ∆ < Ξ r Q . Inparticular, ∆ ⊂ C N ∆ = e ∆ and Q ⊂ e ∆ . Note that ω X ∆ L ≈ N ω X e ∆ L by Harnack’s inequality and the factthat δ ( X ∆ ) ≈ r , δ ( X e ∆ ) ≈ N r , and | X ∆ − X e ∆ | . N r .Recall that ω L ∈ RH p ′ s ( ∂ Ω , ω L ) implies ω L ∈ RH p ′ s ( e ∆ , ω X e ∆ L ) (uniformly). Therefore, usingH ¨older’s inequality and recalling that h ( · ; L , L , X ) denotes the Radon-Nikodym derivative of ω XL withrespect to ω XL , we get ∆ | f ( y ) | d ω X ∆ L ( y ) ≈ N ω X e ∆ L ( ∆ ) ω X e ∆ L ( ∆ ) ∆ | f ( y ) | h ( y ; L , L , X e ∆ ) d ω X e ∆ L ( y ) ≤ ω X e ∆ L ( ∆ ) ω X e ∆ L ( ∆ ) (cid:18) ∆ h ( y ; L , L , X e ∆ ) p ′ s d ω X e ∆ L ( y ) (cid:19) p ′ s (cid:18) ∆ | f ( y ) | ( p ′ s ) ′ d ω X e ∆ L ( y ) (cid:19) p ′ s ) ′ . ω X e ∆ L ( ∆ ) ω X e ∆ L ( ∆ ) ∆ h ( y ; L , L , X e ∆ ) d ω X e ∆ L ( y ) (cid:18) ∆ | f ( y ) | ( p ′ s ) ′ d ω X e ∆ L ( y ) (cid:19) p ′ s ) ′ = (cid:18) ∆ | f ( y ) | ( p ′ s ) ′ d ω X e ∆ L ( y ) (cid:19) p ′ s ) ′ . This, (4.13), and (4.12) yield kN α r u k pL p ( ∆ ,ω X ∆ L ) . α, N sup Q ∈F ∆ ˆ e ∆ (cid:18) sup ∆ ∋ x < r ∆ < Ξ r Q ∆ | f ( y ) | ( p ′ s ) ′ d ω X ∆ L ( y ) (cid:19) p ( p ′ s ) ′ d ω X e ∆ L ( x ) . ˆ e ∆ | f ( x ) | p d ω X e ∆ L ( x ) ≈ N k f k pL p ( N ∆ ,ω X ∆ L ) , where we have used the boundedness of the local Hardy-Littlewood maximal function in the secondterm on L p ( p ′ s ) ′ ( e ∆ , ω X e ∆ L ), which follows from p > ( p ′ s ) ′ and the fact that ω X e ∆ L is doubling in 10 e ∆ .This completes the proof of (b) p . (cid:3) LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 27
Proof of (b) p = ⇒ (a) p ′ . Fix p ∈ (1 , ∞ ) and assume that L is L p ( ω L )-solvable. That is, for somefixed α and some N ≥ C α , N ≥ n , the 1-sided NTA constants, theCDC constant, the ellipticity of L and L , α , N , and p ) such that (3.8) holds for u as in (3.7) for any f ∈ C ( ∂ Ω ) with supp f ⊂ N ∆ . From this and (4.10) we conclude that we can assume that α ≥ c − − c is the Corkscrew constant (cf. Definition 2.1), and we have(4.14) kN α r u k L p ( ∆ ,ω X ∆ L ) . α,α kN α r u k L p (4 ∆ ,ω X ∆ L ) ≤ C α , N k f k L p ( N ∆ ,ω X ∆ L ) , for u as in (3.7) with f ∈ C ( ∂ Ω ) with supp f ⊂ N ∆ and for any ∆ = ∆ ( x , r ), x ∈ ∂ Ω and0 < r < diam( ∂ Ω ) /
4. It is routine to see this estimate also holds with r ≈ diam( ∂ Ω ). Indeed,by splitting f into its positive and negative part we may assume that f ≥
0. In that case if x ∈ ∂ Ω and X ∈ Γ α r ( x ) \ Γ α diam( ∂ Ω ) / ( x ) we have that δ ( X ) ≈ diam( ∂ Ω ) and by (2.31) one has that X ′ : = X ∆ ( x , diam( ∂ Ω ) / ∈ Γ α diam( ∂ Ω ) / ( x ). Harnack’s inequality implies then that u ( X ) ≈ u ( X ′ ) and this showsthat N α r u ( x ) . N α diam( ∂ Ω ) / u ( x ). Further details are left to the interested reader.We claim that for every ∆ = ∆ ( x , r ), x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ), and for every f ∈ C ( ∂ Ω )with supp f ⊂ N ∆ (4.15) (cid:12)(cid:12)(cid:12) ˆ ∆ f ( y ) d ω X ∆ L ( y ) (cid:12)(cid:12)(cid:12) . α, N k f k L p ( N ∆ ,ω X ∆ L ) . To see this let u be the L -solution with datum | f | (see (3.7)). Write X : = X ∆ and e X : = X (2 + α ) − ∆ .Note that δ ( X ) ≈ r , δ ( e X ) ≈ α r , and | X − e X | < r . Hence Harnack’s inequality yields u ( e X ) ≈ α u ( X ). The choice of α guarantees that e X ∈ Γ α (2 + α ) − r ( x ) ⊂ Γ α r ( x ), see (2.31). Let e x ∈ ∂ Ω so that δ ( e X ) = | e X − e x | . Clearly, for every z ∈ ∆ ( e x , αδ ( e X )), | e X − z | ≤ | e X − e x | + | e x − z | < (1 + α ) δ ( e X ) ≤ + α + α r < r , thus e X ∈ Γ α r ( z ) and N α r u ( z ) ≥ u ( e X ) ≈ α u ( X ) , for every z ∈ ∆ ( e x , αδ ( e X )) . Note also that if z ∈ ∆ ( e x , αδ ( e X )) then | z − x | ≤ | z − e x | + | e x − e X | + | e X − x | < ( α + δ ( e X ) + | e X − x | ≤ ( α + | e X − x | ≤ r , hence ∆ ( e x , αδ ( e X )) ⊂ ∆ . Additionally, if z ∈ ∆ then | z − e x | ≤ | z − x | + | x − e X | + | e X − e x | < r + | x − e X | + δ ( e X ) ≤ r + | x − e X | ≤ (cid:16) + + α (cid:17) r ≤ r , and this shows that ∆ ⊂ ∆ ( e x , r ). This together with Lemma 3.26 gives1 . ω X L ( ∆ ) ≤ ω X L ( ∆ ( e x , r )) . α ω X L ( ∆ ( e x , α c r / (2 + α ))) ≤ ω X L ( ∆ ( e x , αδ ( e X )))and the previous estimates readily give (4.15): (cid:12)(cid:12)(cid:12) ˆ ∆ f ( y ) d ω X ∆ L ( y ) (cid:12)(cid:12)(cid:12) ≤ u ( X ) . α u ( e X ) ω X L ( ∆ ( e x , αδ ( e X ))) p ≤ kN α r u k L p ( ∆ ,ω X L ) . α, N k f k L p ( N ∆ ,ω X L ) . To proceed we fix ∆ = ∆ ( x , r ), x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ) /
2. Let F ⊂ ∆ be a Borel set.Since ω X ∆ L and ω X ∆ L are Borel regular, for each ε >
0, there exist compact set K and an open set U such that K ⊂ F ⊂ U ⊂ ∆ satisfying(4.16) ω X ∆ L ( U \ K ) + ω X ∆ L ( U \ K ) < ε. Using Urysohn’s lemma we can construct f F ∈ C c ( ∂ Ω ) such that K ≤ f F ≤ U . Then, by (4.15)(applied with 2 ∆ ) and (4.16) yield ω X ∆ L ( F ) < ε + ω X ∆ L ( K ) ≤ ε + ˆ ∂ Ω f F ( z ) d ω X ∆ L ( z ) ≤ ε + C α, N k f F k L p ( ∆ ,ω X ∆ L ) . ε + C α, N ω X ∆ L ( U ) p < ε + C α, N ( ω X ∆ L ( F ) + ε ) p . Letting ε → + , we obtain that ω X ∆ L ( F ) . α, N ω X ∆ L ( F ) p . Hence, ω X ∆ L ≪ ω X ∆ L in ∆ . By Harnack’sinequality and the fact that we can cover ∂ Ω with surface balls like ∆ we conclude that ω L ≪ ω L in ∂ Ω . We can write h ( · ; L , L , X ) = d ω XL d ω XL ∈ L ( ∂ Ω , ω XL ) which is well-defined ω XL -a.e. in ∂ Ω . Thus, forevery f ∈ C ( ∂ Ω ) with supp f ⊂ ∆ we obtain from (4.15) (cid:12)(cid:12)(cid:12) ˆ ∆ f ( y ) h ( y ; L , L , X ∆ ) d ω X ∆ L ( y ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˆ ∆ f ( y ) d ω X ∆ L ( y ) (cid:12)(cid:12)(cid:12) . α, N k f k L p (2 ∆ ,ω X ∆ L ) . Using the ideas in [1, Lemma 3.38] and with the help of [1, Lemma 3.29], we can then conclude that k h ( · ; L , L , X ∆ ) k L p ′ ( ∆ ,ω X ∆ L ) . α, N . This, Harnack’s inequality, and the fact that ∆ = ∆ ( x , r ) with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ) / k h ( · ; L , L , X ∆ ( x , r ) ) k L p ′ ( ∆ ( x , r ) ,ω X ∆ ( x , r ) L ) . α, N , for every x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ) . This and Remark 3.30 readily imply that ω L ∈ RH p ′ ( ∂ Ω , ω L ) and the proof is complete. (cid:3) Proof of (b) p = ⇒ (b) ′ p . Assume that L is L p ( ω L )-solvable with p ∈ (1 , ∞ ). Fix α > N ≥
1, asurface ball ∆ , and a Borel set S ⊂ N ∆ . Take an arbitrary ε > ω X ∆ L and ω X ∆ L are Borelregular, we can find a closed set F and an open set U such that F ⊂ S ⊂ U ⊂ ( N + ∆ and ω X ∆ L ( U \ F ) + ω X ∆ L ( U \ F ) < ε. Using Urysohn’s lemma we can then construct f ∈ C c ( ∂ Ω ) such that S ≤ f ≤ U . Set, u ( X ) : = ω XL ( S ) , v ( X ) : = ˆ ∂ Ω f ( y ) d ω XL ( y ) , X ∈ Ω . For every M ≥ c − define the truncated cone and truncated non-tangential maximal function Γ α r , M ( x ) : = Γ α r ( x ) ∩ { X ∈ Ω : δ ( X ) ≥ r / M } , N α r , M u ( x ) : = sup X ∈ Γ α r , M ( x ) | u ( X ) | , x ∈ ∂ Ω . Note that if x ∈ ∆ and X ∈ Γ α r , M ( x ) then r / M ≤ δ ( X ) ≤ r , c r ≤ δ ( X ∆ ) ≤ r , and | X − X ∆ | < r .Hence, by the Harnack chain condition and Harnack’s inequality, there is a constant C M depending on M such that ω XL ( U \ F ) ≤ C M ω X ∆ L ( U \ F ) ≤ C M ε, and 0 ≤ u ( X ) = ω XL ( S ) ≤ C M ε + ω XL ( F ) ≤ C M ε + ˆ ∂ Ω f ( y ) d ω XL ( y ) = C M ε + v ( X ) . Thus N α r , M u ( x ) ≤ C M ε + N α r v ( x ) , ∀ x ∈ ∆ . LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 29
Note that our assumption is that L p ( ω L )-solvability holds with the fixed parameters α > N ≥ ⇐⇒ (b) it follows that the L p ( ω L )-solvability holds with α > N +
1. Thus, the fact that f ∈ C c ( ∂ Ω ) with supp f ⊂ U ⊂ ( N + ∆ gives kN α r , M u k L p ( ∆ ,ω X ∆ L ) ≤ C M ε ω X ∆ L ( ∆ ) p + kN α r v k L p ( ∆ ,ω X ∆ L ) ≤ C M ε + C α, N k f k L p (( N + ∆ ,ω X ∆ L ) ≤ C M ε + C α, N ω X ∆ L ( U ) p < C M ε + C α, N ( ω X ∆ L ( S ) + ε ) p = C M ε + C α, N (cid:0) k S k pL p ( N ∆ ,ω X ∆ L ) + ε ) p . We let ε → + and obtain kN α r , M u k L p ( ∆ ,ω X ∆ L ) ≤ C α, N k S k L p ( N ∆ ,ω X ∆ L ) . Since N α r , M u ( x ) ր N α r u ( x )for every x ∈ ∂ Ω as M → ∞ we conclude the desired estimate by simply applying the monotoneconvergence theorem. (cid:3) Proof of (b) ′ = ⇒ (a) . Fix p ∈ (1 , ∞ ) and assume that L is L p ( ω L )-solvable for characteristicfunctions. That is for some α > N ≥ C α, N ≥ n , the1-sided NTA constants, the CDC constant, the ellipticity of L and L , α , N , and p ) such that (3.8) holdsfor u as in (3.7) for any f = S with S being a Borel set S ⊂ N ∆ .Take an arbitrary ∆ = ∆ ( x , r ), x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ). We follow the proof of(b) p = ⇒ (a) p ′ and observe that the same argument we used to obtain (4.15) easily gives, taking f = S with S being a Borel set S ⊂ N ∆ , that(4.17) ω X ∆ L ( S ) = ˆ ∆ S ( y ) d ω X ∆ L ( y ) . α, N k S k L p ( ∆ ,ω X ∆ L ) = ω X ∆ L ( S ) p . This readily implies that ω X ∆ L ≪ ω X ∆ L in ∆ , and since ∆ is arbitrary we conclude that ω L ≪ ω L in ∂ Ω . To proceed, fix B = B ( x , r ) and B = B ( x , r ) with B ⊂ B , x , x ∈ ∂ Ω and 0 < r , r < diam( ∂ Ω ).Write ∆ = B ∩ ∂ Ω and ∆ = B ∩ ∂ Ω . Let S ⊂ ∆ be an arbitrary Borel set. If r ≈ r we have byHarnack’s inequality and Lemma 3.26 part ( a ) ω X ∆ L ( S ) ω X ∆ L ( ∆ ) ≈ ω X ∆ L ( S ) ω X ∆ L ( ∆ ) ≈ ω X ∆ L ( S ) . α, N ω X ∆ L ( S ) p ≈ (cid:16) ω X ∆ L ( S ) ω X ∆ L ( ∆ ) (cid:17) p ≈ (cid:16) ω X ∆ L ( S ) ω X ∆ L ( ∆ ) (cid:17) p , where in the third estimate we have used (4.17) with ∆ in place of ∆ . On the other hand, if r ≪ r wehave by Lemma 3.26 part ( d ) that ω L ≪ ω L with ω X ∆ L ( S ) ω X ∆ L ( ∆ ) ≈ ω X ∆ L ( S ) . α, N ω X ∆ L ( S ) p ≈ (cid:16) ω X ∆ L ( S ) ω X ∆ L ( ∆ ) (cid:17) p , where again we have used (4.17) with ∆ in place of ∆ in the middle estimate. In short we have provedthat ω X ∆ L ( S ) ω X ∆ L ( ∆ ) . α, N (cid:16) ω X ∆ L ( S ) ω X ∆ L ( ∆ ) (cid:17) p , for any Borel set S ⊂ ∆ . Using the fact that the implicit constants do not depend on ∆ (nor on ∆ ) and Lemma 3.26 part ( c ), thisreadily implies that ω X ∆ L ∈ RH q ( ∆ , ω X ∆ L ) for some q ∈ (1 , ∞ ) where q and the implicit constants do notdepend on ∆ , see [10, 22]. Hence, we readily conclude that ω L ∈ RH q ( ∂ Ω , ω L ) (see Definition 3.3).This completes the proof of the present implication. (cid:3) Proof of (a) = ⇒ (d) . Assume that ω L ∈ A ∞ ( ∂ Ω , ω L ). By the classical theory of weights(cf. [10, 22]) and Lemma 3.26 part ( c ) it is not hard to see that ω L ∈ A ∞ ( ∂ Ω , ω L ), hence ω L ∈ RH p ( ∂ Ω , ω L ) for some 1 < p < ∞ . In particular for every Q ∈ D ( ∂ Ω ) and Q ∈ D Q , by Lemma 3.26part ( c ) we have (cid:18) Q h ( y ; L , L , X Q ) p d ω X Q L ( y ) (cid:19) p ≤ C Q h ( y ; L , L , X Q ) d ω X Q L ( y ) = C ω X Q L ( Q ) ω X Q L ( Q ) . Thus, for F ⊂ Q we obtain, by H ¨older’s inequality,(4.18) ω X Q L ( F ) ω X Q L ( Q ) = Q F ( y ) d ω X Q L ( y ) = ω X Q L ( Q ) ω X Q L ( Q ) Q F ( y ) h ( y ; L , L , X Q ) d ω X Q L ( y ) ≤ ω X Q L ( Q ) ω X Q L ( Q ) (cid:16) Q h ( y ; L , L , X Q ) p d ω X Q L ( y ) (cid:17) p (cid:16) ω X Q L ( F ) ω X Q L ( Q ) (cid:17) p ′ . (cid:16) ω X Q L ( F ) ω X Q L ( Q ) (cid:17) p ′ . To continue we need a dyadic version of (3.12): for every Q ∈ D ( ∂ Ω ) and for every ϑ ≥ ϑ weclaim that(4.19) kS ϑ Q u k L q ( Q ,ω XQ L ) ≤ C ϑ kN ϑ Q u k L q ( Q ,ω XQ L ) , < q < ∞ . This estimate can be proved following the argument in [1, Section 5.2] with the following changes.Recall [1, (5.9)] (here we note that the argument in [1, Section 5.2] was done with a fixed value of ϑ su ffi ciently large, but it is routine to see that one can repeat it with this parameter with harmlesschanges) ω X Q L (cid:0)(cid:8) x ∈ Q j : S ϑ, k Q j u ( x ) > β λ, N ϑ Q u ( x ) ≤ γ λ (cid:9)(cid:1) . (cid:16) γβ (cid:17) ϑ ω X Q L ( Q j ) , (4.20)where λ , β , γ > Q j is some dyadic cube (see [1, Section 5.2]); S ϑ, k Q j u is a truncated localized dyadicconical square function with respect to the cones Γ ϑ, k Q j ( x ) : = [ x ∈ Q ′ ∈ D Q ℓ ( Q ′ ) ≥ − k ℓ ( Q ) U ϑ Q ′ ;and k is large enough (eventually k → ∞ ). It should be noted that the implicit constant in theinequality (4.20) does not depend on k . Combining (4.20) with (4.18) we easily arrive at ω X Q L (cid:0)(cid:8) x ∈ Q j : S ϑ, k Q j u ( x ) > β λ, N ϑ Q u ( x ) ≤ γ λ (cid:9)(cid:1) . (cid:16) γβ (cid:17) ϑ p ′ ω X Q L ( Q j ) . (4.21)From this we can derive [1, (5.7)] with ω X Q L in place of ω X Q L and a typical good- λ argument much asin [1, Section 5.2] readily leads to (4.19).With (4.19) at our disposal we can then proceed to obtain (3.12). Fix ∆ = ∆ ( x , r ) with x ∈ ∂ Ω , < r < diam( ∂ Ω ). Let M ≥ F ∆ : = (cid:8) Q ∈ D ( ∂ Ω ) : r / (2 M ) ≤ ℓ ( Q ) < r / M , Q ∩ ∆ , Ø (cid:9) . One has that F ∆ is a pairwise disjoint family and ∆ ⊂ [ Q ∈F ∆ Q ⊂ ∆ , provided M is large enough. LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 31
Write e r : = r / M . Let x ∈ Q ∈ F ∆ and X ∈ Γ α e r ( x ). Let I X ∈ W be so that I X ∋ X and pick Q X ∈ D ( ∂ Ω ) with x ∈ Q X and ℓ ( Q X ) = ℓ ( I X ). Note that ℓ ( Q X ) = ℓ ( I X ) ≤ diam( I X ) ≤ dist( I X , ∂ Ω ) ≤ δ ( X ) ≤ | X − x | < e r = r M ≤ ℓ ( Q ) . This and the fact that x ∈ Q ∩ Q X gives Q X ⊂ Q . On the other hand,dist( I X , Q X ) ≤ | X − x | ≤ (1 + α ) δ ( X ) ≤ (1 + α )(diam( I X ) + dist( I X , ∂ Ω )) ≤ √ n + + α ) ℓ ( I X ) = √ n + + α ) ℓ ( Q X ) . This shows that if we fix ϑ = ϑ ( α ) so that 2 ϑ ≥ √ n + + α ) then I X ∈ W ϑ Q X ⊂ W ϑ, ∗ Q X . As a result, X ∈ I X ⊂ U ϑ Q X and X ∈ Γ ϑ Q ( x ). All these show that for every Q ∈ F ∆ and x ∈ Q ∈ F ∆ we have Γ α e r ( x ) ⊂ Γ ϑ Q ( x ). Thus (4.19) yields kS α e r u k qL q ( ∆ ,ω X ∆ L ) ≤ X Q ∈F ∆ ˆ Q S α e r u ( x ) q d ω X ∆ L ( x ) ≤ X Q ∈F ∆ ˆ Q S ϑ Q u ( x ) q d ω X ∆ L ( x ) . α X Q ∈F ∆ ˆ Q N ϑ Q u ( x ) q d ω X ∆ L ( x ) . To continue let Q ∈ F ∆ , x ∈ Q and X ∈ Γ ϑ, ∗ Q ( x ). Then X ∈ I ∗∗ with I ∈ W ϑ, ∗ Q and x ∈ Q ⊂ Q . As aconsequence, | X − x | ≤ diam( I ∗∗ ) + dist( I , Q ) + diam( Q ) . ϑ ℓ ( I ) ≈ δ ( X ) ≤ κ ℓ ( Q ) < κ e r where we have used (2.23) and the last estimate holds provided M is large enough. This shows that X ∈ Γ α ′ κ e r ( x ) for some α ′ = α ′ ( ϑ ) (hence depending on α ). As a consequence of these, we obtain X Q ∈F ∆ ˆ Q N ϑ Q u ( x ) q d ω X ∆ L ( x ) ≤ ˆ ∆ N α ′ κ e r u ( x ) q d ω X ∆ L ( x ) . α ˆ ∆ N α κ e r u ( x ) q d ω X ∆ L ( x ) ≤ ˆ ∆ N α r u ( x ) q d ω X ∆ L ( x ) , where we have used (4.10) and the last estimate follows provided M is large enough. (cid:3) Proof of (d) = ⇒ (d) ′ . This is trivial since for any arbitrary Borel set S ⊂ ∂ Ω , the solution u ( X ) = ω XL ( S ), X ∈ Ω , belongs to u ∈ W , ( Ω ). (cid:3) Proof of (d) ′ = ⇒ (a) . Assume that (3.12) holds for some fixed α and q ∈ (0 , ∞ ) and for u ( X ) = ω XL ( S ), X ∈ Ω , for any arbitrary Borel set S ⊂ ∂ Ω . By Lemma 4.8 (applied to F ( X ) = |∇ u ( X ) | δ ( X ) (1 − n ) / ), for any α large enough to be chosen we have(4.22) kS α r u k L q ( ∆ ,ω X ∆ L ) . α,α kS α r u k L q (3 ∆ ,ω X ∆ L ) . α ω X ∆ L (15 ∆ ) q ≈ ω X ∆ L ( ∆ ) q , for every ∆ = ∆ ( x , r ) with x ∈ ∂ Ω , < r < diam( ∂ Ω ) /
3, and where we have used that 0 ≤ u ≤ ∂ Ω is bounded, to any diam( ∂ Ω ) / ≤ r < diam( ∂ Ω ). Note that if x ∈ ∆ and X ∈ Γ α diam( ∂ Ω ) ( x ) \ Γ α diam( ∂ Ω ) / ( x ), then14 diam( ∂ Ω ) ≤ | X − x | ≤ (1 + α ) δ ( X ) ≤ (1 + α ) | X − x | < (1 + α ) diam( ∂ Ω ) . Set W x = { I ∈ W : I ∩ ( Γ α diam( ∂ Ω ) ( x ) \ Γ α diam( ∂ Ω ) / ( x )) , Ø } , whose cardinality is uniformly bounded(depending in dimension and α ). Thus, since k u k L ∞ ( Ω ) ≤
1, Caccioppoli’s inequality gives ¨ Γ α diam( ∂ Ω ) ( x ) \ Γ α diam( ∂ Ω ) / ( x ) |∇ u ( X ) | δ ( X ) − n dX . X I ∈W x ℓ ( I ) − n ¨ I |∇ u ( X ) | dX . X I ∈W x ℓ ( I ) − − n ¨ I ∗ | u ( X ) | dX . W x . α . With this in hand and (4.22) applied with r = diam( ∂ Ω ) / < diam( ∂ Ω ) /
3, we readily obtain kS α r u k L q ( ∆ ,ω X ∆ L ) ≤ kS α diam( ∂ Ω ) u k L q ( ∆ ,ω X ∆ L ) ≤ kS α diam( ∂ Ω ) u − S α diam( ∂ Ω ) / u k L q ( ∆ ,ω X ∆ L ) + kS α diam( ∂ Ω ) / u k L q ( ∆ ,ω X ∆ L ) . ω X ∆ L ( ∆ ) q . We next see that given γ ∈ (0 ,
1) there exists β ∈ (0 ,
1) so that for every Q ∈ D ( ∂ Ω ) and for everyBorel set F ⊂ Q we have(4.23) ω X Q L ( F ) ω X Q L ( Q ) ≤ β = ⇒ ω X Q L ( F ) ω X Q L ( Q ) ≤ γ. Indeed, fix γ ∈ (0 ,
1) and Q ∈ D ( ∂ Ω ), and take a Borel set F ⊂ Q so that ω X Q L ( F ) ≤ βω X Q L ( Q ),where β ∈ (0 ,
1) is small enough to be chosen. Applying Lemma 4.4, if we assume that 0 < β < β ,then u ( X ) = ω XL ( S ) satisfies (4.5) and therefore C − q η log ( β − ) q ω X Q L ( F ) ≤ ˆ F S ϑ Q ,η u ( x ) q d ω X Q L ( x ) ≤ ˆ Q S ϑ Q ,η u ( x ) q d ω X Q L ( x ) . (4.24)We claim that there exists α = α ( ϑ , η ) (hence, depending on the allowable parameters) such that(4.25) Γ ϑ Q ,η ( x ) ⊂ Γ α r ∗ Q ( x ) , x ∈ Q , with r ∗ Q = κ r Q (cf. (2.23)). To see this, let x ∈ Q and X ∈ Γ ϑ Q ,η ( x ). Then X ∈ I ∗ for some I ∈ W ϑ , ∗ Q ′ , where Q ′ ⊂ Q ∈ D Q with Q ∋ x and ℓ ( Q ′ ) > η ℓ ( Q ). Then X ∈ T ϑ , ∗ Q ⊂ B ∗ Q ∩ Ω (see(2.23)) and | X − x | ≤ | X − x Q | + | x Q − x | < κ r Q + Ξ r Q ≤ κ r Q : = r ∗ Q , and also | X − x | ≤ diam( I ∗ ) + dist( I , Q ′ ) + diam( Q ) . ϑ ,η ℓ ( I ) ≈ δ ( X ) . Hence, there exists α = α ( ϑ , η ) such that X ∈ Γ α r ∗ Q ( x ), that is, (4.25) holds.To continue, observe first that by (2.14) and the fact that κ ≥ Ξ (cf. (2.23)) we have Q ⊂ ∆ ∗ Q .This, (4.25), Harnack’s inequality, (4.22), and Lemma 3.26 imply(4.26) ˆ Q S ϑ Q ,η u ( x ) q d ω X Q L ( x ) . ˆ ∆ ∗ Q S α r ∗ Q u ( x ) q d ω X Q L ( x ) ≈ ˆ ∆ ∗ Q S α r ∗ Q u ( x ) q d ω X ∆ ∗ Q L ( x ) . α ω X ∆ ∗ Q L (2 ∆ ∗ Q ) ≈ ω X Q L ( Q ) . Combining (4.24) and (4.26) we conclude that ω X Q L ( F ) ω X Q L ( Q ) ≤ C η,ϑ , q log ( β − ) − q . This readily gives (4.23) by choosing β small enough so that C η,ϑ , q log ( β − ) − q < γ . LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 33
Next, we show that (4.23) implies ω L ∈ A ∞ ( ∂ Ω , ω L ). To see this we first obtain a dyadic- A ∞ condition. Fix Q , Q ∈ D with Q ⊂ Q . Remark 3.27 gives for every F ⊂ Q (4.27) 1 C ω X Q L ( F ) ω X Q L ( Q ) ≤ ω X Q L ( F ) ω X Q L ( Q ) ≤ C ω X Q L ( F ) ω X Q L ( Q ) and 1 C ω X Q L ( F ) ω X Q L ( Q ) ≤ ω X Q L ( F ) ω X Q L ( Q ) ≤ C ω X Q L ( F ) ω X Q L ( Q ) , for some C >
1. Thus, given γ ∈ (0 , β ∈ (0 ,
1) so that (4.23) holds with γ/ C in place of γ . Then,(4.28) ω X Q L ( F ) ω X Q L ( Q ) ≤ β C = ⇒ ω X Q L ( F ) ω X Q L ( Q ) ≤ β = ⇒ ω X Q L ( F ) ω X Q L ( Q ) ≤ γ C = ⇒ ω X Q L ( F ) ω X Q L ( Q ) ≤ γ. To complete the proof we need to see that (4.28) gives ω L ∈ A ∞ ( ∂ Ω , ω L ). Fix γ ∈ (0 ,
1) and asurface ball ∆ = B ∩ ∂ Ω , with B = B ( x , r ), x ∈ ∂ Ω , and 0 < r < diam( ∂ Ω ). Take an arbitrarysurface ball ∆ = B ∩ ∂ Ω centered at ∂ Ω with B = B ( x , r ) ⊂ B , and let F ⊂ ∆ be a Borel set suchthat ω X ∆ L ( F ) > γω X ∆ L ( ∆ ). Consider the pairwise disjoint family F = { Q ∈ D : Q ∩ ∆ , Ø , r Ξ <ℓ ( Q ) ≤ r Ξ } where Ξ is the constant in (2.14). In particular, ∆ ⊂ S Q ∈F Q ⊂ ∆ . The pigeon-holeprinciple yields that there is a constant C ′ > ω X ∆ L so that ω X ∆ L ( F ∩ Q ) /ω X ∆ L ( Q ) > γ/ C ′ for some Q ∈ F . Let Q ∈ D be the unique dyadic cube such that Q ⊂ Q and r < ℓ ( Q ) ≤ r . We can then invoke the contrapositive of (4.28) with γ/ C ′ in place of γ to find β ∈ (0 ,
1) such that by Lemma 3.26, and Harnack’s inequality we arrive at ω X ∆ L ( F ) ω X ∆ L ( ∆ ) ≥ ω X ∆ L ( F ∩ Q ) ω X ∆ L ( ∆ ) ≈ ω X ∆ L ( F ∩ Q ) ω X ∆ L ( Q ) ≈ ω X Q L ( F ∩ Q ) ω X Q L ( Q ) > β C . In short, we have obtained that for every γ ∈ (0 ,
1) there exists e β ∈ (0 ,
1) such that ω X ∆ L ( F ) ω X ∆ L ( ∆ ) > γ = ⇒ ω X ∆ L ( F ) ω X ∆ L ( ∆ ) > e β. This and the classical theory of weights (cf. [10, 22]) show that ω L ∈ A ∞ ( ∂ Ω , ω L ), and the proof iscomplete. (cid:3) Proof of (c) = ⇒ (c) ′ . This is trivial since for any arbitrary Borel set S ⊂ ∂ Ω , the solution u ( X ) = ω XL ( S ), X ∈ Ω , belongs to W , ( Ω ) ∩ L ∞ ( Ω ). (cid:3) Proof of (e) = ⇒ (f) . Let ∆ ε = B ε ∩ ∂ Ω , ∆ ′ = B ′ ∩ ∂ Ω , where B ε = B ( x , ε r ) with x ∈ ∂ Ω and0 < r < diam( ∂ Ω ), and B ′ = B ( x ′ , r ′ ) with x ′ ∈ ∆ ε and 0 < r ′ < ε rc /
4, and c is the Corkscrewconstant. Using (3.9) and Lemma 4.1 we easily obtain1 ω X ∆ ε L ( ∆ ′ ) ¨ B ′ ∩ Ω |∇ u ( X ) | G L ( X ∆ ε , X ) dX . k f k ∂ Ω ,ω L ) + | f ∆ , L | ω X ∆ ε L ( ∆ ′ ) ¨ B ′ ∩ Ω |∇ u L , Ω ( X ) | G L ( X ∆ ε , X ) dX . k f k ∂ Ω ,ω L ) + k f k L ∞ ( ∂ Ω ,ω L ) (cid:16) r ′ diam( ∂ Ω ) (cid:17) ρ . k f k ∂ Ω ,ω L ) + k f k L ∞ ( ∂ Ω ,ω L ) ε ρ . Taking the sup over B ε and B ′ we readily arrive at (3.10). (cid:3) Proof of (f) = ⇒ (c) ′ . We first observe that (f) applied with ε = B sup B ′ ω X ∆ L ( ∆ ′ ) ¨ B ′ ∩ Ω |∇ u ( X ) | G L ( X ∆ , X ) dX ≤ C (cid:0) k f k ∂ Ω ,ω L ) + ̺ (1) k f k L ∞ ( ∂ Ω ,ω L ) (cid:1) . k f k L ∞ ( ∂ Ω ,ω L ) , where ∆ = B ∩ ∂ Ω , ∆ ′ = B ′ ∩ ∂ Ω , and the sups are taken respectively over all balls B = B ( x , r ) with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ), and B ′ = B ( x ′ , r ′ ) with x ′ ∈ ∆ and 0 < r ′ < rc /
4, and c is theCorkscrew constant.With this in place we are now ready to establish (c) ′ . Take an arbitrary Borel set S ⊂ ∂ Ω and let u ( X ) = ω XL ( S ), X ∈ Ω . Fix X ∈ Ω and use that ω X L is Borel regular to see that for every j ≥ F j and an open set U j so that F j ⊂ S ⊂ U j and ω X L ( U j \ F j ) < j − . Using Urysohn’slemma we can construct f j ∈ C ( ∂ Ω ) such that F j ≤ f j ≤ U j , and for X ∈ Ω set v j ( X ) : = ˆ ∂ Ω f j ( y ) d ω XL ( y ) . It is straightforward to see that | S ( x ) − f j ( x ) | ≤ U j \ F j ( x ) for every x ∈ ∂ Ω , hence for every compactset K ⊂ Ω and for every X ∈ K we have by Harnack’s inequality | u ( X ) − v j ( X ) | ≤ ˆ ∂ Ω | S ( x ) − f j ( x ) | d ω XL ( x ) ≤ ω XL ( U j \ F j ) ≤ C K , X ω X L ( U j \ F j ) < C K , X j − . Thus v j −→ u uniformly on compacta in Ω . This together with Caccioppoli’s inequality readily implythat ∇ v j −→ ∇ u in L ( Ω ). In particular, ∇ v j −→ ∇ u in L ( K ) for every compact set K ⊂ Ω .Fix ∆ = B ∩ ∂ Ω , ∆ ′ = B ′ ∩ ∂ Ω , with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ), and B ′ = B ( x ′ , r ′ ) with x ′ ∈ ∆ and 0 < r ′ < rc /
4, and c is the Corkscrew constant. Let f j , ∆ , L : = ffl ∆ f j d ω X ∆ L and u L , Ω ( X ) : = ω XL ( ∂ Ω ), X ∈ Ω . For every compact set K ⊂ Ω we then have by (4.29) applied to each f j ω X ∆ L ( ∆ ′ ) ¨ K ∩ B ′ ∩ Ω |∇ u ( X ) | G L ( X ∆ , X ) dX = lim j →∞ ω X ∆ L ( ∆ ′ ) ¨ K ∩ B ′ ∩ Ω |∇ v j ( X ) | G L ( X ∆ , X ) dX . . Taking the sup over B and B ′ we then conclude that ( c ) ′ holds since by the maximum principle one has k u k L ∞ ( Ω ) = (cid:3) Proof of (e) ′ = ⇒ (f) ′ . The argument used to see that (e) = ⇒ (f) can be carried out in thepresent scenario with no changes. (cid:3) Proof of (f) ′ = ⇒ (c) ′ . Let f = S with S ⊂ ∂ Ω a Borel set such that ω XL ( S ) , X ∈ Ω . Note that k f k BMO( ∂ Ω ,ω L ) ≤ k f k L ∞ ( ∂ Ω ,ω L ) =
1. This and the fact that u ( X ) = ω XL ( S ), X ∈ Ω ,verifies k u k L ∞ ( Ω ) =
1, we readily see that (3.10) with ε = (cid:3) Proof of (c) ′ = ⇒ (a) . Let u ( X ) = ω XL ( S ), X ∈ Ω , for an arbitrary Borel set S ⊂ ∂ Ω . Let ϑ ≥ ϑ and η ∈ (0 , ˆ Q S ϑ Q ,η u ( x ) d ω X Q L ( x ) = ˆ Q (cid:18) ¨ Γ ϑ Q ,η ( x ) |∇ u ( Y ) | δ ( Y ) − n dY (cid:19) d ω X Q L ( x ) = ¨ B ∗ Q ∩ Ω |∇ u ( Y ) | δ ( Y ) − n (cid:18) ˆ Q Γ ϑ Q ,η ( x ) ( Y ) d ω X Q L ( x ) (cid:19) dY , LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 35 where we have used that Γ ϑ Q ,η ( x ) ⊂ T ϑ, ∗ Q ⊂ B ∗ Q ∩ Ω (see (2.23)), and Fubini’s theorem. To estimate theinner integral we fix Y ∈ B ∗ Q ∩ Ω and b y ∈ ∂ Ω such that | Y − b y | = δ ( Y ). We claim that(4.31) (cid:8) x ∈ Q : Y ∈ Γ ϑ Q ,η ( x ) (cid:9) ⊂ ∆ ( b y , C ϑ η − δ ( Y )) . To show this let x ∈ Q be such that Y ∈ Γ ϑ Q ,η ( x ). This means that there exists Q ∈ D Q such that x ∈ Q and Y ∈ U ϑ Q ,η . Hence, there is Q ′ ∈ D Q with ℓ ( Q ′ ) > η ℓ ( Q ) such that Y ∈ U ϑ Q ′ and consequently δ ( Y ) ≈ ϑ dist( Y , Q ′ ) ≈ ϑ ℓ ( Q ′ ). As a result, | x − b y | ≤ diam( Q ) + dist( Y , Q ′ ) + δ ( Y ) . ϑ ℓ ( Q ) + δ ( Y ) . ϑ η − δ ( Y ) , thus x ∈ ∆ ( b y , C ϑ η − δ ( Y )) as desired. If we now use (4.31) we conclude that for every Y ∈ B ∗ Q ∩ Ω (4.32) ˆ Q Γ ϑ Q ,η ( x ) ( Y ) d ω X Q L ( x ) ≤ ω X Q L (cid:0) ∆ ( b y , C ϑ η − δ ( Y )) (cid:1) . ϑ,η ω X Q L (cid:0) ∆ ( b y , δ ( Y )) (cid:1) . Write B = c − B ∗ Q , B ′ = B ∗ Q , ∆ = B ∩ ∂ Ω , ∆ ′ = B ′ ∩ ∂ Ω . Assuming that r B = c − κ r Q < diam( ∂ Ω )we have by Lemma 3.26 part (b) and Harnack’s inequality(4.33) ω X Q L (cid:0) ∆ ( b y , δ ( Y )) (cid:1) ≈ ω X ∆ L (cid:0) ∆ ( b y , δ ( Y )) (cid:1) ≈ δ ( Y ) n − G L ( X ∆ , Y ) , Y ∈ B ∗ Q ∩ Ω = B ′ ∩ Ω . If we then combine (4.30), (4.32), and (4.33) we conclude that (c) ′ and Lemma 3.26 yield(4.34) ˆ Q S ϑ Q ,η u ( x ) d ω X Q L ( x ) . ϑ,η ¨ B ′ ∩ Ω |∇ u ( Y ) | G L ( X ∆ , Y ) dY . ω X ∆ L ( ∆ ′ ) k u k L ∞ ( Ω ) . ω X Q L ( Q ) . Note that this estimate corresponds to (4.26) for q =
2. Hence the same argument we use in (d) ′ = ⇒ (a)applies in this scenario. Note however, that we have assumed that 16 c − κ r Q < diam( ∂ Ω ) and thiscauses that (4.28) is valid under this restriction. If ∂ Ω is unbounded then the same argument applies.When ∂ Ω is bounded we can replace the family F by F ′ consisting on all Q ′ ∈ D with Q ′ ⊂ Q forsome Q ∈ F and ℓ ( Q ′ ) = − M ℓ ( Q ) where M is large enough so that 2 − M < Ξ c / (8 κ ). This guaranteesthat 16 c − κ r Q ′ < diam( ∂ Ω ) for every Q ′ ∈ F ′ and thus (4.28) holds for every Q ′ ∈ F ′ . At this pointthe rest of the argument can be carried out mutatis mutandis , details are left to the reader. (cid:3) Proof of (a) = ⇒ (c) . Note that we have already proved that (a) implies (d), in particular weknow that (3.12) holds with q = α ≥ c − . Our goal is to see that the latter estimate implies(c). With this goal in mind consider u ∈ W , ( Ω ) ∩ L ∞ ( Ω ) satisfying Lu = Ω .Fix B = B ( x , r ) with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ) and B ′ = B ( x ′ , r ′ ) with x ′ ∈ ∆ and 0 < r ′ < rc / ∆ = B ∩ ∂ Ω , ∆ ′ = B ′ ∩ ∂ Ω . Note that 2 r ′ < rc / < diam( ∂ Ω ) and we can now invoke Lemma 4.7and (3.12) with q = ω X ∆ L ( ∆ ′ ) ¨ B ′ ∩ Ω |∇ u ( X ) | G L ( X ∆ , X ) dX . ˆ ∆ ′ S C α r ′ u ( y ) d ω X ∆ ′ L ( y ) + sup {| u ( Y ) | : Y ∈ B ′ , δ ( Y ) ≥ r ′ / C } . ˆ ∆ ′ N α ′ r ′ u ( y ) d ω X ∆ ′ L ( y ) + k u k L ∞ ( Ω ) . k u k L ∞ ( Ω ) . Taking the sup over B and B ′ we have then shown (3.11). (cid:3) Proof of (a) = ⇒ (e) . Fix f ∈ C ( ∂ Ω ) ∩ L ∞ ( ∂ Ω ) and let u be its associated solution as in (3.7).Let u L , Ω ( X ) : = ω XL ( ∂ Ω ), X ∈ Ω . Fix B = B ( x , r ) with x ∈ ∂ Ω and 0 < r < diam( ∂ Ω ) and B ′ = B ( x ′ , r ′ )with x ′ ∈ ∆ and 0 < r ′ < rc /
4. Let ∆ = B ∩ ∂ Ω , ∆ ′ = B ′ ∩ ∂ Ω . Let ϕ ∈ C ( R ) with [0 , ≤ ϕ ≤ [0 , and ϕ ∆ ′ : = ϕ ( | · − x ′ | / r ′ ) so that ∆ ′ ≤ ϕ ∆ ′ ≤ ∆ ′ . Recall that for every surface ball e ∆ we write f e ∆ , L : = e ∆ f d ω X e ∆ L . Then, f − f ∆ , L = ( f − f ∆ ′ , L ) + ( f ∆ ′ , L − f ∆ , L ) = ( f − f ∆ ′ , L ) ϕ ∆ ′ + ( f − f ∆ ′ , L )(1 − ϕ ∆ ′ ) + ( f ∆ ′ , L − f ∆ , L ) = : h loc + h glob + ( f ∆ ′ , L − f ∆ , L ) . Hence,(4.35) v ( X ) : = u ( X ) − f ∆ , L u L , Ω ( X ) = ˆ ∂ Ω ( f ( y ) − f ∆ , L ) d ω XL ( y ) = ˆ ∂ Ω h loc ( y ) d ω XL ( y ) + ˆ ∂ Ω h glob ( y ) d ω XL ( y ) + ( f ∆ ′ , L − f ∆ , L ) u L , Ω ( X ) = : v loc ( X ) + v glob ( X ) + ( f ∆ ′ , L − f ∆ , L ) u L , Ω ( X ) . Note that h loc , h glob ∈ C ( ∂ Ω ) ∩ L ∞ ( ∂ Ω ).Let us observe that we have already proved that (a) implies (d), in particular we know that (3.12)holds with q = α ≥ c − . Hence, since 2 r ′ < rc / < diam( ∂ Ω ) and we can now invokeLemma 4.7 and (3.12) with q = ω X ∆ L ( ∆ ′ ) ¨ B ′ ∩ Ω |∇ v loc ( X ) | G L ( X ∆ , X ) dX (4.36) . ˆ ∆ ′ S C α r ′ v loc ( y ) d ω X ∆ ′ L ( y ) + sup {| v loc ( Y ) | : Y ∈ B ′ , δ ( Y ) ≥ r ′ / C } . ˆ ∆ ′ N α ′ r ′ v loc ( y ) d ω X ∆ ′ L ( y ) + (cid:16) ˆ ∂ Ω | h loc ( y ) | d ω X ∆ ′ L ( y ) (cid:17) . ˆ ∆ ′ N α ′ r ′ v loc ( y ) d ω X ∆ ′ L ( y ) + (cid:16) ˆ ∂ Ω | h loc ( y ) | d ω X ∆ ′ L ( y ) (cid:17) = : I + I . Regarding I we note that by Lemma 3.26 part ( a ) there holds I ≤ (cid:16) ˆ ∆ ′ | f ( y ) − f ∆ ′ , L | d ω X ∆ ′ L ( y ) (cid:17) . k f k ∂ Ω ,ω L ) . (4.37)To estimate I we first observe that since ω L ∈ A ∞ ( ∂ Ω , ω L ), there is q ∈ (1 , ∞ ) so that ω L ∈ RH q ( ∂ Ω , ω L ). Note that by Jensen’s inequality we may assume that q < RH q ( ∂ Ω , ω L ) ⊂ RH q ( ∂ Ω , ω L ) if q ≤ q ). Note that we have already proved that (a) q implies (b) q ′ , hence (3.8) holdswith p = q ′ >
2. This, H ¨older’s inequality and the fact that h loc ∈ C ( ∂ Ω ) with supp h loc ⊂ ∆ ′ readilylead to(4.38) I ≤ kN α ′ r ′ v loc k L q ′ (4 ∆ ′ ,ω X ∆ ′ L ) ω X ∆ ′ L (4 ∆ ′ ) q ′ / ′ . k h loc k L q ′ (8 ∆ ′ ,ω X ∆ ′ L ) . (cid:16) ˆ ∆ ′ | f ( y ) − f ∆ ′ , L | q ′ d ω X ∆ ′ L ( y ) (cid:17) q ′ . k f k ∂ Ω ,ω L ) , where the last estimate uses Lemma 3.26 part ( a ) and John-Nirenberg’s inequality (cf. (3.34)).We next turn our attention to the estimate involving v glob . Note that LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 37 | h glob | ≤ | f − f ∆ ′ , L | ∂ Ω \ ∆ ′ = ∞ X k = | f − f ∆ ′ , L | k + ∆ ′ \ k ∆ ′ ≤ ∞ X k = | f − f ∆ ′ , L | ( ϕ k − ∆ ′ − ϕ k − ∆ ′ ) = : X k ≥ k r ′ ≤ diam( ∂ Ω ) h glob , k , with the understanding that the sum runs from k = ∂ Ω is unbounded.Fix k ≥ k r ′ ≤ diam( ∂ Ω ) and note that h glob , k ∈ C ( ∂ Ω ) with supp h glob , k ⊂ k + ∆ ′ \ k − ∆ ′ .Thus, for every X ∈ B ′ ∩ Ω , by Lemma 3.26 part ( f ) we have v glob , k ( X ) : = ˆ ∂ Ω h glob , k ( y ) d ω XL ( y ) . (cid:16) δ ( X )2 k − r ′ (cid:17) ρ v glob , k ( X k − ∆ ′ ) . (4.39)Next we estimate v glob , k ( X k − ∆ ′ ) , k ≥ , via a telescopic argument. Indeed applying Harnack’s in-equality, that ω L ∈ RH q ( ∂ Ω , ω L ), Lemma 3.26, and John-Nirenberg’s inequality (cf. (3.34)) we arriveat v glob , k ( X k − ∆ ′ ) ≤ ˆ k + ∆ ′ | f ( y ) − f ∆ ′ , L | d ω X k − ∆ ′ L ( y ) . ˆ k + ∆ ′ | f ( y ) − f ∆ ′ , L | d ω X k + ∆ ′ L ( y ) . (cid:16) k + ∆ ′ | f ( y ) − f ∆ ′ , L | q ′ d ω X k + ∆ ′ L ( y ) (cid:17) q ′ ≤ (cid:16) k + ∆ ′ | f ( y ) − f k + ∆ ′ , L | q ′ d ω X k + ∆ ′ L ( y ) (cid:17) q ′ + k + X j = | f j + ∆ ′ , L − f j ∆ ′ , L |≤ (cid:16) k + ∆ ′ | f ( y ) − f k + ∆ ′ , L | q ′ d ω X k + ∆ ′ L ( y ) (cid:17) q ′ + k + X j = j ∆ ′ | f ( y ) − f j + ∆ ′ , L | d ω X j ∆ ′ L ( y ) . k + X j = (cid:16) j + ∆ ′ | f ( y ) − f j + ∆ ′ , L | q ′ d ω X j + ∆ ′ L ( y ) (cid:17) q ′ . k k f k BMO( ∂ Ω ,ω L ) . This and (4.39) give for every X ∈ B ′ ∩ Ω ˆ ∂ Ω | h glob ( y ) | d ω XL ( y ) ≤ X k ≥ k r ′ ≤ diam( ∂ Ω ) ˆ ∂ Ω h glob , k ( y ) d ω XL ( y ) = X k ≥ k r ′ ≤ diam( ∂ Ω ) v glob , k ( X ) . X k ≥ k (cid:16) δ ( X )2 k − r ′ (cid:17) ρ k f k BMO( ∂ Ω ,ω L ) ≈ (cid:16) δ ( X ) r ′ (cid:17) ρ k f k BMO( ∂ Ω ,ω L ) . If we next write W B ′ : = { I ∈ W : I ∩ B ′ , Ø } and pick Z I , B ′ ∈ I ∩ B ′ , the previous estimate gives forevery I ∈ W B ′ ¨ I |∇ v glob ( X ) | dX . ℓ ( I ) − ¨ I ∗ v glob ( X ) dX ≤ ℓ ( I ) − ¨ I ∗ (cid:16) ˆ ∂ Ω | h glob ( y ) | d ω XL ( y ) (cid:17) dX ≈ ℓ ( I ) n − (cid:16) ˆ ∂ Ω h glob ( y ) d ω Z I , B ′ L ( y ) (cid:17) . ℓ ( I ) n − (cid:16) ℓ ( I ) r ′ (cid:17) ρ k f k ∂ Ω ,ω L ) . Thus, Lemma 3.26 gives ¨ B ′ ∩ Ω |∇ v glob ( X ) | G L ( X ∆ , X ) dX . X I ∈W B ′ ω X ∆ L ( Q I ) ℓ ( I ) − n ¨ I |∇ v glob ( X ) | dX . k f k ∂ Ω ,ω L ) X I ∈W B ′ ω X ∆ L ( Q I ) (cid:16) ℓ ( I ) r ′ (cid:17) ρ . k f k ∂ Ω ,ω L ) X k :2 − k . r ′ (cid:16) − k r ′ (cid:17) ρ X I ∈W B ′ : ℓ ( I ) = − k ω X ∆ L ( Q I ) , where Q I ∈ D ( ∂ Ω ) is so that ℓ ( Q I ) = ℓ ( I ) and contains b y I ∈ ∂ Ω such that dist( I , ∂ Ω ) = dist( b y I , I ). Itis easy to see that for every k with 2 − k . r ′ , the family { Q I } I ∈W B ′ ,ℓ ( I ) = − k has bounded overlap and alsothat Q I ⊂ C ∆ ′ for every I ∈ W B ′ , where C is some harmless dimensional constant. Hence,(4.40) ¨ B ′ ∩ Ω |∇ v glob ( X ) | G L ( X ∆ , X ) dX . k f k ∂ Ω ,ω L ) X k :2 − k . r ′ (cid:16) − k r ′ (cid:17) ρ ω X ∆ L ( C ∆ ′ ) . k f k ∂ Ω ,ω L ) ω X ∆ L ( ∆ ′ ) . To continue we pick k ≥ r < k r ′ ≤ r . Note that 2 k + ∆ ′ and ∆ have comparableradius and x ′ ∈ ∆ ∩ k + ∆ ′ . Hence, Lemma 3.26 and Harnack’s inequality yield | f ∆ ′ , L − f ∆ , L | ≤ k X k = | f k ∆ ′ , L − f k + ∆ , L | + | f k + ∆ ′ , L − f ∆ , L | (4.41) ≤ k X k = k ∆ ′ | f ( y ) − f k + ∆ ′ , L | d ω X k ∆ ′ L ( y ) + ∆ | f ( y ) − f k + ∆ ′ , L | d ω X ∆ L ( y ) . k X k = k + ∆ ′ | f ( y ) − f k + ∆ ′ , L | d ω X k + ∆ ′ L ( y ) . k k f k BMO( ∂ Ω ,ω L ) ≤ (1 + log( r / r ′ )) k f k BMO( ∂ Ω ,ω L ) . This and Lemma 4.1 imply(4.42) 1 ω X ∆ L ( ∆ ′ ) ¨ B ′ ∩ Ω (cid:12)(cid:12) ( f ∆ ′ , L − f ∆ , L ) ∇ u L , Ω ( X ) (cid:12)(cid:12) G L ( X ∆ , X ) dX . (1 + log( r / r ′ )) (cid:16) r ′ diam( ∂ Ω ) (cid:17) ρ k f k ∂ Ω ,ω L ) ≤ (1 + log( r / r ′ )) (cid:16) r ′ r (cid:17) ρ k f k ∂ Ω ,ω L ) . k f k ∂ Ω ,ω L ) . Here we note in passing that if diam( ∂ Ω ) = ∞ (or if both ∂ Ω and Ω are bounded) then the left-handside of the previous estimate vanishes as we know that u L , Ω ≡ ¨ B ′ ∩ Ω |∇ v ( X ) (cid:12)(cid:12) G L ( X ∆ , X ) dX . ¨ B ′ ∩ Ω |∇ v loc ( X ) (cid:12)(cid:12) G L ( X ∆ , X ) dX + ¨ B ′ ∩ Ω |∇ v glob ( X ) (cid:12)(cid:12) G L ( X ∆ , X ) dX + ¨ B ′ ∩ Ω (cid:12)(cid:12) ( f ∆ ′ , L − f ∆ , L ) ∇ u L , Ω ( X ) (cid:12)(cid:12) G L ( X ∆ , X ) dX LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 39 . k f k ∂ Ω ,ω L ) ω X ∆ L ( ∆ ′ ) . This completes the proof. (cid:3)
Remark . It is not di ffi cult to see that in (3.9) one can replace f ∆ , L by f ∆ ′ , L . Indeed, this is whatwe have essentially done in the proof: much as in (4.41) one has that | f ∆ , L − f ∆ ′ , L | . (1 + log( r / r ′ )) k f k BMO( ∂ Ω ,ω L ) . With this we can proceed as in (4.42) to see that1 ω X ∆ L ( ∆ ′ ) ¨ B ′ ∩ Ω (cid:12)(cid:12) ( f ∆ , L − f ∆ ′ , L ) ∇ u L , Ω ( X ) (cid:12)(cid:12) G L ( X ∆ , X ) dX . k f k ∂ Ω ,ω L ) . Hence, (3.9) with f ∆ , L is equivalent to (3.9) with f ∆ ′ , L .On the other hand, when Ω is unbounded and ∂ Ω bounded, in (3.9) one can replace f ∆ , L by f ∂ Ω , L : = ∂ Ω f d ω X Ω L , where X Ω ∈ Ω satisfy δ ( X Ω ) ≈ diam( ∂ Ω ) (say, X Ω = X ∆ ( x , r ) with x ∈ ∂ Ω and r ≈ diam( ∂ Ω )). To see this, one proceeds as in (4.41) to see that | f ∆ , L − f ∂ Ω , L | . (1 + log(diam( ∂ Ω ) / r )) k f k BMO( ∂ Ω ,ω L ) . This and Lemma 4.1 readily give1 ω X ∆ L ( ∆ ′ ) ¨ B ′ ∩ Ω (cid:12)(cid:12) ( f ∆ , L − f ∂ Ω , L ) ∇ u L , Ω ( X ) (cid:12)(cid:12) G L ( X ∆ , X ) dX . (1 + log(diam( ∂ Ω ) / r )) (cid:16) r ′ diam( ∂ Ω ) (cid:17) ρ k f k ∂ Ω ,ω L ) ≤ (1 + log(diam( ∂ Ω ) / r )) (cid:16) r diam( ∂ Ω ) (cid:17) ρ k f k ∂ Ω ,ω L ) . k f k ∂ Ω ,ω L ) . Hence, (3.9) with f ∆ , L is equivalent to (3.9) with f ∂ Ω , L .4.16. Proof of (a) = ⇒ (e) ′ . The proof is almost the same as the previous one with the followingmodifications. We work with f = S with S ⊂ ∂ Ω an arbitrary Borel set. We replace ϕ by [0 , anduse in (4.36) that Lemma 4.7 is also valid for the associated v loc since it belongs to W , ( Ω ) ∩ L ∞ ( Ω ).Also, in (4.37) we need to invoke that (a) q = ⇒ (b) q ′ = ⇒ (b) ′ q ′ . The rest of the proof remains thesame, details are left to the interested reader. (cid:3)
5. P roof of T heorem = ⇒ (c) = ⇒ (d), (b) ′ = ⇒ (c) ′ = ⇒ (d) ′ are trivial. Also, since for anyBorel set S ⊂ ∂ Ω the solution u ( X ) = ω XL ( S ) belongs to W , ( Ω ) ∩ L ∞ ( Ω ), it is also straightforward that(b) = ⇒ (b) ′ , (c) = ⇒ (c) ′ , and (d) = ⇒ (d) ′ .We next observe that for every α >
0, 0 < r < r ′ , and ̟ ∈ R , if F ⊂ ∂ Ω is a bounded set and v ∈ L ( Ω ), then(5.1) sup x ∈ F ¨ Γ α r ′ ( x ) \ Γ α r ( x ) | v ( Y ) | δ ( Y ) ̟ dY < ∞ . To see this we first note that since F is bounded we can find R large enough so that F ⊂ B (0 , R ). Then,if x ∈ F one readily sees that Γ α r ′ ( x ) \ Γ α r ( x ) ⊂ B (0 , r ′ + R ) ∩ n Y ∈ Ω : r + α ≤ δ ( Y ) ≤ r ′ o = : K . Note that K ⊂ Ω is a compact set. Then, since v ∈ L ( Ω ), we conclude thatsup x ∈ F ¨ Γ α r ′ ( x ) \ Γ α r ( x ) | v ( Y ) | δ ( Y ) ̟ dY ≤ max (cid:26) r ′ , + α r (cid:27) | ̟ | ¨ K | v ( Y ) | dY < ∞ . (5.2)Using then (5.1) it is also trivial to see that (d) = ⇒ (c) and (d) ′ = ⇒ (c) ′ . Hence we are left withshowing (a) = ⇒ (b) and (c) ′ = ⇒ (a) . Proof of (a) = ⇒ (b) . Assume that ω L ≪ ω L . Let ϑ ≥ ϑ large enough to be chosen (thischoice will depend on α ). Fix an arbitrary Q ∈ D k where k ∈ Z is taken so that 2 − k = ℓ ( Q ) < diam( ∂ Ω ) / M , where M > κ c − , κ is taken from (2.23), and c is the Corkscrew constant. Let X : = X M ∆ Q be a Corkscrew point relative to M ∆ Q so that X < B ∗ Q by the choice of M . ByLemma 3.26 part ( a ) and Harnack’s inequality, there exists C > ω X L ( Q ) ≥ C − . Set ω : = ω X L , ω : = C ω X L ( Q ) ω X L , G : = G L ( X , · ) , and G : = C ω X L ( Q ) G L ( X , · ) . (5.4)By assumption, we have ω ≪ ω and by (5.3),(5.5) 1 ≤ ω ( Q ) ω ( Q ) = C ω X L ( Q ) ≤ C . For N > C , we let F + N : = { Q j } ⊂ D Q \{ Q } , respectively, F − N : = { Q j } ⊂ D Q \{ Q } , be the collectionof descendants of Q which are maximal (and therefore pairwise disjoint) with respect to the propertythat ω ( Q j ) ω ( Q j ) < N , respectively ω ( Q j ) ω ( Q j ) > N . (5.6)Write F N : = F + N ∪ F − N and note that F + N ∩ F − N = Ø. By maximality, there holds1 N ≤ ω ( Q ) ω ( Q ) ≤ N , ∀ Q ∈ D F N , Q . (5.7)Denote, for every N > C , E ± N : = [ Q ∈F ± N Q , E N : = E + N ∪ E − N , E N : = Q \ E N , (5.8)and Q = (cid:18) \ N > C E N (cid:19) ∪ (cid:18) [ N > C E N (cid:19) = : E ∪ (cid:18) [ N > C E N (cid:19) . (5.9)By Lemma 2.26, Ω ϑ F N , Q is a bounded 1-sided NTA satisfying the CDC for any ϑ ≥ ϑ . As in [28,Proposition 6.1] E N ⊂ F N : = ∂ Ω ∩ ∂ Ω ϑ F N , Q ⊂ Q \ [ Q j ∈F N int( Q j ) . Hence, F N \ E N ⊂ (cid:18) Q \ [ Q j ∈F N int( Q j ) (cid:19) \ (cid:18) Q \ [ Q j ∈F N Q j (cid:19) ⊂ ∂ Q ∪ (cid:18) [ Q j ∈F N ∂ Q j (cid:19) . LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 41
This, [1, Lemma 2.37], and Lemma 3.26 imply ω ( F N \ E N ) = . (5.10)Next, we are going to show(5.11) ω ( E ) = . Let x ∈ E ± N + . Then there exists Q x ∈ F ± N + such that x ∈ Q x . By (5.6), we have ω ( Q x ) ω ( Q x ) < N + < N if Q x ∈ F + N + or ω ( Q x ) ω ( Q x ) > N + > N if Q x ∈ F − N + . By the maximality of the cubes in F ± N , one has Q x ⊂ Q ′ x for some Q ′ x ∈ F ± N with x ∈ Q ′ x ⊂ E ± N .Thus, { E + N } N , { E − N } N and { E N } N are decreasing sequences of sets. This, together with the fact that ω ( E ± N ) ≤ ω ( Q ) ≤ C ω ( Q ) ≤ C and ω ( E ± N ) ≤ ω ( Q ) ≤
1, implies that(5.12) ω (cid:18) \ N > C E ± N (cid:19) = lim N →∞ ω ( E ± N ) and ω (cid:18) \ N > C E ± N (cid:19) = lim N →∞ ω ( E ± N ) . By (5.6) and (5.8), ω ( E + N ) = X Q ∈F + N ω ( Q ) < N X Q ∈F + N ω ( Q ) = N ω ( E + N ) ≤ N , which together with (5.12) yields ω (cid:18) \ N > C E + N (cid:19) = lim N →∞ ω ( E + N ) = . In view of the fact that by assumption ω ≪ ω , we then conclude that(5.13) 0 = ω (cid:18) \ N > C E + N (cid:19) = lim N →∞ ω ( E + N ) . On the other hand, (5.6) yields ω ( E − N ) = X Q ∈F − N ω ( Q ) < N X Q ∈F − N ω ( Q ) = N ω ( E − N ) ≤ C N , and hence,(5.14) ω (cid:18) \ N > C E − N (cid:19) = lim N →∞ ω ( E − N ) = . Since { E N } N is a decreasing sequence of sets with ω ( E N ) ≤ ω ( Q ) ≤
1, (5.13) and (5.14) readilyimply (5.11): ω ( E ) = lim N →∞ ω ( E N ) ≤ lim N →∞ ω ( E + N ) + lim N →∞ ω ( E − N ) = . Now we turn our attention to the square function estimates in L q ( F N , ω ) for q ∈ (0 , ∞ ). Let u ∈ W , ( Ω ) ∩ L ∞ ( Ω ) be a weak solution of Lu = Ω . To continue, we observe that if Q ∈ D Q is sothat Q ∩ E N , Ø, then necessarily Q ∈ D F N , Q , otherwise, Q ⊂ Q ′ ∈ F N , hence Q ⊂ Q \ E N which is acontradiction. As a result, ω ( Q ) ω ( Q ) ≈ N , ∀ x ∈ E N , Q ∈ D Q , Q ∋ x . By the (dyadic) Lebesgue di ff erentiation theorem with respect to ω , along with the fact that ω ≪ ω (cf. (5.4)), we conclude that d ω / d ω ( x ) ≈ N ω -a.e. x ∈ E N , hence also for ω -a.e. x ∈ E N . Thus, ˆ E N S ϑ Q u ( x ) q d ω ( x ) = ˆ E N S ϑ Q u ( x ) q d ω d ω ( x ) d ω ( x ) ≈ N ˆ E N S ϑ Q u ( x ) q d ω ( x ) . ˆ Q S ϑ Q u ( x ) q d ω ( x ) . ˆ Q N ϑ Q u ( x ) q d ω ( x ) . k u k qL ∞ ( Ω ) ω ( Q ) . k u k qL ∞ ( Ω ) , where in the third estimate we have used (4.19) with ω L = ω L (see also [1, Theorem 5.3]) which holdssince ω L ∈ A ∞ ( ∂ Ω , ω L ). This and (5.10) imply S ϑ Q u ∈ L q ( F N , ω ) . (5.15)Now, note that for fixed α >
0, we can find ϑ su ffi ciently large depending on α such that for any r ≪ − k ,(5.16) Γ α r ( x ) ⊂ Γ ϑ Q ( x ) , ∀ x ∈ Q . Indeed, let Y ∈ Γ α r ( x ). Pick I ∈ W so that I ∋ Y , hence ℓ ( I ) ≈ δ ( Y ) ≤ | Y − x | < r ≪ − k = ℓ ( Q ).Pick Q I ∈ D Q such that x ∈ Q I and ℓ ( Q I ) = ℓ ( I ) ≪ ℓ ( Q ). Thus,dist( I , Q I ) ≤ | Y − x | < (1 + α ) δ ( Y ) ≤ C (1 + α ) ℓ ( I ) = C (1 + α ) ℓ ( Q I ) . Recalling (2.18), if we take ϑ ≥ ϑ large enough so that 2 ϑ ≥ C (1 + α ), then Y ∈ I ∈ W ϑ Q I ⊂ W ϑ, ∗ Q I .The latter gives that Y ∈ U ϑ Q I ⊂ Γ ϑ Q ( x ) and consequently (5.16) holds. We would like to mention thatthe dependence of ϑ on α implies that all the sawtooth regions Ω ϑ F N , Q above as well as all the implicitconstants depend on α .Next, (5.16) readily yields that S α r u ( x ) ≤ S ϑ Q u ( x ) for every x ∈ Q . This, together with (5.15),implies that S α r u ∈ L q ( F N , ω ). If we next take an arbitrary X ∈ Ω , by Harnack’s inequality (albeitwith constants depending on X ) and by (5.1), then we have(5.17) S α r u ∈ L q ( F N , ω XL ) , for any r > . Note also that by (5.11) and Harnack’s inequality(5.18) ω XL ( E ) = . To complete the proof, we perform the preceding operation for an arbitrary Q ∈ D k . Therefore,invoking (5.8), (5.9), and (5.10) with Q k ∈ D k , we conclude, with the induced notation, that(5.19) ∂ Ω = [ Q k ∈ D k Q k = (cid:18) [ Q k ∈ D k E k (cid:19) [ (cid:18) [ Q k ∈ D k [ N > C E kN (cid:19) = (cid:18) [ Q k ∈ D k E k (cid:19) [ (cid:18) [ Q k ∈ D k [ N > C F kN (cid:19) = : F ∪ (cid:18) [ k , N F kN (cid:19) . where ω XL ( F ) = F kN = ∂ Ω ∩ ∂ Ω ϑ F kN , Q k where each Ω ϑ F kN , Q k ⊂ Ω is a bounded 1-sidedNTA domain satisfying the capacity density condition. Combining (5.19) and (5.17) with F kN in placeof F N , the proof of (a) = ⇒ (b) is complete. (cid:3) Proof of (c) ′ = ⇒ (a) . Let α be so that (4.25) holds. Suppose that (c) ′ holds where throughoutit is assumed that α ≥ α . In order to prove that ω L ≪ ω L on ∂ Ω , by Lemma 2.13 and the fact that byHarnack’s inequality ω XL ≪ ω YL and ω XL ≪ ω YL for any X , Y ∈ Ω , it su ffi ces to show that for any given Q ∈ D , F ⊂ Q , ω X Q L ( F ) = = ⇒ ω X Q L ( F ) = . (5.20) LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 43
Consider then F ⊂ Q with ω X Q L ( F ) =
0. Lemma 4.4 applied to F gives a Borel set S ⊂ Q such that u ( X ) : = ω XL ( S ), X ∈ Ω , satisfies S α r ∗ Q u ( x ) ≥ S ϑ Q ,η u ( x ) = ∞ , ∀ x ∈ F , (5.21)where the first inequality follows from (4.25) and the fact that α ≥ α , and r ∗ Q = κ r Q . By assumptionand (5.1) we have that S α r ∗ Q u ( x ) < ∞ for ω X Q L -a.e. x ∈ ∂ Ω . Hence, ω X Q L ( F ) = ′ = ⇒ (a) is complete. (cid:3)
6. P roof of T heorems and Theorem 6.1.
Let Ω ⊂ R n + , n ≥ , be a 1-sided NTA domain (cf. Definition 2.5) satisfying the capacitydensity condition (cf. Definition 2.10). There exists e α > (depending only on the 1-sided NTA andCDC constants) such that the following holds. Assume that L u = − div( A ∇ u ) and L u = − div( A ∇ u ) are real (not necessarily symmetric) elliptic operators such that A − A = A + D, where A , D ∈ L ∞ ( Ω ) are real matrices satisfying the following conditions: (i) There exist α ≥ e α and r > such that ¨ Γ α r ( x ) a ( X ) δ ( X ) − n − dX < ∞ , for ω L -a.e. x ∈ ∂ Ω , (6.2) where a ( X ) : = sup Y ∈ B ( X ,δ ( X ) / | A ( Y ) | , X ∈ Ω . (ii) D ∈ Lip loc ( Ω ) is antisymmetric and there exist α ≥ e α and r > such that ¨ Γ α r ( x ) | div C D ( X ) | δ ( X ) − n dX < ∞ , for ω L -a.e. x ∈ ∂ Ω . (6.3) Then ω L ≪ ω L . Assuming this result momentarily we deduce Theorems 1.10 and 1.12:
Proof of Theorem 1.10.
For L and L as in the statement set e A = A , e A = A , e A = A − A , and D = e A − e A = e A + D . Note that (6.2) follows at once from (1.11), and also that (6.3) holds automatically.With all these in hand Theorem 6.1 gives ω L = ω e L ≪ ω e L = ω L . (cid:3) Proof of Theorem 1.12.
Set A = A , A = A ⊤ , e A =
0, and D = A − A ⊤ so that A − A = e A + D . Observethat D ∈ Lip loc ( Ω ) is antisymmetric, (6.2) holds trivially, and (6.3) agrees with (1.13). Thus, Theorem6.1 implies that ω L ≪ ω L ⊤ .On the other hand, ω L ≪ ω L sym follows similarly if we set A = A , A = ( A + A ⊤ ) / e A = D = ( A − A ⊤ ) / ω L ⊤ ≪ ω L follows from what has been proved by switching the roles of L and L ⊤ and thefact that F α r ( x ; A ) < ∞ for ω L ⊤ -a.e. x ∈ ∂ Ω . (cid:3) Before proving Theorem 6.1 we need the following auxiliary which adapts [31, Lemma 4.44] and[1, Lemma 3.11] to our current setting:
Lemma 6.4.
Let Ω ⊂ R n + be a 1-sided NTA domain (cf. Definition 2.5) satisfying the capacity densitycondition (cf. Definition 2.10). Given Q ∈ D , a pairwise disjoint collection F ⊂ D Q , and N ≥ let F N be the family of maximal cubes of the collection F augmented by adding all the cubes Q ∈ D Q such that ℓ ( Q ) ≤ − N ℓ ( Q ) . There exist Ψ ϑ N ∈ C ∞ c ( R n + ) and a constant C ≥ depending only ondimension n, the 1-sided NTA constants, the CDC constant, and ϑ , but independent of N, F , and Q such that the following hold: ( i ) C − Ω ϑ F N , Q ≤ Ψ ϑ N ≤ Ω ϑ, ∗F N , Q . ( ii ) sup X ∈ Ω |∇ Ψ ϑ N ( X ) | δ ( X ) ≤ C. ( iii ) Setting (6.5) W ϑ N : = [ Q ∈ D F N , Q W ϑ, ∗ Q , W ϑ, Σ N : = (cid:8) I ∈ W ϑ N : ∃ J ∈ W \ W ϑ N with ∂ I ∩ ∂ J , Ø (cid:9) , one has (6.6) ∇ Ψ ϑ N ≡ in [ I ∈W ϑ N \W ϑ, Σ N I ∗∗ , and there exists a family { b Q I } I ∈W ϑ, Σ N so that (6.7) C − ℓ ( I ) ≤ ℓ ( b Q I ) ≤ C ℓ ( I ) , dist( I , b Q I ) ≤ C ℓ ( I ) , X I ∈W ϑ, Σ N b Q I ≤ C . Proof.
The proof combines ideas from [31, Lemma 4.44], [1, Lemma 3.11], and [29, Appendix A.2].The parameter ϑ ≥ ϑ will remain fixed in the proof and then constants are allowed to depend on it.To ease the notation we will omit the superscript ϑ everywhere in the proof. Recall that given I , anyclosed dyadic cube in R n + , we set I ∗ = (1 + λ ) I and I ∗∗ = (1 + λ ) I . Let us introduce e I ∗ = (1 + λ ) I so that(6.8) I ∗ ( int( e I ∗ ) ( e I ∗ ⊂ int( I ∗∗ ) . Given I : = [ − , ] n + ⊂ R n + , fix φ ∈ C ∞ c ( R n + ) such that 1 I ∗ ≤ φ ≤ e I ∗ and |∇ φ | . λ ). For every I ∈ W = W ( Ω ) we set φ I ( · ) = φ (cid:0) · − X ( I ) ℓ ( I ) (cid:1) so that φ I ∈ C ∞ ( R n + ), 1 I ∗ ≤ φ I ≤ e I ∗ and |∇ φ I | . ℓ ( I ) − (with implicit constant depending only on n and λ ).For every X ∈ Ω , we let Φ ( X ) : = P I ∈W φ I ( X ). It then follows that Φ ∈ C ∞ ( Ω ) since for everycompact subset of Ω , the previous sum has finitely many non-vanishing terms. Also, 1 ≤ Φ ( X ) ≤ C λ for every X ∈ Ω since the family { e I ∗ } I ∈W has bounded overlap by our choice of λ . Hence we can set Φ I = φ I / Φ and one can easily see that Φ I ∈ C ∞ c ( R n + ), C − λ I ∗ ≤ Φ I ≤ e I ∗ and |∇ Φ I | . ℓ ( I ) − . Withthis at hand, set Ψ N ( X ) : = X I ∈W N Φ I ( X ) = P I ∈W N φ I ( X ) P I ∈W φ I ( X ) , X ∈ Ω . We first note that the number of terms in the sum defining Ψ N is bounded depending on N . Indeed,if Q ∈ D F N , Q then Q ∈ D Q and 2 − N ℓ ( Q ) < ℓ ( Q ) ≤ ℓ ( Q ) which implies that D F N , Q has finitecardinality with bounds depending on dimension and N (here we recall that the number of dyadicchildren of a given cube is uniformly controlled). Also, by construction W ∗ Q has cardinality dependingonly on the allowable parameters. Hence, W N . C N < ∞ . This and the fact that each Φ I ∈ C ∞ c ( R n + ) LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 45 yield that Ψ N ∈ C ∞ c ( R n + ). Note also that (6.8) and the definition of W N givesupp Ψ N ⊂ [ I ∈W N e I ∗ = [ Q ∈ D F N , Q [ I ∈W ∗ Q e I ∗ ⊂ int (cid:16) [ Q ∈ D F N , Q [ I ∈W ∗ Q I ∗∗ (cid:17) = int (cid:16) [ Q ∈ D F N , Q U ∗ Q (cid:17) = Ω ∗F N , Q . This, the fact that W N ⊂ W , and the definition of Ψ N immediately give that Ψ N ≤ Ω ∗F N , Q . Onthe other hand, if X ∈ Ω N = Ω F N , Q , then there exists I ∈ W N such that X ∈ I ∗ , in which case Ψ N ( X ) ≥ Φ I ( X ) ≥ C − λ . All these imply ( i ). Note that ( ii ) follows by observing that for every X ∈ Ω wehave |∇ Ψ N ( X ) | ≤ X I ∈W N |∇ Φ I ( X ) | . X I ∈W ℓ ( I ) − e I ∗ ( X ) . δ ( X ) − , where we have used that if X ∈ e I ∗ then δ ( X ) ≈ ℓ ( I ) and also that the family { e I ∗ } I ∈W has boundedoverlap.To see ( iii ) fix I ∈ W N \ W Σ N and X ∈ I ∗∗ , and set W X : = { J ∈ W : φ J ( X ) , } so that I ∈ W X . Wefirst note that W X ⊂ W N . Indeed, if φ J ( X ) , X ∈ e J ∗ . Hence X ∈ I ∗∗ ∩ J ∗∗ and our choice of λ gives that ∂ I meets ∂ J , this in turn implies that J ∈ W N since I ∈ W N \ W Σ N . All these yield Ψ N ( X ) = P J ∈W N φ J ( X ) P J ∈W φ J ( X ) = P J ∈W N ∩W X φ J ( X ) P J ∈W X φ J ( X ) = P J ∈W N ∩W X φ J ( X ) P J ∈W N ∩W X φ J ( X ) = . Hence Ψ N (cid:12)(cid:12) I ∗∗ ≡ I ∈ W N \ W Σ N . This and the fact that Ψ N ∈ C ∞ c ( R n + ) immediately givethat ∇ Ψ N ≡ S I ∈W N \W Σ N I ∗∗ .We are left with showing the last part of ( iii ) and for that we borrow some ideas from [29, AppendixA.2]. Fix I ∈ W Σ N and let J be so that J ∈ W \ W N with ∂ I ∩ ∂ J , Ø, in particular ℓ ( I ) ≈ ℓ ( J ).Since I ∈ W Σ N there exists Q I ∈ D F N , Q . Pick Q J ∈ D so that ℓ ( Q J ) = ℓ ( J ) and it contains any fixed b y ∈ ∂ Ω such that dist( J , ∂ Ω ) = dist( J , b y ). Then, as observed in Section 2.3, one has J ∈ W ∗ Q J . But, since J ∈ W \ W N , we necessarily have Q J < D F N , Q = D F N ∩ D Q . Hence, W Σ N = W Σ , N ∪ W Σ , N ∪ W Σ , N where W Σ , N : = { I ∈ W Σ N : Q ⊂ Q J } , W Σ , N : = { I ∈ W Σ N : Q J ⊂ Q ∈ F N } , W Σ , N : = { I ∈ W Σ N : Q J ∩ Q = Ø } . For later use it is convenient to observe that(6.9) dist( Q J , I ) ≤ dist( Q J , J ) + diam( J ) + diam( I ) ≈ ℓ ( J ) + ℓ ( I ) ≈ ℓ ( I ) . Let us first consider W Σ , N . If I ∈ W Σ , N we clearly have ℓ ( Q ) ≤ ℓ ( Q J ) = ℓ ( J ) ≈ ℓ ( I ) ≈ ℓ ( Q I ) ≤ ℓ ( Q )and since Q I ∈ D Q dist( I , x Q ) ≤ dist( I , Q I ) + diam( Q ) ≈ ℓ ( I ) . In particular, W Σ , N .
1. Thus if we set b Q I : = Q J it follows from (6.9) that the two first conditions in(6.7) hold and also P I ∈W Σ , N b Q I ≤ W Σ , N . W Σ , N and W Σ , N , we proceed as follows. For any I ∈ W Σ , N ∪ W Σ , N wepick b Q I ∈ D so that b Q I ∋ x Q J and ℓ ( b Q I ) = − M ′ ℓ ( Q J ) with M ′ ≥ M ′ ≥ Ξ (cf. (2.14)). Note that b Q I ⊂ ∆ Q J ⊂ Q J which, together with (6.9), implies(6.10) dist( I , b Q I ) ≤ dist( I , Q J ) + diam( Q J ) . ℓ ( I )and(6.11) diam( b Q I ) ≤ Ξ r b Q I ≤ Ξ ℓ ( b Q I ) = − M ′ + Ξ ℓ ( Q J ) ≤ Ξ − ℓ ( Q J ) . Hence, the first two conditions in (6.7) hold for I ∈ W Σ , N ∪ W Σ , N .To see that the last condition in (6.7) holds, we start with the family W Σ , N . For any I ∈ W Σ , N there is a unique Q j ∈ F N such that Q J ⊂ Q j . But, since Q I ∈ D F N , Q then necessarily Q I Q j and Q I \ Q j , Ø. This and the fact that 2 ∆ Q J ⊂ Q J ⊂ Q j imply2 Ξ − ℓ ( Q J ) ≤ dist( x Q J , ∂ Ω \ Q j ) ≤ dist( x Q J , Q I \ Q j ) ≤ diam( Q J ) + dist( Q J , J ) + diam( J ) + diam( I ) + dist( I , Q I ) + diam( Q I ) ≈ ℓ ( J ) ≈ ℓ ( I ) . Thus, 2 Ξ − ℓ ( Q J ) ≤ dist( x Q J , ∂ Ω \ Q j ) ≤ C ℓ ( J ). Suppose next that I , I ′ ∈ W Σ , N are so that b Q I ∩ b Q I ′ , Ø(it could even happen that they are indeed the same cube) and assume without loss of generality that b Q I ′ ⊂ b Q I , hence ℓ ( I ′ ) ≤ ℓ ( I ). Let Q j , Q j ′ ∈ F N be so that Q J ⊂ Q j and Q J ′ ⊂ Q j ′ . Then, x Q J ∈ b Q I and x Q J ′ ∈ b Q I ′ ⊂ b Q I ⊂ Q J . As a consequence, x Q J ′ ∈ Q J ′ ∩ Q J ⊂ Q j ∩ Q ′ j and this forces Q j = Q j ′ (since F N is a pairwise disjoint family). This and (6.11) readily imply2 Ξ − ℓ ( Q J ) ≤ dist( x Q J , ∂ Ω \ Q j ) ≤ | x Q J − x Q J ′ | + dist( x Q J ′ , ∂ Ω \ Q j ) ≤ diam( b Q I ) + dist( x Q J ′ , ∂ Ω \ Q j ′ ) ≤ diam( b Q I ) + C ℓ ( J ′ ) ≤ Ξ − ℓ ( Q J ) + C ℓ ( J ′ )and therefore Ξ − ℓ ( Q J ) ≤ C ℓ ( J ′ ). This in turn gives ℓ ( I ) ≈ ℓ ( J ) ≈ ℓ ( J ′ ) ≈ ℓ ( I ′ ). Note also that since I touches J , I ′ touches J ′ , and b Q I ∩ b Q I ′ , Ø we obtaindist( I , I ′ ) ≤ diam( J ) + dist( J , Q J ) + diam( Q J ) + diam( Q J ′ ) + dist( Q J ′ , J ′ ) + diam( J ′ ) ≈ ℓ ( J ) + ℓ ( J ′ ) ≈ ℓ ( I ) . As a result, for fixed I ∈ W Σ , N there is a uniformly bounded number of I ′ ∈ W Σ , N with b Q I ∩ b Q I ′ , Ø,thus P I ∈W Σ , N b Q I . W Σ , N . Let I ∈ W Σ , N . Then, Q ∩ ∆ Q J ⊂ Q ∩ Q J = Ø andtherefore 2 Ξ − ℓ ( Q J ) ≤ dist( x Q J , Q ). Besides, since Q I ⊂ Q , we havedist( x Q J , Q ) ≤ diam( Q J ) + dist( Q J , J ) + diam( J ) + diam( I ) + dist( I , Q I ) + diam( Q I ) ≈ ℓ ( J ) ≈ ℓ ( I ) . Thus, 2 Ξ − ℓ ( Q J ) ≤ dist( x Q J , Q ) ≤ C ℓ ( J ). Suppose next that I , I ′ ∈ W Σ , N are so that b Q I ∩ b Q I ′ , Ø(it could even happen that they are indeed the same cube) and assume without loss of generality that b Q I ′ ⊂ b Q I , hence ℓ ( J ′ ) ≤ ℓ ( J ). Then, since x Q J ∈ b Q I and x Q J ′ ∈ b Q I ′ ⊂ b Q I we get from (6.11) that2 Ξ − ℓ ( Q J ) ≤ dist( x Q J , Q ) ≤ | x Q J − x Q J ′ | + dist( x Q J ′ , Q ) ≤ diam( b Q I ) + C ℓ ( J ′ ) ≤ Ξ − ℓ ( Q J ) + C ℓ ( J ′ ) , and therefore Ξ − ℓ ( Q J ) ≤ C ℓ ( J ′ ). This yields ℓ ( I ) ≈ ℓ ( J ) ≈ ℓ ( J ′ ) ≈ ℓ ( I ′ ). Note also that since I touches J , I ′ touches J ′ , and b Q I ∩ b Q I ′ , Ø we obtaindist( I , I ′ ) ≤ diam( J ) + dist( J , Q J ) + diam( Q J ) + diam( Q J ′ ) + dist( Q J ′ , J ′ ) + diam( J ′ ) ≈ ℓ ( J ) + ℓ ( J ′ ) ≈ ℓ ( I ) . LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 47
Consequently, for fixed I ∈ W Σ , N there is a uniformly bounded number of I ′ ∈ W Σ , N with b Q I ∩ b Q I ′ , Ø.As a result, P I ∈W Σ , N b Q I .
1. This completes the proof of ( iii ) and hence that of Lemma 6.4. (cid:3)
We are now ready to prove Theorem 6.1.
Proof of Theorem 6.1.
We use some ideas from [8, Section 4] and [6, Section 4]. Let u ∈ W , ( Ω ) ∩ L ∞ ( Ω ) be a weak solution of L u = Ω and assume that k u k L ∞ ( Ω ) =
1. Applying Theorem 1.9(c) = ⇒ (a) to u , we are reduced to showing that for some r > S α r u ( x ) < ∞ , for ω L -a.e. x ∈ ∂ Ω , where α is given in Theorem 1.9. By (5.16) and Lemma 2.13, it su ffi ces to see that for every fixed Q ∈ D k and for some fixed large ϑ (which depends on α and hence solely on the 1-sided NTA andCDC constants) one has Q = [ N ≥ b E N , ω X L ( b E ) = S ϑ Q u ∈ L ( b E N , ω L ) , ∀ N ≥ , (6.12)where X is given at the beginning of Section 5.1. Fix then Q ∈ D k and write ω : = ω X L , ω : = ω X L , G : = G L ( X , · ) , and G : = G L ( X , · ) . (6.13)Much as in (4.25) (with η = − / so that Γ ϑ, ∗ Q = Γ ϑ, ∗ Q ,η ) there exist e α > C (depending on the1-sided NTA and CDC constants) such that if we set e r : = C r Q >
0, then(6.14) Γ ϑ, ∗ Q ( x ) : = [ x ∈ Q ∈ D Q U ϑ, ∗ Q ⊂ Γ e α e r ( x ) , x ∈ Q . As a result,(6.15) S Q γ ϑ ( x ) : = X x ∈ Q ∈ D Q γ ϑ Q : = X x ∈ Q ∈ D Q ¨ U ϑ, ∗ Q a ( X ) δ ( X ) − n − dX + ¨ U ϑ, ∗ Q | div C D ( X ) | δ ( X ) − n dX . ¨ Γ ϑ, ∗ Q ( x ) a ( X ) δ ( X ) − n − dX + ¨ Γ ϑ, ∗ Q ( x ) | div C D ( X ) | δ ( X ) − n dX ≤ ¨ Γ α { e r , r } ( x ) a ( X ) δ − n − dX + ¨ Γ α { e r , r } ( x ) | div C D ( X ) | δ − n dX < ∞ , for ω L -a.e. x ∈ Q , where we have used the fact that the family { U ϑ, ∗ Q } Q ∈ D has bounded overlap, that α , α ≥ e α and the last estimate follows from (6.2), (6.3), and (5.1).Given N > C ( C is the constant that appeared in Section 5.1), let F N ⊂ D Q be the collection ofmaximal cubes (with respect to the inclusion) Q j ∈ D Q such that X Q j ⊂ Q ∈ D Q γ ϑ Q > N . (6.16)Write E : = \ N > C ( Q \ E N ) , E N : = Q \ [ Q j ∈F N Q j , Q = E ∪ ( Q \ E ) = E ∪ (cid:16) [ N > C E N (cid:17) . (6.17)Let us observe that(6.18) S Q γ ϑ ( x ) ≤ N , ∀ x ∈ E N . Otherwise, there exists a cube Q x ∋ x such that P Q x ⊂ Q ∈ D Q γ ϑ Q > N , hence x ∈ Q x ⊂ Q j for some Q j ∈ F N , which is a contradiction. Note that if x ∈ E , then for every N > C there exists Q Nx ∈ F N such that Q Nx ∋ x . By the definitionof F N , we then have S Q γ ϑ ( x ) = X x ∈ Q ∈ D Q γ ϑ Q ≥ X Q Nx ⊂ Q ∈ D Q γ ϑ Q > N . On the other hand, if x ∈ Q \ E N + there exists Q x ∈ F N + such that x ∈ Q x . By (6.16) one has X Q x ⊂ Q ∈ D Q γ ϑ Q > ( N + > N , and the maximality of the cubes in F N gives that Q x ⊂ Q ′ x for some Q ′ x ∈ F N with x ∈ Q ′ x ⊂ Q \ E N .This shows that { Q \ E N } N is a decreasing sequence of sets, and since Q \ E N ⊂ Q for every N weconclude that(6.19) ω ( E ) = lim N →∞ ω ( Q \ E N ) ≤ lim N →∞ ω ( { x ∈ Q : S Q γ ϑ ( x ) > N } ) = ω ( { x ∈ Q : S Q γ ϑ ( x ) = ∞} ) = , where the last equality uses (6.15). This and (6.17) imply that to get (6.12) we are left with proving(6.20) S ϑ Q u ∈ L ( E N , ω ) , ∀ N > C . With this goal in mind, note that if Q ∈ D Q is so that Q ∩ E N , Ø, then necessarily Q ∈ D F N , Q ,otherwise, Q ⊂ Q ′ ∈ F N , hence Q ⊂ Q \ E N . Recalling (6.13) and the fact X < B ∗ Q , we useLemma 3.26 and Harnack’s inequality to conclude that ˆ E N S ϑ Q u ( x ) d ω ( x ) = ˆ E N ¨ S x ∈ Q ∈ D Q U ϑ Q |∇ u ( Y ) | δ ( Y ) − n dYd ω ( x )(6.21) . X Q ∈ D Q ℓ ( Q ) − n ω ( Q ∩ E N ) ¨ U ϑ Q |∇ u ( Y ) | dY ≤ X Q ∈ D F N , Q ℓ ( Q ) − n ω ( Q ) ¨ U ϑ Q |∇ u ( Y ) | dY . X Q ∈ D F N , Q ¨ U ϑ Q |∇ u ( Y ) | G ( Y ) dY . ¨ Ω ϑ F N , Q |∇ u ( Y ) | G ( Y ) dY , where we have used that the family { U ϑ Q } Q ∈ D has bounded overlap. To estimate the last term we makethe following claim ¨ Ω ϑ F N , Q |∇ u ( Y ) | G ( Y ) dY . ω ( Q ) + X Q ∈ D F N , Q γ ϑ Q ω ( Q ) , (6.22)where the implicit constant is independent of N .Assuming this momentarily we note that(6.23) X Q ∈ D F N , Q γ ϑ Q ω ( Q ) = ˆ Q X x ∈ Q ∈ D F N , Q γ ϑ Q d ω ( x ) LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 49 ≤ ˆ E N S Q γ ϑ ( x ) d ω ( x ) + X Q j ∈F N X Q ∈ D F N , Q γ ϑ Q ω ( Q ∩ Q j ) ≤ N ω ( Q ) + X Q j ∈F N X Q ∈ D F N , Q γ ϑ Q ω ( Q ∩ Q j ) , where the last estimate follows from (6.18). In order to control the last term we fix Q j ∈ F N . Note thatif Q ∈ D F N , Q is so that Q ∩ Q j , Ø then necessarily Q j ( Q ⊂ Q . Write b Q j for the dyadic parent of Q j , that is, b Q j is the unique dyadic cube containing Q j with ℓ ( b Q j ) = ℓ ( Q j ). By the fact that Q j is themaximal cube so that (6.16) holds one obtains X b Q j ⊂ Q ∈ D Q γ ϑ Q = X Q j ( Q ∈ D Q γ ϑ Q ≤ N . As a result,(6.24) X Q j ∈F N X Q ∈ D F N , Q γ ϑ Q ω ( Q ∩ Q j ) = X Q j ∈F N ω ( Q j ) X Q j ( Q ∈ D Q γ ϑ Q ≤ N X Q j ∈F N ω ( Q j ) ≤ N ω (cid:18) [ Q j ∈F N Q j (cid:19) ≤ N ω ( Q ) . Collecting (6.21), (6.22), (6.23), and (6.24), we deduce that ˆ E N ( S ϑ Q u ( x )) d ω ( x ) ≤ C N ω ( Q ) ≤ C N . This shows (6.20) and completes the proof of Theorem 6.1 modulo proving (6.22).Let us then establish (6.22). For every M ≥
4, we consider the pairwise disjoint collection F N , M given by the family of maximal cubes of the collection F N augmented by adding all the cubes Q ∈ D Q such that ℓ ( Q ) ≤ − M ℓ ( Q ). In particular, Q ∈ D F N , M , Q if and only if Q ∈ D F N , Q and ℓ ( Q ) > − M ℓ ( Q ).Moreover, D F N , M , Q ⊂ D F N , M ′ , Q for all M ≤ M ′ , and hence Ω ϑ F N , M , Q ⊂ Ω ϑ F N , M ′ , Q ⊂ Ω ϑ F N , Q . Then themonotone convergence theorem implies ¨ Ω ϑ F N , Q |∇ u | G dX = lim M →∞ ¨ Ω ϑ F N , M , Q |∇ u | G dX = : lim M →∞ K N , M . (6.25)Write E ( X ) : = A ( X ) − A ( X ) and pick Ψ N , M from Lemma 6.4. By Leibniz’s rule,(6.26) A ∇ u · ∇ u G Ψ N , M = A ∇ u · ∇ ( u G Ψ N , M ) − A ∇ ( u Ψ N , M ) · ∇G + A ∇ ( Ψ N , M ) · ∇G u − A ∇ ( u ) · ∇ ( Ψ N , M ) G − E∇ ( u ) · ∇ ( G Ψ N , M ) . Note that u ∈ W , ( Ω ) ∩ L ∞ ( Ω ), G ∈ W , ( Ω \ { X } ), and that Ω ϑ, ∗∗F N , M , Q is a compact subset of Ω awayfrom X since X < B ∗ Q and (2.23). Hence, u ∈ W , ( Ω ϑ, ∗∗F N , M , Q ) and u G Ψ N , M ∈ W , ( Ω ϑ, ∗∗F N , M , Q ). Thesetogether with the fact that L u = Ω give ¨ Ω A ∇ u · ∇ ( u G Ψ N , M ) dX = ¨ Ω ϑ, ∗∗F N , M , Q A ∇ u · ∇ ( u G Ψ N , M ) dX = . (6.27) On the other hand, Lemma 3.16 (see in particular (3.22)) implies that G ∈ W , ( Ω ϑ, ∗∗F N , M , Q ) and L ⊤ G = Ω \ { X } . Thanks to the fact that u Ψ N , M ∈ W , ( Ω ϑ, ∗∗F N , M , Q ), we then obtain ¨ Ω A ∇ ( u Ψ N , M ) · ∇G dX = ¨ Ω ϑ, ∗F N , M , Q A ⊤ ∇G · ∇ ( u Ψ N , M ) dX = . (6.28)By Lemma 6.4, the ellipticity of A and A , (6.26)–(6.28), the fact that k u k L ∞ ( Ω ) =
1, and our assumption E = A − A = − ( A + D ) we then arrive at e K N , M : = ¨ Ω |∇ u | G Ψ N , M dX . ¨ Ω A ∇ u · ∇ u G Ψ N , M dX (6.29) . ¨ Ω |∇ Ψ N , M | |∇G | dX + ¨ Ω |∇ u | |∇ Ψ N , M | G dX + (cid:12)(cid:12)(cid:12)(cid:12) ¨ Ω A ∇ ( u ) · ∇ ( G Ψ N , M ) dX (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ¨ Ω D ∇ ( u ) · ∇ ( G Ψ N , M ) dX (cid:12)(cid:12)(cid:12)(cid:12) = : I + I + I + I . We estimate each term in turn. Regarding I we use Lemma 6.4, Caccioppoli’s and Harnack’sinequalities, and Lemma 3.26:(6.30) I . X I ∈W ϑ, Σ N , M ¨ I ∗ |∇ Ψ N , M | |∇G | dX . X I ∈W ϑ, Σ N , M ℓ ( I ) − | I | (cid:18) ¨ I ∗ |∇G | dX (cid:19) . X I ∈W ϑ, Σ N , M ℓ ( I ) n − G ( X ( I )) . X I ∈W ϑ, Σ N , M ω ( b Q I ) . ω (cid:18) [ I ∈W ϑ, Σ N , M b Q I (cid:19) ≤ ω ( C ∆ Q ) . ω ( Q ) , where the implicit constants do not depend on N nor M . We estimate I similarly:(6.31) I . X I ∈W ϑ, Σ N , M ¨ I ∗ |∇ Ψ N , M | |∇ u | G dX . X I ∈W ϑ, Σ N , M ℓ ( I ) − | I | G ( X ( I )) (cid:18) ¨ I ∗ |∇ u | dX (cid:19) . X I ∈W ϑ, Σ N , M ℓ ( I ) n − G ( X ( I )) . ω ( Q ) . Concerning I we use that A ∈ L ∞ ( Ω ) and k u k L ∞ ( Ω ) = I . ¨ Ω | A | |∇ u | |∇G | Ψ N , M dX + ¨ Ω |∇ u | |∇ Ψ N , M | Ψ N , M G dX = : I ′ + I ′′ . (6.32)Observe that I ∗∗ ⊂ { Y ∈ Ω : | Y − X | < δ ( X ) / } for every X ∈ I ∗ , and hence sup I ∗∗ | A | ≤ inf I ∗ a . ByCauchy-Schwarz inequality, Caccioppoli’s and Harnack’s inequalities, and Lemma 3.26 we have I ′ . X Q ∈ D F N , Q X I ∈W ϑ, ∗ Q sup I ∗∗ | A | (cid:18) ¨ I ∗∗ |∇ u | Ψ N , M dX (cid:19) (cid:18) ¨ I ∗∗ |∇G | dX (cid:19) (6.33) . X Q ∈ D F N , Q X I ∈W ϑ, ∗ Q (cid:18) ¨ I ∗∗ |∇ u | Ψ N , M dX (cid:19) (cid:16) sup I ∗∗ | A | G ( X ( I )) ℓ ( I ) n − (cid:17) LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 51 . X Q ∈ D F N , Q X I ∈W ϑ, ∗ Q (cid:18) ¨ I ∗∗ |∇ u | G Ψ N , M dX (cid:19) (cid:18) ω ( Q ) ¨ I ∗ a ( X ) δ ( X ) − n − dX (cid:19) . (cid:18) ¨ Ω |∇ u | G Ψ N , M dX (cid:19) (cid:18) X Q ∈ D F N , Q ω ( Q ) ¨ U ϑ, ∗ Q a ( X ) δ ( X ) − n − dX (cid:19) ≤ e K N , M (cid:16) X Q ∈ D F N , Q γ ϑ Q ω ( Q ) (cid:17) , where we used the fact that the family { I ∗∗ } I ∈W has bounded overlap. Additionally, as in (6.30)(6.34) I ′′ . (cid:18) ¨ Ω |∇ u | G Ψ N , M dX (cid:19) (cid:18) ¨ Ω |∇ Ψ N , M | G dX (cid:19) . e K N , M (cid:18) X I ∈W ϑ, Σ N , M ℓ ( I ) n − G ( X ( I )) (cid:19) . e K N , M ω ( Q ) . Finally, to bound I , we note that u ∈ W , ( Ω ), G Ψ N , M ∈ W , ( Ω ) and supp( G Ψ N , M ) ⊂ Ω ϑ, ∗F N , M , Q is compactly contained in Ω . Then [8, Lemma 4.1] and Lemma 3.26 imply that I = (cid:12)(cid:12)(cid:12)(cid:12) ¨ Ω div C D · ∇ ( u ) G Ψ N , M dX (cid:12)(cid:12)(cid:12)(cid:12) (6.35) . (cid:18) ¨ Ω |∇ u | G Ψ N , M dX (cid:19) (cid:18) ¨ Ω | div C D | G Ψ N , M dX (cid:19) . e K N , M (cid:18) X Q ∈ D F N , Q X I ∈W ϑ, ∗ Q G ( X ( I )) ¨ I ∗∗ | div C D | dX (cid:19) . e K N , M (cid:18) X Q ∈ D F N , Q X I ∈W ϑ, ∗ Q ω ( Q ) ¨ I ∗∗ | div C D ( X ) | δ ( X ) − n dX (cid:19) . e K N , M (cid:18) X Q ∈ D F N , Q ω ( Q ) ¨ U ϑ, ∗ Q | div C D ( X ) | δ ( X ) − n dX (cid:19) ≤ e K N , M (cid:16) X Q ∈ D F N , Q γ ϑ Q ω ( Q ) (cid:17) . Gathering (6.29)–(6.35) and using Young’s inequality we obtain e K N , M . ω ( Q ) + e K N , M ω ( Q ) + e K N , M (cid:16) X Q ∈ D F N , Q γ ϑ Q ω ( Q ) (cid:17) ≤ C ω ( Q ) + C X Q ∈ D F N , Q γ ϑ Q ω ( Q ) + e K N , M , where the implicit constants are independent of N and M . Note that e K N , M < ∞ because supp Ψ N , M ⊂ Ω ϑ, ∗F N , M , Q , which is a compact subset of Ω and u ∈ W , ( Ω ). Thus, the last term can be hidden and we eventually obtain K N , M ≤ e K N , M . ω ( Q ) + X Q ∈ D F N , Q γ ϑ Q ω ( Q ) . This estimate (whose implicit constant is independent of N and M ) and (6.25) readily yield (6.22) andthe proof is complete. (cid:3) R eferences [1] M. Akman, S. Hofmann, J.M. Martell, and T. Toro, Perturbation of elliptic operators in 1-sided NTA domains satisfyingthe capacity density condition , http://arxiv.org/abs/1901.08261 . 3, 4, 5, 6, 9, 10, 13, 26, 28, 30, 41, 42, 43, 44[2] J. Azzam, S. Hofmann, J.M. Martell, S. Mayboroda, M. Mourgoglou, X. Tolsa, and A. Volberg, Rectifiability of har-monic measure . Geom. Funct. Anal. (2016), 703–728. 2[3] J. Azzam, S. Hofmann, J.M. Martell, M. Mourgoglou, and X. Tolsa, Harmonic measure and quantitative connectivity:geometric characterization of the L p -solvability of the Dirichlet problem , Invent. Math. (2020), 881–993. 2[4] J. Azzam, S. Hofmann, J.M. Martell, K. Nystr¨om, and T. Toro, A new characterization of chord-arc domains , J. Eur.Math. Soc. (2017), 967–981. 2, 3[5] C.J. Bishop and P.W. Jones, Harmonic measure and arclength , Ann. of Math. (2) (1990), 511–547. 2[6] M. Cao, J.M. Martell, and A. Olivo,
Elliptic measures and square function estimates on 1-sided chord-arc domains , http://arxiv.org/abs/2007.06992 . 5, 20, 47[7] J. Cavero, S. Hofmann, and J.M. Martell, Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Smalland large perturbation for symmetric operators , Trans. Amer. Math. Soc. (2019), 2797–2835. 3[8] J. Cavero, S. Hofmann, J.M. Martell, and T. Toro,
Perturbations of elliptic operators in 1-sided chord-arc domains. PartII: Non-symmetric operators and Carleson measure estimates , Trans. Amer. Math. Soc. (2020), no. 11, 7901–7935.3, 7, 20, 43, 47, 51[9] M. Christ,
A T ( b ) theorem with remarks on analytic capacity and the Cauchy integral , Colloq. Math. / (1990),no. 2, 601–628. 9[10] R.R. Coifman and C. Fe ff erman, Weighted norm inequalities for maximal functions and singular integrals , Studia Math. (1974), 241–250. 23, 25, 29, 30, 33[11] D. Cruz-Uribe, J.M. Martell, and C. P´erez, Weights, extrapolation and the theory of Rubio de Francia , Operator Theory:Advances and Applications, Vol. , Birkh¨auser / Springer Basel AG, Basel, 2011. 24[12] B.E.J. Dahlberg,
Estimates of harmonic measure , Arch. Rational Mech. Anal. (1977), 275–288. 2[13] B.E.J. Dahlberg, On the absolute continuity of elliptic measures , Amer. J. Math. (1986), 1119–1138. 3[14] G. David, M. Engelstein, and S. Mayboroda,
Square functions, non-tangential limits and harmonic measure in co-dimensions larger than one , Duke Math. J. (2020), to appear. 2[15] G. David, J. Feneuil, and S. Mayboroda,
Dahlberg’s theorem in higher co-dimension , J. Funct. Anal. (2019), 2731–2820. 2[16] G. David and D. Jerison,
Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals , IndianaUniv. Math. J. (1990), 831–845. 2[17] G. David and S. Mayboroda, Harmonic measure is absolutely continuous with respect to the Hausdor ff measure on alllow-dimensional uniformly rectifiable sets , http://arxiv.org/abs/arXiv:2006.14661 Necessary and su ffi cient conditions for absolute continuity of elliptic-harmonicmeasure , Ann. of Math. (2), (1984), 121–141. 3[19] R. Fe ff erman, A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator , J.Amer. Math. Soc. (1989), 127–135. 3[20] R. Fe ff erman, C.E. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations , Ann. ofMath. (2), (1991), 65–124. 3[21] J. Feneuil and B. Poggi,
Generalized Carleson perturbations of elliptic operators and applications , http://arxiv.org/abs/arXiv:2011.06574 . 7[22] J. Garc´ıa-Cuerva and J.L. Rubio de Francia, Weighted norm inequalities and related topics , North-Holland MathematicsStudies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985, Notas de Matem´atica [Mathematical Notes], 104.23, 29, 30, 33[23] F.W. Gehring,
The L p -integrability of the partial derivatives of a quasiconformal mapping , Acta Math. (1973),265–277. 25[24] M. Gr¨uter and K.-O. Widman, The Green function for uniformly elliptic equations , Manuscripta Math. (1982), no. 3,303–342. 17 LLIPTIC OPERATORS IN 1-SIDED NTA DOMAINS SATISFYING THE CAPACITY DENSITY CONDITION 53 [25] J. Heinonen, T. Kilpel¨ainen, and O. Martio,
Nonlinear potential theory of degenerate elliptic equations , Dover Publica-tions, Inc., Mineola, NY, 2006, Unabridged republication of the 1993 original. 9[26] S. Hofmann and P. Le,
BMO solvability and absolute continuity of harmonic measure , J. Geom. Anal. (2018), no. 4,3278–3299. 2, 3, 4, 5, 16[27] S. Hofmann, P. Le, J.M. Martell, and K. Nystr¨om, The weak-A ∞ property of harmonic and p-harmonic measures impliesuniform rectifiability , Anal. PDE (2017), 513–558. 9[28] S. Hofmann and J.M. Martell, Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poissonkernels in L p , Ann. Sci. ´Ec. Norm. Sup´er. (2014), 577–654. 2, 3, 11, 12, 13, 40[29] S. Hofmann, J.M. Martell, and S. Mayboroda, Uniform rectifiability, Carleson measure estimates, and approximationof harmonic functions , Duke Math. J. (2016), no. 12, 2331–2389. 44, 45[30] S. Hofmann, J.M. Martell, S. Mayboroda, T. Toro, and Z. Zhao,
Uniform rectifiability and elliptic operators satisfyinga Carleson measure condition , http://arxiv.org/abs/arXiv:2008.04834 A ∞ implies NT A for a class of variable coe ffi cient elliptic operators , J. Di ff er-ential Equations (2017), 6147–6188. 3, 43, 44[32] S. Hofmann, J.M. Martell, and T. Toro, General divergence form elliptic operators on domains with ADR boundaries,and on 1-sided NTA domains , Book in preparation. 7, 11, 12, 14, 16, 17[33] S. Hofmann, J.M. Martell, and I. Uriarte-Tuero,
Uniform rectifiability and harmonic measure, II: Poisson kernels in L p imply uniform rectifiability , Duke Math. J. (2014), 1601–1654. 2, 3[34] S. Hofmann, D. Mitrea, M. Mitrea, and A.J. Morris, L p -square function estimates on spaces of homogeneous type andon uniformly rectifiable sets , Mem. Amer. Math. Soc. (2017), no. 1159. 9[35] S. Hofmann, M. Mitrea, and M. Taylor, Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains , Int. Math. Res. Not. IMRN
Systems of dyadic cubes in a doubling metric space , Colloq. Math. (2012), no. 1, 1–33.9[37] T. Hyt¨onen and A. Kairema,
What is a cube? , Ann. Acad. Sci. Fenn. Math. (2013), no. 2, 405–412. 9[38] D.S. Jerison and C.E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains , Adv.Math. (1982), 80–147. 2, 8[39] C.E. Kenig, B. Kirchheim, J. Pipher, and T. Toro, Square functions and the A ∞ property of elliptic measures , J. Geom.Anal. (2014), no. 3, 2383–2410. 7, 20[40] C.E. Kenig, H. Koch, J. Pipher, and T. Toro, A new approach to absolute continuity of elliptic measure, with applicationsto non-symmetric equations , Adv. Math. (2000), no. 2, 231–298. 7, 20[41] C.E. Kenig and J. Pipher,
The Dirichlet problem for elliptic equations with drift terms , Publ. Mat. (2001), 199–217.3[42] J.L. Lewis, Uniformly fat sets , Trans. Amer. Math. Soc. (1988), 177–196. 9[43] J.M. Martell and C. Prisuelos-Arribas, W eighted Hardy spaces associated with elliptic operators. Part I: Weighted norminequalities for conical square functions, Trans. Amer. Math. Soc. (2017), no. 6, 4193–4233. 23[44] S. Mayboroda and Z. Zhao, Square function estimates, the BMO Dirichlet problem, and absolute continuity of harmonicmeasure on lower-dimensional sets , Anal. PDE (2019), 1843–1890. 2[45] E. Milakis, J. Pipher, and T. Toro, Harmonic analysis on chord arc domains , J. Geom. Anal. (2013), 2091–2157. 3[46] E. Milakis, J. Pipher, and T. Toro, Perturbations of elliptic operators in chord arc domains , Harmonic analysis andpartial di ff erential equations, 143–161, Contemp. Math., , Amer. Math. Soc., Providence, RI, 2014. 3[47] F. Riesz and M. Riesz, ¨Uber die randwerte einer analtischen funktion , Compte Rendues du Quatri`eme Congr`es desMath´ematiciens Scandinaves, Stockholm 1916, Almqvists and Wilksels, Uppsala, 1920. 2[48] Z. Zhao, BMO solvability and A ∞ condition of the elliptic measures in uniform domains , J. Geom. Anal. (2018),no. 2, 866–908. 9 M ingming C ao , I nstituto de C iencias M atem ´ aticas CSIC-UAM-UC3M-UCM, C onsejo S uperior de I nvestigaciones C ien - t ´ ıficas , C / N icol ´ as C abrera , 13-15, E-28049 M adrid , S pain Email address : [email protected] ´O scar D om ´ ınguez , D epartamento de A n ´ alisis M atem ´ atico y M atem ´ atica A plicada , F acultad de M atem ´ aticas , U niversi - dad C omplutense de M adrid , P laza de C iencias
3, E-28040 M adrid , S pain . Email address : [email protected] J os ´ e M ar ´ ıa M artell , I nstituto de C iencias M atem ´ aticas CSIC-UAM-UC3M-UCM, C onsejo S uperior de I nvestigaciones C ient ´ ıficas , C / N icol ´ as C abrera , 13-15, E-28049 M adrid , S pain Email address : [email protected] P edro T radacete , I nstituto de C iencias M atem ´ aticas CSIC-UAM-UC3M-UCM, C onsejo S uperior de I nvestigaciones C ient ´ ıficas , C / N icol ´ as C abrera , 13-15, E-28049 M adrid , S pain Email address ::