10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Unique Continuation at the Boundary for Harmonic Functions in C 1 Domains and Lipschitz Domains with Small Constant XAVIER TOLSA ICREA, Universitat Autònoma de Barcelona, and Centre de Recerca Matemàtica Abstract Let   R be a C domain, or more generally, a Lipschitz domain with small n 1 local Lipschitz constant. In this paper it is shown that if u is a function harmonic in  and continuous in  S , which vanishes in a relatively open subset †  @; moreover, the normal derivative @ u vanishes in a subset of † with positive sur- face measure; then u is identically zero. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. 1 Introduction In Rn , with n  3, there are examples of harmonic functions in the half-space RC , C 1 up to the boundary, such that the function and its normal derivative vanish n simultaneously on a set of positive measure of @RnC . This was shown by Bourgain and Wolff in [5]. The same result was generalized later to arbitrary C 1; domains by Wang [19]. A related conjecture that is still open is the following: Conjecture. Let   Rn be a Lipschitz domain and let †  @ be relatively open with respect to @. Let u be a function harmonic in  and continuous in  x. Suppose that u vanishes in † and the normal derivative @ u vanishes in a subset of † with positive surface measure. Then u  0 in x. Note that the assumption that u vanishes continuously in † implies that r u ex- ists  -a.e. as a nontangential limit in †, and moreover r u D .@ u/  2 L2loc . j† /. Here  stands for the .n 1/-dimensional surface measure and  is the outer unit normal. See Appendix A for more details. The preceding conjecture is an open problem that is already mentioned in Fang- Hua Lin’s work [12]. It was later stated explicitly as a conjecture in the works by Adolfsson, Escauriaza, and Kenig [1, 2]1. The conjecture is known to be true in the plane, and also in higher dimensions if one assumes the function u to be positive. In the first case, this can be deduced from the subharmonicity of log jr uj, 1 For an accurate historical account, see the recent work [10]. Communications on Pure and Applied Mathematics, Vol. LXXVI, 0305–0336 (2023) © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Peri- odicals LLC. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial- NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 306 X. TOLSA and in the second one it is possible to use standard techniques in connection with harmonic measure and the comparison principle. In this paper I show that the conjecture is true for Lipschitz domains with small local Lipschitz constant. The precise result is the following: T HEOREM 1.1. Let   Rn be a Lipschitz domain, let B be a ball centered in @, and suppose that † D B \ @ is a Lipschitz graph with slope at most 0 , where 0 is some positive small enough constant depending only on n. Let u be a function harmonic in  and continuous in  x . Suppose that u vanishes in † and the normal derivative @ u vanishes in a subset of † with positive surface measure. Then u  0 in x. As an immediate corollary, it follows that the above conjecture holds for C 1 domains. Note that, up now, the result stated in Theorem 1.1 (and in the conjecture) was only known in the case of Dini domains (i.e., Lipschitz domains whose outer nor- mal is Dini continuous), by results of Adolfsson and Escauriaza [1] and Kukavica and Nyström [11], and also in the case of convex Lipschitz domains, by Adolfsson, Escauriaza, and Kenig [2]. Previously, the case of C 1;1 domains had been solved by F.-H. Lin [12]. See also [16] for a recent contribution in the particular case of convex domains where the recent geometric techniques introduced by Naber and Valtorta [17] are applied to study the strata of the set where @ u vanishes. The proof of Theorem 1.1 is based on the study of the doubling properties of L2 averages of the harmonic function u by means of the so-called Almgren’s fre- quency function, which is analogous to the works mentioned in the previous para- graph. The strategy in this paper consists in studying the behavior of the frequency function at points in  approaching the boundary. This strategy is closer to the one of Kukavica and Nyström in [11] than to the one of Adolfsson and Escauri- aza [1], which is based on the use of a clever change of variables that transforms the Laplace equation into an elliptic PDE in divergence form with nonconstant coefficients and improves the domain, in a sense. The main novelty in the arguments to prove Theorem 1.1 is the application of some combinatorial techniques developed by Logunov and Malinnikova in the works [13–15] in connection with the nodal sets of harmonic functions and the Nadirashvili and Yau conjectures. In particular, one of the main technical results in this paper, the Key Lemma 3.1, uses some ideas inspired by [13] to bound the set where the frequency function is large. With the Key Lemma 3.1 in hand, in the last section of the paper a probabilistic argument is used to show that R @B.x;12r/ u2 d lim inf R 2 <1 r !0 @B.x;r/ u d for almost all points x 2 †. By a lemma due to Adolfsson and Escauriaza [1, lemma 0.2], this suffices to show that @ u cannot vanish in a subset of † with positive measure. 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 307 In the case that  is a Dini domain, in [1, 11] it is also proven that if u is harmonic in  and vanishes continuously in † (where † is as in Theorem 1.1), then j@ uj is a local B2 weight in † with respect to surface measure (i.e., it satisfies a local reverse Hölder inequality with exponent 2). This follows from the local uniform bound of Almgren’s frequency function proven in [1, 11], which in turn implies a local uniform doubling condition for L2 averages of the function u on surface balls. Then, as shown in [2, theorem 1], this doubling condition ensures that j@ uj is a local B2 weight. In the case of Lipschitz domains with small constant, the proof of Theorem 1.1 in this paper does not ensure that the frequency function is locally uniformly bounded (or even pointwise bounded!) in †, and thus one cannot deduce that j@ uj is a local B2 weight. In a similar vein, under the Dini assumption, in [1] it is shown that the dimension of the set where @ u vanishes in † has dimension at most n 2. This follows by arguments developed previously in [12] in the case of C 1;1 domains, which are based on the monotonicity of the frequency function in †. For C 1 domains or Lipschitz domains with small constant, one cannot derive any bound on the Hausdorff dimension smaller than n 1 from the arguments in this paper, as far as I know. Finally, it is worth mentioning a corollary about harmonic measure that follows easily from Theorem 1.1: C OROLLARY 1.2. Let   Rn be a Lipschitz domain, let B be a ball centered in @, and suppose that † D B \ @ is a Lipschitz graph with small enough slope. Let ! p , ! q be the harmonic measures for  with respective poles in p; q 2 . Suppose that there exists some subset E  † with positive harmonic measure such that ! p jE D ! q jE Then p D q: Note that saying that ! p jE D ! q jE is the same as saying that ! p .F / D ! q .F / for all Borel sets F  E . The corollary follows by applying the theorem to u D g.  ; p/ x "/ [ B.q; g.  ; q/ in  n .B.p; x "//, where g.  ;  / is the Green function of  and " > 0 is small enough so that B.p; 2"/ [ B.q; 2"/  . Observe that u is harmonic in the Lipschitz domain " WD  n .B.p; x "/ [ B.q; x "//, it is continuous in " , and it vanishes identically in @  @" . Further, by Dahlberg’s classical theorem [6], it follows that the har- monic measures ! p and ! q are mutually absolutely continuous with the surface measure  on @, with d! p d! q d D @ g.  ; p/; d D @ g.  ; q/; 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 308 X. TOLSA so that @ u vanishes in the whole E . So " and u satisfy the assumptions of Theorem 1.1, and thus u  0 in " . This implies that p D q . Otherwise, letting " ! 0 we infer that u.x/ ! 1 as x ! p . 2 The Frequency Function As usual in harmonic analysis, in the whole paper, the letters C; c are used to denote positive constants that just depend on the dimension n and whose values may change at different occurrences. On the other hand, constants with subscripts, such as C0 , retain their values in different occurrences. The notation A . B is equivalent to A  C B , and A  B is equivalent to A . B . A. In the whole paper, unless otherwise stated, we assume that  and † are as in Theorem 1.1. We consider a function u harmonic in  and continuous in  x which vanishes in †, and we assume that u is not constant in . We extend u by 0 out of x , so that u is continuous across †. Without loss of generality we assume that † is a Lipschitz graph with respect to the horizontal axes and that  \ B lies above † \ B . For 0 < "  21 r.B/, we denote †" D † C "en and " D  C " en , where en D .0; : : : ; 0; 1/. For x 2 Rn and r > 0, we denote Z 1 h.x; r/ D u2 d: .@B.x; r// @B.x;r/ For a ball B.x; r/ that intersects , the Almgren frequency function (or just fre- quency function) associated with u is defined by F .x; r/ D r@r log h.x; r/: L EMMA 2.1. Let x 2 Rn and let I  .0; 1/ be a closed bounded interval. Sup- pose that B.x; r/ \  ¤ ¿ and B.x; r/  2B for all r 2 I (where B is as Theorem 1.1). Then h.x;  / is of class C 1 in I and Z 2 y x @r h.x; r/ D u.y/r u.y/  d.y/ .@Br / @B.x;r/ r Z 2 (2.1) D .@Br / @B.x;r/\  u@ u d Z 2 D .@Br / B.x;r/\ jr uj2 dy for a.e. r 2 I: D 2 R .@Br / B.x;r/\ jr uj dy holds for all 2 I. Further, the identity @r h.x; r/ 2 r Also, R @ h.x; r/ 2r B.x;r/ jr uj2 dy (2.2) F .x; r/ D r r D R 2 for all r 2 I: h.x; r/ @B.x;r/ u d 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 309 Note that, for B.x; r/ as in the lemma, we have Z jr uj2 dy < 1: B.x;r/\ Indeed, write u D uC u and, for any " > 0, let uC C " D max.u ; "/ ", u" D max.u ; "/ ", and v" D uC " u" . It is immediate to check that uC " and 1;2 0 0 u" belong to W .B.x; r // for some r > r , and moreover they are subharmonic in B.x; r 0 /. As a consequence, by Caccioppoli’s inequality, Z Z Z jr u j2 dy . C.r; r 0 / " ju j2 dy  C.r; r 0 / " juj2 dy: B.x;r/ B.x;r 0 / B.x;r 0 / Letting " ! 0, we deduce that Z jr uj2 dy < 1: Further, it follows easily that v" ! u in W 1;2 .B.x; r//, and so u 2 W 1;2 .B.x; r// too. An immediate corollary of the lemma and, in particular, of the third identity in (2.1), is that @r h.x; r/  0 and thus h.x; r/ is nondecreasing with respect to r . P ROOF OF L EMMA 2.1. The calculations in the lemma are quite straightfor- ward and well-known in the case when u is sufficiently smooth up to the boundary. In the general case when we only assume u to be continuous up to the boundary, we have to be a little more careful, and so we will show here all the details. Notice first that the second R identity in (2.1) is immediate. Concerning the third one, for a.e. r 2 I we have @B.x;r/ jr uj2 d < 1, and so Z Z 2 2 u @ u d .@Br / @B.x;r/\  D "lim u @ u d !0 .@Br / @.B.x;r/\" /  because Z 2 lim "!0 .@Br / B.x;r/\@" u @ u d D 0: This follows easily from the fact that u vanishes continuously in †, while (2.3) lim "!0 r u.x C "en / ! r u.x/ in L2loc . j† / with r u D .@ u/ 2 L2loc . j† / defined as a nontangential limit, as shown in Theorem A.1. Then, by Green’s theorem, using that u is C 1 in a neighborhood of 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 310 X. TOLSA " \ B.x; r/ for any " > 0 sufficiently small and for a.e. r 2 I , we obtain Z Z 1 2 u @ u d D @ .u2 /d .@Br / @.B.x;r/\" /  @.B.x;r/\" / Z 1 D .@Br / B.x;r/\" .u2 /dy Z 2 D .@Br / B.x;r/\" jr uj2 dy: So letting " ! 0, taking into account that u 2 Wloc 1;2 .B/, the third identity in (2.1) follows. To show the first identity in (2.1), observe that, for all Œa; b  I , writing .@Br / D cn r n 1 , we have Z b Z 2 y x u.y/r u.y/  d.y/dr a .@Br / @B.x;r/ r Z bZ y x D 2cn 1 u.y/ r u.y/  j y x jn d.y/dr a @B.x;r/ Z y x D 2cn 1 u.y/ r u.y/  jy x jn dy A.x;a;b/\ Z   y x D cn 1 divy u.y/2 jy x jn dy; A.x;a;b/\ where A.x; a; b/ stands for the open annulus centered at x with inner radius a and outer radius b . Since u 2 Wloc 1;2 .B/ and is smooth in a neighborhood of B.x; b/ \ " , by the divergence theorem, we have Z   1 y x cn u.y/2 A.x;a;b/\ divy jy x jn dy Z   y x D cn "lim 1 !0 A.x;a;b/\" y div u.y/ 2 jy x jn dy  Z Z  1 1 D "lim !0 .@Bb / @B.x;b/\" 2 u d .@Ba / @B.x;a/\" 2 u d Z .y/  .y x/ C "lim !0 cn 1 u.y/2 jy x jn d.y/: A.x;a;b/\@" Since u vanishes continuously up to the boundary †, the last limit on the right-hand side above vanishes. Therefore, Z b Z 2 y x (2.4) u.y/ r u.y/  d.y/dr D h.x; b/ h.x; a/: a .@Br / @B.x;r/\ r 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 311 On the other hand, we have already shown that, for a.e. r 2 I, Z Z 2 y x 2 u.y/r u.y/  d.y/ D jr uj2 dy; .@Br / @B.x;r/\ r .@Br / B.x;r/\" and so this term is continuous in r , as u 2 Wloc 1;2 .B/. Then, the first identity in (2.1) follows from (2.4) and the fundamental theorem of calculus. The identity (2.2) is an immediate consequence of the definition of F .x; r/ and (2.1). □ The following lemma is already known. It is essentially contained (but not stated in this way) in [2]. For the reader’s convenience we include the detailed proof here. L EMMA 2.2. Let x 2 Rn and let I  .0; 1/ be a closed bounded interval. Sup- pose that B.x; r/ \  ¤ ¿ and B.x; r/  2B for all r 2 I , where B is as in Theorem 1.1. Then F .x;  / is absolutely continuous in I and, for a.e. r 2 I , @r F .x; r/ Z Z 4r D H.x; r/2 juj d 2 @ u 2 d @B.x;r/\ @B.x;r/\ Z 2  (2.5) u @ u d @B.x;r/ Z 2 C H.x; .y x/  .y/ @ u.y/ 2 d.y/; r/ B.x;r/\@ where Z H.x; r/ D .@Br / h.x; r/ D u2 d: @B.x;r/ In particular, if .y x/  .y/  0 for  -a.e. y 2 B.x; r/ \ @, then @r F .x; r/  0: P ROOF. Denote Z I .x; r/ D jr uj2 dy: B.x;r/ Since Z rZ I .x; r/ D jr uj2 d.y/dt; 0 @B.x;r/ it follows that I .x;  / is absolutely continuous with respect to r . As H.x;  / is of class C 1 and bounded away from 0 in I , we deduce that F .x;  / is also absolutely continuous in I . By (2.1), we have 2r I .x; r/ @r F .x; r/ D @r H.x; r/ 0 0 D 2 .I .x; r/ C r I .x; r//HH.x;.x;r/r/2 r I .x; r/ H .x; r/ ; 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 312 X. TOLSA where the symbol 0 denotes the derivative with respect to r . Observe that, by (2.1), for a.e. r 2 I , H0 .x; r/ D @r .@Br / h.x; r/ C .@Br / h0 .x; r/ D .n 1/.@Br / r h.x; r/ C 2 I .x; r/ D .n 1/ r H.x; r/ C 2 I .x; r/: Therefore, 2 @r F .x; r/ D r H.x; r/ I 0 .x; r/ (2.6) H.x; r/2  .n 2/H.x; r/ I .x; r/ 2r I .x; r/2 : To calculate I 0 .x; r/ we take into account that Z I 0 .x; r/ D jr uj2 d @B.x;r/ Z y x D r  .y/ jr u.y/j2 d.y/ @.B.x;r/\/ Z y x r  .y/jr u.y/j2 d.y/: B.x;r/\@ By the Rellich-Necas identity with vector field .y/ D y x , y 2 , we have div.jr uj2 / D 2 div..  r u/ r u/ C .n 2/ jr uj2 in : Integrating in B.x; r/ \ " (with " as in the proof of Lemma 2.1), applying the divergence theorem in this domain, and then letting " ! 0, taking into account (2.3), we derive Z .y x/  .y/ jr u.y/j2 d.y/ @.B.x;r/\/ Z D2 .y x/  r u.y/ @ u.y/d.y/ C .n 2/ I .x; r/ @.B.x;r/\/ Z Z D 2r j@ uj2 d C 2 .y x/  .y/ j@ u.y/j2 d.y/ @B.x;r/\ B.x;r/\@ C .n 2/ I .x; r/: Thus, Z n 2 I 0 .x; r/ D 2 @ u 2 d C I .x; r/ @B.x;r/\ r Z C 1r .y x/  .y/ @ u.y/ 2 d.y/: B.x;r/\@ 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 313 Plugging the last calculation for I 0 .x; r/ into (2.6), we obtain @r F .x; r/  Z 4r D H.x; H .x; r/ @ u 2 d r/2 @B.x;r/\ Z H.x; r/ .y x/  .y/ @ u.y/ 2 d.y/ (2.7) C 2r B.x;r/\@  I .x; r/2 : Observe now that I .x; r/ can be written in the following way: Z Z 1 1 I .x; r/ D .u2 /dy D @ .u2 /d 2 B.x;r/\ 2 @.B.x;r/\/  Z D u @ u d: @B.x;r/ Plugging the last identity into (2.7), we get (2.5). The last assertion in the lemma follows from the fact that, by Cauchy-Schwarz, Z Z Z 2 2 juj2 d @ u d u @ u d  0; @B.x;r/\ @B.x;r/\ @B.x;r/ and from the condition that .y x/  .y/  0 for  -a.e. y 2 B.x; r/ \ @, which implies that Z .y x/  .y/ @ u.y/ 2 d.y/  0: □ B.x;r/\@ It is immediate to check that saying that @r F .x; r/  0 a.e. in an interval is equivalent to saying that the function  f .t/ D log h x; e t is convex in t D log r for r in that interval, i.e., f 00 .log r/  0 a.e. in the interval. L EMMA 2.3. Given x 2 Rn , let I  .0; 1/ be an interval such that h.x; r/ > 0 and @r F .x; r/  0 for a.e. r 2 I . Given a > 1, if both r; ar 2 I , then h.x; ar/ (2.8) F .x; r/  loga  F .x; ar/: h.x; r/ Another way of writing the preceding estimate is the following: for R D ar    F .x;R/ R F .x;r/ (2.9) h.x; r/ r  h.x; R/  h.x; r/ Rr : P ROOF. By the convexity of the function f defined above, we have f .log ar/ f .log r/ f 0 .log r/   f 0 .log ar/: log ar log r It is immediate to check that this is equivalent to (2.8). □ 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 314 X. TOLSA From now on, we say that an interval I  .0; 1/ is admissible for x 2 Rn if h.x; r/ > 0 and @r F .x; r/  0 for a.e. r 2 I . L EMMA 2.4. Let x; y 2 Rn and r > 0, 2 .0; 1=10/, such that jx y j  r . Let I be an open interval admissible for x and y such that both r; 2.1 C 1=2 /r 2 I . Suppose that B.x; 5r/ \ @  †. Then  (2.10) F .y; r/  .1 C C 1=2 / F x; 2.1 C 1=2 /r C C 1=2 ; for some absolute constant C > 0. P ROOF. Let x; y; r; be as above and let  2 .0; 1/ to be chosen below. Since h.y;  / is nondecreasing, we deduce that « « h.y; r/ D u2 d  u2 d m: @B.y;r/ A.y;r;.1C/r/ Analogously, « h.y; r/  u2 d m: A.y;.1 /r;r/ The same estimates are valid interchanging y with x and/or r with 2r . Then, by (2.8), we have ª 2 h.y; 2r/ A.y;2r;.2C/r/ u d m F .y; r/  log2  log2 ª 2 : h.y; r/ A.y;.1 /r;r/ u d m Observe now that Ay2 WD A.y; 2r; .2 C /r/  A.x; .2 /r; .2 C  C /r/ DW A2x ; and Ay1 WD A.y; .1 /r; r/  A.x; .1  C /r; .1 /r/ DW A1x : Thus, ª ª ! u2 d m Ay2 A2 u2 d m m.A2x / m.Ay1 / F .y; r/  log2 ª 2  log2 ª x 2  1 2 Ay1 u d m A1x u d m m.Ax / m.Ay / ª 2 @B.x;.2C C /r/ u d m  log2 ª 2 C C; ; @B.x;.1  C /r/ u d m where we denoted m.A2 / m.A1 / C; D log2 m.Ax1 / m.Ay2 / : x y 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 315 We choose  D 1=2 . Using just that  , we get ª u2 d m F .y; r/  log2 ª@B.x;.2C2/r/ 2 C C; @B.x;.1 /r/ u d m ª log 21C2 u2 d m D log 2 log 21C2 ª@B.x;.2C2/r/u2 d m C C; :  @B.x;.1 /r/ Since .2 C 2/r 2 I (by assumption), by Lemma 2.3 we have ª @B.x;.2 C2/r/ u2 d m log 2C2 ª 2  F .x; .2 C 2/r/: 1  @B.x;.1 /r/ u d m It is also immediate to check that log 21C2 log 2   1 C C D 1 C C 1=2 : Hence, F .y; r/  .1 C C 1=2 /F .x; .2 C 2 1=2 /r/ C C; : It only remains to show that C; C 1=2 . To this end, observe that m.A2x / D .2 C .2CC // n .2 n /n m.Ay2 / 2n D .2 C / C n.2 C / .2 C O. 2 / 2n C n2n 1 C O. n n 1 2/ C /n 2n D 1 C n.2 C / n2n 1C C n2 1 C O. 2 / n 1 n : O. 2 / It follows that  m A2x  1 C DC 1=2 : m.Ay2 Almost the same arguments show that  m Ay1  1 C DC 1=2 : m A1x Therefore,   m A2x m Ay1 C; D log2  C log2  . 1=2 ; m Ay2 m A1x as wished. □ 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 316 X. TOLSA 3 The Key Lemma To prove Theorem 1.1 we consider an arbitrary ball B0 centered in † such that M 2 B0  B , where B is the ball in Theorem 1.1 and M  1 will be fixed below. We denote †0 D @ \ MB0 . We will show that if u is a non-zero (i.e., not identically zero) function harmonic in  and continuous in  S which vanishes in †, then the normal derivative @ u cannot vanish in a subset of †0 \B0 with positive surface measure. Clearly, this suffices to prove Theorem 1.1. Let H0 be the horizontal hyperplane through the origin. By the hypotheses in the theorem, we can assume that @ \ MB0 is a Lipschitz graph with respect to the hyperplane H0 with slope at most 0  1, and that  \ MB0 is above the graph. We consider the following Whitney decomposition of : we have a family W of dyadic cubes in Rn with disjoint interiors such that [ Q D ; Q 2W and moreover there are some constants ƒ > 20 and D0  1 such the following holds for every Q 2 W: (i) 10Q  ; (ii) ƒQ \ @ ¤ ¿; (iii) there are at most D0 cubes Q0 2 W such that 10Q \ 10Q0 ¤ ¿. Further, for such cubes Q0 , we have 21 `.Q0 /  `.Q/  2`.Q0 /. Above, we denote by `.Q/ the side length of Q. From the properties (i) and (ii) it is clear that dist.Q; @/  `.Q/. We assume that the Whitney cubes are small enough so that 1 (3.1) diam.Q/ < dist.Q; @/: 20 The arguments to construct a Whitney decomposition satisfying the properties above are standard but we include the detailed arguments in Lemma B.1 below for the convenience of the reader. Let denote the orthogonal projection on H0 . By translating the usual dyadic lattice if necessary, we can assume that there exists some cube R0 2 W such that .B0 /  .R0 / and `.R0 /  C r.B0 / and moreover R0  M2 B0 , for M big enough. Next we need to define some “generations” of cubes in W. We let DW0 .R / D 0 fR0 g. For k  1 we define DW .R0 / as follows. Let k (3.2) J.R0 / D f .Q/ W Q 2 W such that .Q/  .R0 / and Q is below R0 g: Observe that J.R0 / is a family of .n 1/-dimensional dyadic cubes in H0 , all of them contained in .R0 /. Let Jk .R0 /  J.R0 / be the subfamily of .n 1/- dimensional dyadic cubes in H0 with side length equal to 2 k `.R0 /. To each Q0 2 Jk .R0 / we assign some Q 2 W such that .Q/ D Q0 , .Q/  .R0 /, and such that Q is below R0 (see Lemma B.2 for more details), and we write 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 317 s.Q0 / D Q. Notice there may be more than one possible choice for Q. However, the choice is irrelevant. Anyway, for definiteness we take the cube Q that is closest to R0 among all the possible choices. Then we define Dk .R0 / D fs.Q0 / W Q0 2 J .R0 /g: W k Next we let [ DW .R0 / D k .R /g: DW 0 k 0 Notice that, for each k , the family f .Q/ W Q 2 DWk .R /g is a partition of 0 .R0 /. Finally, for each R 2 DW k .R / and j  1 we denote 0 j k Cj DW .R/ D fQ 2 DW .R0 / W .Q/  .R/g: By the properties of the Whitney cubes, it is easy to check that Q 2 DW .R0 / ) dist.Q; †0 / D dist.Q; @/  `.Q/: From now on, for any cube Q, we denote by xQ its center. Further, we denote by mn 1 the .n 1/-dimensional Lebesgue measure on the hyperplane H0 . K EY L EMMA 3.1. Under the assumptions of Theorem 1.1, let R0 be as above and let N0 > 1 be big enough. There exists some absolute constant 0 > 0 such that for all A  1 big enough the following holds, assuming also 0 small enough and M big enough. Let R 2 DW .R0 / satisfy F .xR ; A `.R//  N0 . There exists some positive integer K D K.A/ big enough such that if we let  GK .R/ D Q 2 DW K .R/ W F .x ; A`.Q//  1 F .x ; A `.R// ; Q 2 R then: S  (a) mn 1 Q2GK .R/ .Q/  0 mn 1 . .R//. (b) For all Q 2 DW K .R/, it holds F .xQ ; A`.Q//  .1 C CA 1=2 / F .xR ; A`.R//: A key point in the lemma is that 0 does not depend on A. On the other hand, 0 , M , and K depend on A. The constant N0 is also an absolute constant independent of the other parameters. The general strategy for the proof of the Key Lemma is similar to one of the Hyperplane Lemma 4.1 from [13]. The main differences stem from the fact that in the lemma above we wish to estimate the frequency function in points that are close to @, and then we have to be more careful and more precise with the monotonicity properties of the frequency function. A basic tool for the proof of the Key Lemma 3.1 is the following result on quantitative Cauchy uniqueness: 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 318 X. TOLSA T HEOREM 3.2. Let v be a function harmonic in the half ball B1C D fx 2 Rn W jx j < 1; xn > 0g and C 1 smooth up to the boundary. Let € the following subset of @B1C : € D fx 2 Rn W jx j < 3=4; xn D 0g: Suppose that Z jvj2 d m  1 BC 1 and sup jv j C sup jr v j  "; € € for some " 2 .0; 1/. Then sup jvj  C " ; B.1=2;1=4/ where C; are positive absolute constants. This result appears in [12, Lemma 4.3] and it is proven in much greater gener- ality in [3, Theorem 1.7]. Remark 3.3. We claim that, for T > 0 and R 2 DW .R0 /, if x 2  satisfies dist.x; R/  T `.R/ and dist.x; @/  T 1 `.R/; then the interval .0; A`.R// is admissible for x , assuming M  T; A and that 0 is small enough, depending on T and A. This property will be essential for the proof of the Key Lemma. To prove the claim it suffices to check that, in the situation above, (3.3) .y x/  .y/  0  -a.e. y 2 B.x; r/ \ @, 0 < r  A`.R/, since then Lemma 2.2 ensures that @r F .x; r/  0 for 0 < r  A`.R/. To prove (3.3), let x 0 2 † be the point such that .x/ D .x 0 /, so that x x 0 is orthogonal to H0 . Denote by Hx 0 the hyperplane parallel to H0 through x 0 . Given y as in (3.3), let y 0 be the orthogonal projection of y on Hx 0 ; see Figure 3.1. So y 0 x 0 is orthogonal to x x 0 . Moreover, since the slope of † is at most 0 , jy y 0 j  0 jy 0 x 0 j  0 jy x j and j.y/ . en /j  0 : Then we write     .y x/  .y/ D .y y 0 / C .y 0 x 0 / C .x 0 x/  en C .en C .y//  jx x 0 j jx 0 x j jen C .y/j jy y 0 j jy 0 x 0 j jen C .y/j  jx 0 x j 0 jx x 0 j 0 jy x j 0 jy x j D .1 0 / jx 0 x j 20 jy x j: Using now that, for 0  1=2, we have jx 0 x j  dist.x; @/, we get .y x/  .y/  c dist.x; @/ 20 jy x j  c T 1 `.R/ 2 0 A `.R/  0; assuming 0  2c .AT / 1 , which proves (3.3). 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 319 Ω x y y′ x′ Hx ′ ν(y) Σ F IGURE 3.1. The domain  and the Lipschitz graph †. The points x; y satisfy .y x/  .y/  0. Proof of the Key Lemma 3.1. For any Q 2 W, we consider its associated cylinder: C .Q/ D .Q//:1. Let R 2 DW .R0 / be as in the lemma and let A  1. For some j  1 to be fixed below (independent of A), let L be a hyperplane parallel to H0 such that dist.L; †0 \ C .R// D 2 j `.R/: Notice that there are two possible choices for L. If 0 is small enough (and so † flat enough) depending on j , then (3.4) dist.x; @ \ C .10R//  2 j `.R/ for all x 2 L \ C .10R/. Then we choose L so that L \ C .10R/  . Let J denote the family of cubes from W which intersect L \ C . 12 R/. By the properties of Whitney cubes and (3.4), it is clear that `.Q/  2 j `.R/ and .Q/  .R/ for all Q 2 J . Let Adm.2ƒQ/ be the set of x 2  \ 2ƒQ such that the interval .0; diam.25ƒQ// is admissible for x . Recall that ƒ is one of the constants in the definition of Whit- ney cubes. We assume 0 small enough so that 3Q  Adm.2ƒQ/2. Then by Lemma 2.4, (3.5) sup F .x; diam.5ƒQ//  C0 F .xQ ; diam.20ƒQ// C C0 ; x 2Adm.2ƒQ/ where C0 is an absolute constant. S . Instead, a big portion of 2ƒQ 2 Notice that 2ƒQ 6 Adm.2ƒQ/ because 2ƒQ intersects Rn n  is contained in Adm.2ƒQ/ if †0 is flat enough. 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 320 X. TOLSA Claim. There exists some Q 2 J such that F .xR ; A`.R// (3.6) F .xQ ; diam.20ƒQ//  4C0 if j is big enough and we assume that 0 is small enough depending on j , and also that N0 is big enough. Remark again that the choice of j will not depend on the constant A. To prove the claim we intend to apply a rescaled version of Theorem 3.2 to a suitable half ball BC centered at ´R , the orthogonal projection of xR on L. We take  BC D x 2 B.´R ; `.R/=4/ W xn > .´R /n ; so that BC  , assuming that †0 \ C .R/ is below L. We also consider the point źR D ´R C .0; : : : ; 0; `.R/=8/: Notice that źR 2 BC (in fact, B.źR ; `.R/=8/  BC ). Aiming for a contradiction, suppose that F .xQ ; diam.20ƒQ// > 4CN for all 0 Q 2 J , where N D F .xR ; A`.R//. For each Q 2 J , by the subharmonicity of juj and (2.9), we have « sup juj . juj d  h.xQ ; diam.20ƒQ//1=2 2Q @B.xQ ;diam.3Q//   diam.20ƒQ/ F .xQ ;diam.20ƒQ//=2  h.xQ ; `.R//1=2 `.R/ : Here we applied the property described in Remark 3.3, allowing the smallness of the slope constant 0 to depend on j . Below we will make repeated use of this property, often without further reference. To estimate h.xQ ; `.R// we take into account that « « h.xQ ; `.R//  juj2 d m . juj2 d m  h.źR ; C1 `.R//; A.xQ ;`.R/;2`.R// B.źR ;C1 `.R// since A.xQ ; `.R/; 2`.R//  B.źR ; C1 `.R// for some fixed C1 > 1. Further, by (2.9), h.źR ; C1 `.R//  h.źR ; `.R/=16/ .16 C1 /F .źR ;C1 `.R// « . .16 C1 /F .źR ;C1 `.R// juj2 d m B.źR ;`.R/=8/ « . .16 C1 /F .źR ;C1 `.R// juj2 d m; BC recalling that B.źR ; `.R/=8/  BC . Thus, (3.7)   « ź diam.20ƒQ/ F .xQ ;diam.20ƒQ// sup juj . .16 C1 / 2 F . R ;C1 `.R// juj2 d m: 2Q `.R/ BC 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 321 Observe now that, by Lemma 2.4, F .źR ; C1 `.R//  C F .xR ; C `.R// C C; for a suitable absolute constant C > 2C1 . So for A and N0 big enough (both independent of j , just larger than some absolute constant), (3.8) F .źR ; C1 `.R//  C F .xR ; A`.R// C C  C 0 N: Therefore, recalling also the assumption F .xQ ; diam.20ƒQ// > 4C N , by (3.7) we 0 get   « 0 diam.20ƒQ/ N=4C0 sup juj2 . .16 C1 /C N `.R/ juj2 d m 2Q BC « D2 jcN CC 00 N juj2 d m BC (here we took into account that diam.20ƒQ/  `.R/ for j larger than some abso- lute constant). Also, by standard interior estimates for harmonic functions, 2j « 1 2. 2 00 sup jr uj2 . 2 sup ju j 2 2 jcN CC N juj2 d m: 3Q `.Q/ 2Q `.R/ BC 2 From the last two estimates we deduce that if j is big enough and N0 (and thus N ) also big enough, then there exists some c 0 > 0 such that «  0 sup juj2 C `.R/2 jr uj2 . 2 jc N juj2 d m: 3Q 2 BC Since the cubes 32 Q with Q 2 J cover the flat part of the boundary of BC , which we denote by € , it is clear that «  0N sup juj C `.R/ jr uj . 2 2 2 2 jc juj2 d m: € BC Applying now a rescaled version of Theorem 3.2 to the half ball BC , we infer that « 0N sup juj . 2 2 jc juj2 d m: B.źP ;`.R/=16/ BC Consequently, « h.źR ; `.R/=16/ . 2 2jc 0 N 2 juj 2jc 0 N dm . 2 h.źR ; `.R//; B.źR ;`.R// for some fixed > 0. By Lemma 2.3, this implies that h.źR ; `.R// F .źR ; `.R//  log16  c jN ; h.źR ; `.R/=16/ for some fixed c > 0. However, for j big enough this contradicts the fact that F .źR ; `.R// . N , which follows from (3.8). So the proof of the claim is con- cluded. 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 322 X. TOLSA Now we are ready to introduce the set GK .R/. Fix Q0 2 J such that (3.6) holds for Q0 . Notice that, by (3.5), (3.9) N sup F .x; ƒ`.Q0 //  C0 F .xQ0 ; diam.20ƒQ0 // C C0  4 C C0  N2 ; x 2Adm.2ƒQ0 / since N  N0 and we assume N0 big enough. Now we just define j Ck GK .R/ D fQ 2 DW .R/ W .Q/  .Q0 /g; with k D dlog2 Ae. So we have GK .R/  DW K .R/ with K D j C k and it holds `.Q/  2 k `.Q0 / for every Q 2 GK .R/. The property (a) in the lemma follows easily from (3.9). Indeed, if P 2 GK .R/, then taking into account that xP 2 Adm.2ƒQ0 / for 0 small enough (depending on A), N F .xP ; A`.P //  F .xP ; `.Q0 //  F .xP ; ƒ`.Q0 //  : 2 Notice also that  [  mn 1 .Q/ D `.Q0 /n 1  .2 j `.R//n 1 ; Q2GK .R/ and recall that j is independent of A. So (a) holds with 0  2 j.n 1/ . The property (b) is an easy consequence of Lemma 2.4. Indeed, for any P 2 P xR j . `.R/, taking  A in (2.10), we deduce DWK .R/, since jx 1  F .xP ; A`.P //  F .xP ; A`.R/=3/  .1 C CA 1=2 / F xR ; A`.R/ C CA 1=2  .1 C 2CA 1=2 / N: □ 4 Proof of Theorem 1.1 Our next objective is to prove the following result: L EMMA 4.1. Under the assumptions of Theorem 1.1, let R0 2 W be as in Sec- tion 3. Then, h.x; 12r/ lim inf < 1 for  -a.e. x 2 †0 \ C .R0 /. r !0 h.x; r/ Recall that C .R0 / is the cylindernn C .R0 / D 1 . .R //; 0 and it contains B0 , by the assumption just after (3.1). The proof of the preceding lemma will use the following version of the law of large numbers, due to Etemadi [7]: T HEOREM 4.2. Let fXk gk 1 be a sequence of nonnegative random variables with finite second moments such that: 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 323 (a) supk 1 EXk < 1, (b) E.Xj Xk /  EXj EXk for j ¤ k , and P k 1 k 2 VarXk < 1. (c) 1 Let Sm D X1 C    C Xm . Then lim Sm ESm D 0 almost surely: m!1 m P ROOF OF L EMMA 4.1. Let †0 W C .R0 / ! C .R0 / \ †0 denote the projection on C .R0 / \ †0 in the direction orthogonal to the horizontal hyperplane H0 . We consider the measure  D †0 #.mn 1 jC .R0 /\H0 /: This is the image measure (or push-forward measure) of the .n 1/-dimensional Lebesgue measure on C .R0 / \ H0 to C .R0 / \ †0 . Obviously,  is mutually absolutely continuous with the surface measure  on C .R0 / \ †0 . Next we consider the families of cubes from W defined by  Tj0 jK (4.1) D Q 2 DW .R0 / W F .xQ ; A`.Q//  N0 ; j  0; where the constants K , A, and N0 are given by the Key Lemma 3.1 (the precise large value of A will be chosen below). We also denote R†0 D †0 .R0 / and consider the following subset of R†0 : [ Tj D †0 .Q/; j  0: Q2Tj0 We will prove the following: Claim. We have    R†0 n lim sup Tj D 0: j !1 Let us see first that the lemma follows from this claim. Indeed, if x 2 Tj , then there exists some cube Q 2 Tj0 such that x 2 †0 .Q/. By construction, F .xQ ; A`.Q//  N0 and thus, by (2.9), h.xQ ; A`.Q//  48N0 h.xQ ; A`.Q/=48/: For A big enough, we have B.x; A`.Q/=24/  32 B.xQ ; A`.Q/=48/ and 3 B.x; A`.Q/=2/  B.x ; A`.Q//: 2 Q Then, by the subharmonicity of juj in a neighborhood of †0 and standard argu- ments, h.x; A`.Q/=24/ & h.xQ ; A`.Q/=48/ and h.x; A`.Q/=2/ . h.xQ ; A`.Q//: 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 324 X. TOLSA Hence, for each x 2 Tj , h.x; A`.Q/=2/ h.x; A`.Q/=24/ . h.xh.xQ; A`.Q/=48/ ; A`.Q//  48N0 ; Q with `.Q/  2 jK `.R /. 0 Consequently, if x 2 lim supj !1 Tj , then a sequence of radii rj ! 0 exists such that h.x; 12rj / h.x; rj / . 48N0 ; which implies that h.x; 12r/ lim inf < 1; r !0 h.x; r/ and yields the lemma, assuming the claim. To prove the claim above we need to introduce some additional notation. For j  0 and K as in the Key Lemma 3.1, we denote zj .R†0 / D D jK  †0 DW .R0 / ; or more precisely, zj .R†0 / D D  †0 .Q 0 / W Q0 2 DjK .R0 / : W We also set [ z .R†0 / D D zj .R†0 /: D j 0 For any R 2 D zj .R†0 / such that R D †0 .R0 / for some R0 jK 2 DW .R0 /, in case that F .xR0 ; A `.R0 //  N0 , we consider the good set [ G.R/ D †0 .Q 0 /; Q0 2GK .R0 / with GK .R0 / as in the Key Lemma 3.1. In case that F .xR0 ; A `.R0 // < N0 , we let G.R/ D †0 .R 0 / D R: Finally, we write  0 / W Q0 2 DjK .R0 /; F .xQ0 ; A`.Q0 //  N0 ; Tj D †0 .Q W or in other words, Tj D †0 .Tj /; 0 with Tj0 defined in (4.1) S To prove the claim we have to show that j h Tj has full -measure in R†0 for every h  0. To this end, for any fixed h we define the following functions fj , j  h: X fj D fQ ; zj .R†0 / Q2D 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 325 D 0 if Q is contained in some “cube” Qz 2 j h Tj and, otherwise, S where fQ .Q/ fQ D  Q : .G.Q// G.Q/ It is immediate toR check that the functions fj have zero -mean, and they are orthogonal, i.e., fi fj d D 0 if i ¤ j (taking into account that fj has zero -mean in each Q 2 Dzj .R†0 / and is constant in each P 2 Dzj C1 .R†0 /). Observe also that the functions fj are uniformly bounded, due to the fact that .G.Q//  0 .Q/ in the latter case by the Key Lemma. So their L2 ./ norms are uniformly bounded too. We consider the probability measure jR†0 =.R†0 / and the random variables Xj D fj C 1, j  h. Notice that they are nonnegative and the assumptions in Theorem 4.2 are satisfied. Indeed, (a) and (c) follow from the uniform boundedness of the functions fj , and the zero mean of each fj and the mutual orthogonality of the fj ’s imply that E.Xi Xj / D E.Xi / E.Xj / if i ¤ j . Applying the theorem then we infer that m 1 X (4.2) lim fj .x/ D 0 for -a.e. x 2 R†0 ; m!1 m j DhC1 using the fact that EXj D 1 for all j . We will show that m  [ 1 X (4.3) x 2 R†0 Tj ) lim fj .x/ ¤ 0: m!1 m j h j DhC1 S Clearly, by (4.2), this implies that R†0 n j h Tj has null -measure and finishes the proof of the claim. We prove S (4.3) by contradiction. Suppose P that there exists some point x 2 R†0 n j h Tj such that limm!1 m1 jmDhC1 fj .x/ D 0. Denote by Qj the “cube” from D zj .R†0 / that contains x . Since Qi 62 Ti for any i  h, by definition we have .Qj / fj .x/ D  .x/ 1 for any j  h. .G.Qj // G.Qj / Then (4.2) tells us that, for any " > 0, m X .Qj /  .G.Qj // G.Qj / .x/ m  "m j DhC1 for any m big enough. In particular, choosing " D 1 we infer that, for some 2 m0 D m0 .x/, m X .Qj / m G.Qj / .x/  for any m  m0 . .G.Qj // 2 j DhC1 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 326 X. TOLSA Since .G.Qj //  0 .Qj /, we get m X  m (4.4) G.Qj / .x/  0 for m  m0 . 2 j DhC1 For each j  h, let Qj0 2 DW .R0 / be such that Qj D †0 .Qj0 /. Recall that the Key Lemma (we can apply this because F .xQj0 ; A`.Qj0 //  N0 ) asserts that 1 F .xQj0 C1 ; A`.Qj0 C1 //  F .xQj0 ; A`.Qj0 // if x 2 G.Qj / 2 and otherwise just ensures that F .xQj0 C1 ; A`.Qj0 C1 //  .1 C CA 1=2 / F .xQj0 ; A`.Qj0 // if x 2 Qj n G.Qj /: These estimates and (4.4) imply that  0 m=2 1 F .xQm0 C1 ; A`.Q0 mC1 //  .1 C CA 1=2 /m for m  m0 . 2 However, if A is chosen big enough (recall that A is independent of 0 and can be taken arbitrarily big in the Key Lemma 3.1), this implies that 0 // ! 0 as m ! 1, F .xQm0 ; A`.Qm S which cannot happen because x 62 j h Tj , recalling the definition of Tj . This concludes the proof of the claim and of the lemma. □ The proof of Theorem 1.1 will follow as a straightforward consequence of Lemma 4.1 and the next result of Adolfsson and Escauriaza: L EMMA 4.3 ([1, lemma 0.2]). Let D  Rn be a Lipschitz domain and let V be a relatively open subset of @D . Let v be a nonzero function harmonic in D and continuous in Dx that vanishes identically in V , and whose normal derivative @ v vanishes in a subset E  V of positive surface measure. Then, for every point x 2 V that is a density point of E (with respect to surface measure), we have R jvjd m B.x;r/\D (4.5) lim R r !0 B.x;6r/\D jv jd m D 0: Actually, the identity (4.5) is not stated explicitly in lemma 0.2 in [1]. Instead, it is said that v vanishes to infinite order in x . However, a quick inspection of the proof shows that the authors actually prove (4.5), which in turn implies that v van- ishes to infinite order in x . The lemma above relies on [2, lemma 1 and theorem 1]. Though the proof of [2, lemma 1] is not correct—as explained in [4, para- graph before lemma 5]—one can replace that lemma either by [1, lemma 2.2] or 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 327 by more quantitative arguments involving [4, lemma 4] and well-known proper- ties of harmonic functions.3 For the reader’s convenience I provide an alternative self-contained proof in the Appendix C. P ROOF OF T HEOREM 1.1. As explained at the beginning of Section 3, it suf- fices to show that @ u cannot vanish in a subset of positive surface measure of †0 \ C .R0 / (since this set contains the ball B0 ). For the sake of contradiction, suppose that @ u vanishes in a subset E  †0 \ C .R0 / of positive surface measure. By Lemma 4.3, for any x 2 †0 \ C .R0 / that is a density point of E , R jujd m B.x;6r/ lim R r !0 B.x;r/ jujd m D 1: By the subharmonicity of juj, for r small enough, « 1=2 « h.x; r=2/1=2 D juj2 d . jujd m: @B.x;r=2/ B.x;r/ Also, by Cauchy-Schwarz and the fact that h.x;  / is nondecreasing in r , « « 1=2 jujd m  juj2 d m  h.x; 6r/1=2 : B.x;6r/ B.x;6r/ Therefore, ª h.x; 6r/1=2 B.x;6r/ jujd m lim inf & lim inf ª D 1: r !0 h.x; r=2/1=2 r !0 B.x;r/ jujd m Consequently, h.x; 12r/ lim r !0 h.x; r/ D 1; which contradicts Lemma 4.1. □ Appendix A Existence of Nontangential Limits for r u Let   Rn be a Lipschitz domain, and let  denote the surface measure on @. For  -a.e. x , there exists a tangent hyperplane to @ in x and the outer unit normal .x/ is well-defined. For an aperture parameter a 2 .0; 1/ we consider the one-sided inner cone with axis in the direction of .x/ defined by  XaC .x/ D y 2 Rn W .x y/  .x/ > ajy xj : Analogously, we consider the outer cone  Xa .x/ D y 2 Rn W .y x/  .x/ > ajy xj : 3 I thank Luis Escauriaza for informing me about this fact. 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 328 X. TOLSA For a given function f W Rn n @ ! R and a fixed parameter a 2 .0; 1/, we define the nontangential limits fC;a .x/ D lim f .y/; f ;a .x/ D lim f .y/; XaC .x/3y !x Xa .x/3y !x whenever they exist. Although the following result is already known (see [9, theorem 5.19], I have not been able to find an easy argument in the literature and thus I provide a detailed proof based on Dahlberg’s theorem on harmonic measure [6]. T HEOREM A.1. Let   Rn be a Lipschitz domain, let B be an open ball centered in @, and let † D B \ @ be a Lipschitz graph. Let u be a function harmonic in  and continuous in  x . Suppose that u vanishes in †. Then, for any a 2 .0; 1/, .r u/C;a exists  -a.e. and belongs to L2loc . j† /. Further, .r u/C;a has vanishing tangential component. That is, .r u/C;a D .@ u/  . In addition, assuming that  \ B is above †, lim "!0 r u. C " en / ! .@ u/  in L2loc . j† /: Also, in the sense of distributions, .u/jB D @ u  j† : P ROOF. We extend u by 0 out of  and denote uC D max.u; 0/; u D min.u; 0/; C so that u and u are continuous and subharmonic in B . This implies that, in the sense of distributions, in B , u D uC u is a signed Radon measure supported on †. First we claim that (A.1) .u/jB D  ! j† ; 1 where  2 Lloc .†/ and ! is the harmonic measure for  with respect to some fixed pole p 2 . To prove the lemma we may assume B small enough so that  n 2B ¤ ¿ and that p 2  n 2B . To prove the claim, let B 0 be an open ball concentric with B such that Bx0  B . We will show that that there exists some constant C2 depending on B 0 and p such that for any compact set K  †, it holds that (A.2) jhu; K ij  C2 !.K/: By duality, this implies (A.1). Given an arbitrary " 2 .0; 21 dist.K; Rn n B 0 //, let fQi gi 2I be a lattice of cubes that cover Rn , with diameter equal to "=2. Let f'i gi 2I be a partition of unity of Rn , so that each 'i is supported in 2Qi and kr j 'i k1 . `.Qi / j for j D 0; 1; 2. Then we have D X E D X E (A.3) hu; K i D u; 'i u; 'i K ; i 2I 0 i 2I 0 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 329 where I 0 is the collection of indices i 2 I such that 2Qi \ K ¤ ¿. Since .u/jB is a signed Radon measure, D X E u; 'i K  juj; U" .K/nK ! 0 as " ! 0, i 2I 0 where U" .K/ is the "-neighborhood of K . Concerning the other term in (A.3), we have Z D E X X 1 X u; 'i  hu; 'i i . `.Qi /2 2Qi jujd m: i 2I 0 i 2I 0 i 2I 0 Since juj is subharmonic and continuous in B and vanishes in B n , by the bound- ary Harnack principle (see theorem 5.1 from [8], for example), we have ju.x/j  C3 g.x; p/ for all x 2 B 0 \ , where g.  ;  / is the Green function of  and C3 depends on u, p , and B 0 , but not on K . Thus, D E Z X 1 X u; 'i . g.x; p/dx; i 2I 0 i 2I 0 `.Qi /2 2Qi with the implicit constant depending on u, p , and B 0 . By standard estimates for the harmonic measure (see (4.3) and (4.4) from [8], for example), we have !.4Qi / g.x; p/ . for all x 2 2Qi \ , i 2 I . `.Qi /n 2 Therefore, D X E X u; 'i . !.4Qi /  !.U4" .K//: i 2I 0 i 2I 0 Letting " ! 0, we have !.U4" .K// ! !.K/ and thus (A.2) follows, which implies the claim (A.1) . Next recall that by Dahlberg’s theorem, harmonic measure on a Lipschitz do- main  is a B2 weight with respect to the surface measure  . In particular, the density function d! 2 d belongs to Lloc ./. Therefore, in the sense of distributions, .u/jB D h  j† for some h 2 L2loc ./. Our next objective consists in showing that .r u/C;a exists  -a.e., and moreover .r u/C;a D .@ u/  2 L2loc . j† /. To this end, consider an arbitrary open ball Bz centered in † such that 4Bz  B . Let ' be a C 1 function that equals 1 on 2Bz and vanishes out of 3Bz , and let v D ' u. Observe that (A.4) v D E  .' v/ D E  .' u C u ' C 2 r u  r '/; where E is the fundamental solution of the Laplacian. Note also that r u 2 L2loc .B/, by Caccioppoli’s inequality. 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 330 X. TOLSA For a finite Borel measure , let R be the .n 1/-dimensional Riesz transform of . That is, Z x y R.x/ D jx y jn d.y/; whenever the integral makes sense. From the identity (A.4), we deduce that, for all x 62 †,   r v.x/ D cn R.' g  j† /.x/ C R.u ' m/.x/ C 2 R.r u  r ' m .x/ (recall that m is the Lebesgue measure in Rn ). Observe that R.u ' m/ and R.r u r 'm/ are continuous functions in Bz. On the other hand, the nontangential limit .R.' g  j† //;a .x/ exists for  -a.e. x 2 @, by the classical jump formulas for the Riesz transforms (see [18], for example). Taking also into account that r v D r u in Bz, it follows then that .r u/;a .x/ exists for  -a.e. x 2 † \ Bz. By the L2 ./ boundedness of the (principal value) Riesz transform operator R.  / on Lipschitz graphs, we deduce that .r u/;a 2 L2 . j†\Bz /. Since u  0 in c , it is clear that .r u/ ;a  0 in † \ Bz . As the tangen- tial component of R.' g  j† /.x/ is continuous across @ for  -a.e. x 2 @, again by the jump formulas for the Riesz transforms, we deduce that the tangential component of .r u/C;a coincides with the tangential component of .r u/ ;a  -a.e. in † \ Bz , and thus .r u/C;a  0 in † \ Bz , which is equivalent to saying that .r u/C;a D .@ u/ in † \ Bz. It remains to prove that .u/jB D @ u  j† in the sense of distributions. To this end, let be a C 1 function supported in Bz . Without loss of generality we may assume that † \ Bz is a Lipschitz graph with respect to the horizontal axes and that  \ Bz lies above † \ Bz . For 0 < "  r.B/ z , denote †" D † C " en and " D  C " en , where en D .0; : : : ; 0; 1/. Then we have Z Z hu; i D u  d m D lim "!0 Bz \" u  dm Z Z (A.5) D "lim !0 u @ d lim "!0 Bz \@" @ u d Bz \@" Z D0 @ u d: † The last identity follows from the fact that, in a neighborhood of † \ Bz , as " ! 0, u.  C " en / converges uniformly to 0 and r u.  C " en / converges to .r u/C;a in L2 . j†\Bz / (this is proven by arguments analogous to the ones above for the  - a.e. existence of the limit .r u/C;a .x/ in †). From (A.5), we deduce that u D @ u  j† in Bz, and thus also in B . □ Appendix B Whitney Cubes In this appendix we prove some of the properties of the Whitney cubes con- structed at the beginning of Section 3. 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 331 L EMMA B.1. Let  ¨ Rn be open. Then there exists a family W of dyadic cubes with disjoint interiors such that [ Q D ; Q 2W and moreover there are some constants ƒ > 20 and D0  1 such the following holds for every Q 2 W: (i) 10Q   and diam.Q/ < 20 1 dist.Q; @/; (ii) ƒQ \ @ ¤ ¿; (iii) there are at most D0 cubes Q0 2 W such that 10Q \ 10Q0 ¤ ¿. Further, for such cubes Q0 , we have 12 `.Q0 /  `.Q/  2`.Q0 /. P ROOF. We assume the dyadic cubes to be half-open and half-closed. Let c0 2 .0; 12 / be some constant to be fixed below. Denote d.x/ D dist.x; @/, and let W be the family of all dyadic cubes Q   that satisfy (B.1) `.Q/  c0 inf d.x/ x 2Q and moreover are maximal. It is immediate to check that the cubes from W cover  and are disjoint, because they are maximal. For all Q 2 W, since c0 d.x/  `.Q/ for all x 2 Q, it follows that dist.Q; @/  c0 1 `.Q/: Taking c0 small enough, the properties (i) and (ii) follow. Let Q; Q0 2 W satisfy 10Q \ 10Q0 ¤ ¿. Let Q00 the dyadic parent of Q0 , which is also contained in , by (i). By the definition of W, there exists x 00 2 Qx00 such that `.Q00 /  c0 d.x 00 /. Fix also any x 2 Q. From the condition 10Q \ 10Q0 ¤ ¿, it follows that jx x 00 j  C .`.Q/ C `.Q0 //, where C is some constant depending just on n. Then we have `.Q/  c0 d.x/  c0 d.x 00 / C c0 jx x 00 j  2 `.Q0 / C c0 C.`.Q/ C `.Q0 //: For c0 small enough we deduce that `.Q/  2:5 `.Q0 /, which implies that `.Q/  2 `.Q0 / because `.Q/=`.Q0 / D 2k for some k 2 Z. Reversing the roles of Q and Q0 , we deduce that 21 `.Q0 /  `.Q/  2`.Q0 /. From this property and standard volume considerations it follows easily that there are at most D0 cubes Q0 2 W such that 10Q \ 10Q0 ¤ ¿, with D0 depending just on n. □ L EMMA B.2. Let  be a Lipschitz domain, and let †, B0 , †0 , H0 , , and R0 be as in Section 3. Also, let J.R0 / be as in (3.2) and, for k  1, let Jk .R0 /  J.R0 / be the subfamily of .n 1/-dimensional dyadic cubes in H0 with side length equal to 2 k `.R0 /. Then, for each Q0 2 Jk .R0 / there exists some cube Q 2 W such that .Q/ D Q0 , .Q/  .R0 /, and such that Q is below R0 . P ROOF. Let Q0 2 Jk .R0 / and denote by x 0 its center, so that x 0 2 H0 \ .R0 / too. Let Lx 0 be the line orthogonal to H0 through x 0 . Let x be the intersection 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 332 X. TOLSA of Lx 0 with the lower face of R0 , and let x 00 D Lx 0 \ †0 . Let S be the segment .x; x 00 /, which lies on Lx 0 . Consider the sequence of dyadic Whitney cubes fRj gj 1  W that intersect S , so that [ S D Lx 0 \ Rj ; j 1 and assume that the sequence is ordered in such a way that, for all j  0, Rj and Rj C1 are neighbors and Rj C1 is below Rj . The sequence of side lengths `.Rj / tends to 0 as j ! 1 because dist.Rj ; @/ ! 0 as j ! 1. Also, for any j  0, `.Rj /=`.Rj C1 / equals 1, 21 , or 2, by the property (iii) in Lemma B.1. This implies that, for some j  1, `.Rj / D 2 k `.R0 /. Indeed, we claim that the cube Rj such that `.Rj /  2 k `.R0 / and j is minimal does the job. To check this, notice that, by the minimality of j , `.Rj 1 /  2 k C1 `.R0 /. So property (iii) of Lemma B.1 implies that `.Rj /  2 k `.R0 / and the claim follows. Notice now that .Rj / D Q0 , because both .Rj / and Q0 are .n 1/-dimensional dyadic cubes in H0 with side length 2 k `.R0 / and both contain x 0 . □ Appendix C An Alternative Proof of Lemma 4.3 We assume that we are under the assumptions of Lemma 4.3. So given a Lips- chitz domain D  Rn and a relatively open subset V of @D , we consider a nonzero function v that is harmonic in D and continuous in D x , vanishing identically in V , and whose normal derivative @ v also vanishes in a subset E  V of positive sur- face measure. We have to show that, for every point x 2 V that is a density point of E (with respect to surface measure), it holds that R B.x;r/\D jvjd m (C.1) lim R r !0 B.x;6r/\D jv jd m D 0: To this end, for such point x , given " 2 .0; 1/, let r0 > 0 be small enough so that B.x; r0 / \ @D  V and .@D \ B.x; r/ n E/  " .@D \ B.x; r// for all 0 < r  r0 . We fix r 2 .0; r0 =3/. Without loss of generality, we assume that x D 0 and denote Br D B.0; r/, We also assume that @D \ B2r is a Lipschitz graph with respect to the horizontal axes, and that D is above the graph. As usual, we understand that v has been extended by 0 in D c . As shown in Theorem A.1, .v/jBr D @ v  j@D\Br DW : Thus, g WD v E   is harmonic in Br : We intend to apply the three-ball inequality to the function g . In order to do this, first we need to estimate the total variation of the signed measure . We apply the 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 333 Rellich-Necas identity 2.  r v/v X D 2 div..  r v/r v/ div. jr vj2 / C jr vj2 div 2 @i j @i v @j v; i;j with a vector field D ' en , where ' is a smooth function supported on B.3=2/r and identically 1 on Br . Integrating the above identity in B2r \ D with respect to Lebesgue measure and applying the divergence theorem, we obtain Z   0D 2 '.y/ @n v.y/ @ v.y/ '.y/ jr v.y/j2 .en  .y// d.y/ @D \B2r Z Z X C @n ' jr v j2 dy 2 @i 'n @i v @n v dy: B2r B2r i In fact, to be more precise, since v need not be smooth up to @D , first we apply the divergence theorem in the domain B2r \ .en C D/ with  > 0, and then we let  ! 0 as in the proof of Lemma 2.2. Taking into account that r v D .@ v/ on @D \ B2r , we get Z '.y/ j@ v.y/j2 .en  .y//d.y/ @D \B2r Z Z X D @n ' jr v j2 dy C2 @i 'n @i v @n v dy: B2r B2r i Thus, recalling that Br  '  B.3=2/r and applying Caccioppoli’s inequality, Z Z Z 1 1 j@ v.y/j2 d.y/ . r B3r jr v j2 dy . 3 r B2r jvj2 dy: @D \Br 2 Now, by Cauchy-Schwarz, we derive Z kk D j@ v.y/jd.y/ @D \Br Z 1=2  j@ v.y/j2 d.y/ .Br \ @D n E/1=2 @D \Br Z 1=2 . " 12 r n2 2 jvj2 dy : B2r Let x 0 D 10 r e . Then there is some c > 0 depending just on the Lipschitz n 2 x . We denote r 0 D c2 r . Observe character of D such that B.x 0 ; 2c2 r/  Br n D 0 now that for any s 2 Œr ; r, Z ZZ 1 (C.2) jE  jd . d.y/d jj.´/ . kk s: @B.x 0 ;s/ @B.x 0 ;s/ jy ´jn 2 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 334 X. TOLSA Thus, the function4 w WD E   satisfies « Z 1=2 1 1 jwjd . r n 2 kk . "1=2 jvj2 dy @B.x 0 ;s/ r n=2 B2r (C.3) « 1=2  "1=2 jvj2 dy ; B2r with the implicit constants, from now on, possibly depending on c2 and thus on the Lipschitz character of D . Since g D w in B.x 0 ; 2r 0 /, and g is harmonic in B.x 0 ; 2r 0 / and continuous in B.x 0 ; 2r 0 /, we deduce that, for all ´ 2 B.x 0 ; r 0 /, « « 1=2 jg.´/j . jgjd . "1=2 jvj2 dy : @B.x 0 ;2r 0 / B2r Hence, « « hg .x 0 ; r 0 / WD jgj2 d ." jvj2 dy: @B.x 0 ;r 0 / B2r Next we estimate hg .x 0 ; 34 r/. We write « « « jgjd  jvjd C jwjd: @B.x 0 ; 54 r/ @B.x 0 ; 45 r/ @B.x 0 ; 45 r/ Since jv j is subharmonic in B2r and continuous in B2r , we have « jv.´/j . jvjdy for all y 2 Br . B2r Therefore, « « « 1=2 « 1=2 jgjd . jvjdy C "1=2 jvj2 dy . jvj2 dy : @B.x 0 ; 45 r/ B2r B2r B2r Since g is harmonic in B.x 0 ; 45 r/ and continuous in B.x 0 ; 45 r/, we have, for all ´ 2 B.x 0 ; 43 r/, « « 1=2 jg.´/j . jgjd . jvj2 dy ; @B.x 0 ; 45 r/ B2r and so « hg .x 0 ; 34 r/ . jvj2 dy: B2r Recall now that, by the three-ball inequality, given 2 .0; 1/ and r2 > r1 > 0 such that B.x 0 ; r2 /  Br , it holds that  hg x 0 ; r1 r21  hg .x 0 ; r1 / hg .x 0 ; r2 /1 : 4 In the case n D 2 it is better to choose w D E   1 2 .R2 / log r D 1 jx j 2 log r  , and then (C.2) and (C.3) follow analogously. 10970312, 2023, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cpa.22025 by Readcube (Labtiva Inc.), Wiley Online Library on [21/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License UNIQUE CONTINUATION AT THE BOUNDARY 335 In fact, this inequality is an immediate consequence of the convexity of the function log hg .x 0 ; e t /, proven in Lemma 2.2 in a more general situation. Applying this inequality with r1 D r 0 , r2 D 34 r , and such that .r 0 / . 34 r/1 D 23 r; i.e., log 89 D ; log 4c32 we infer that « hg .x 0 ; 2 r/ . " 3 jvj2 dy: B2r Hence, using Cauchy-Schwarz and again (C.3), « « « jvjd  jgjd C jwjd @B.x 0 ; 23 r/ @B.x 0 ; 32 r/ @B.x 0 ; 23 r/ « 1=2 . hg .x 0 ; 2 r/1=2 C "1=2 3 jvj2 dy B2r « 1=2 ." =2 jvj2 dy : B2r Since jv j is subharmonic in B3r and continuous in B3r and B.1=2/r  B.x 0 ; 23 r/, we get « « « 1=2 « jvjdy . jvjd . 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