John domain and the weak boundary Harnack principle Hiroaki Aikawa Department of Mathematics, Hokkaido University Summer School in Conformal Geometry, Potential Theory, and Applications NUI Maynooth, Ireland 23 27 June 2009 1
Contents 1. Introduction 3 2. Weak boundary Harnack principle for a John domain 9 3. Domar s argument 18 4. Union of convex sets 24 References 28 2
1. Introduction Exposition of [AHL06]: H. Aikawa, K. Hirata, and T. Lundh, Martin boundary points of a John domain and unions of convex sets, J. Math. Soc. Japan 58 (2006), no. 1, 247 274. Let E {x = (x 1,..., x n ) : x n = 0} be closed. D = R n \ E is called a Denjoy domain. D Let P be the family of positive harmonic functions in D vanishing on D. Benedicks [Ben80] proved the following: E Contents 3
dim P = 1 or 2. i.e., 1 or 2 minimal Martin boundary points at. Criterion in terms of harmonic measure β E (x) = ω(x, K x, K x \ E). K x : cube center at x, side α x. 0 x 1 K x dim P = 1 dim P = 2 x 1 x 1 β E (x) x n 1 dx 1 dx n 1 =. β E (x) x n 1 dx 1 dx n 1 <. Monotonicity: If E E, dim P E = 2, then dim P E = 2. Contents 4
Location Topics Authors C 2 surface dim P 2 Ancona [Anc84] Hyperplane Harmonic Measure Benedicks [Ben80] Lipschitz surface dim P 2 WBHP Ancona [Anc84] Real line Lebesgue Measure Segawa [Seg88] Hyperplane Lebesgue Measure Gardiner [Gar89] C 1,1 surface Harmonic Measure Chevallier [Che89] C 1,α surface Harmonic Measure Ancona [Anc90] Lipschitz surface Non Monotonicity Ancona [Anc90] Real line Quasi-conformal Segawa [Seg90] Sectorial Harmonic Measure Cranston- Salisbury [CS93] Quasi-Sectorial Schrödinger Equation Lömker [Löm00] Contents 5
Weak boundary Harnack principle. Ancona [Anc84]. B(x, r), S (x, r) the open ball and the sphere with center at x and radius r. P ξ : kernel functions h at ξ, i.e., h > 0 harmonic on D, h = 0 q.e on D, bounded outside ξ. E S : Lipschitz surface. h 0, h 1, h 2 P ξ. Then ( h0 (y + R h 0 (x) A ) h 1 (y + R )h 1(x) + h 0(y R ) ) h 2 (y R )h 2(x) for x D B(ξ, R) \ B(ξ, R/2). y + R y R Contents 6
If h 0, h 1, h 2 P ξ, then i s.t. Hence dim P ξ 2. h i A j i h j ; Contents 7
Sectorial domain. Cranston-Salisbury [CS93]. y j R If h 0,..., h N P ξ. Then N h 0 (y j R h 0 (x) A ) h j (y j R )h j(x) j=1 for x D B(ξ, R) \ B(ξ, R/2); i s.t. h i A h j ; Hence dim P ξ N. Quasi-sectorial domain (higher dimension) Lömker [Löm00]. Contents 8 j i
2. Weak boundary Harnack principle for a John domain John domain. twisted cone condition: x D, γ : x x0 s.t. δ D (y) c J l(γ(x, y)) for all y γ, x 0 y twisted cone D x Contents 9
Denjoy domain Sectorial domain Quasi-Sectorial = John domain Theorem 1 Let D be a John domain with John constant c J. Let ξ D. Then (i) dim P ξ N(c J ) <. (ii) If c J > 3/2, then dim P ξ 2. Remark 1 c J > 3/2 is sharp for all n 2. Contents 10
x 0 ξ Quasihyperbolic metric: k D (x, y) = inf γ γ ds(z) δ D (z). where inf is taken over all curves γ connecting x to y in D. k D (x, y) length of Harnack chain. Contents 11
If h > 0 is harmonic on D, then exp( Ak D (x, y)) h(x) h(y) exp(ak D(x, y)) Local reference points: y 1 R,..., y N R S (ξ, R) D s.t. δ D(y i R ) R and min {k D R (x, y i R )} A log R i=1,...,n for x B(ξ, ηr) D, where D R = D B(ξ, AR). δ D (x) + A ξ y j R If h P ξ, then 0-extension to D c is subharmonic in R n \ {ξ}. Contents 12
Lemma 1 (Domar [Dom57]) Let u 0 be subharmonic in Ω s.t. I = (log + u) n 1+ε dx < Ω for ε > 0. Then u(x) exp(ai 1/ε dist(x, Ω) n/ε ). Lemma 2 τ > 0 s.t. D B(ξ,R) ( R δ D (x) ) τ dx AR n. Contents 13
Lemma 3 Let h P ξ for ξ D. Then h(x) A x ξ λ. By the tract argument [FH76], dim P ξ N(c J ) <. Proof. By local reference points ( ) λ R N h(x) A δ D (x) i=1 h(y i R ). Apply Domar s argument (Lemma 1) to Ω = B(ξ, AR) \ B(ξ, A 1 R) with the help of Lemma 2. Then (1) h(x) A N h(y i R ) on S (ξ, R), i=1 Contents 14
and hence on D \ B(ξ, R) by the maximum principle. Since δ D (y i R ) R, we have h(y i R ) AR λ. Hence i.e. h(x) A x ξ λ on D. h(x) AR λ on D \ B(ξ, R), Contents 15
By the box argument introduced by Bass-Burdzy [BB91] (see [Aik01, Lemma 2]) we have ω(x, D S (ξ, AR), D B(ξ, AR)) AR 2 n N G R (x, y i R ) i=1 for x D B(ξ, R), where G R is the Green function for D B(ξ, A R). Combine with (1). Then h(x) AR 2 n N N G R (x, y i R ) i=1 j=1 h(y j R ). Apply this inequality to h(x) = G R (x, y). Then G R (x, y) AR 2 n N N G R (x, y i R ) G R (y j R, y). i=1 j=1 Contents 16
Ancona s ingenious tricks [Anc84] and [Anc07] erase cross terms: N G R (x, y) AR 2 n G R (x, y i R )G R(y i R, y). Weak boundary Harnack principle Let h 0, h 1,..., h N P ξ. Then N h 0 (y i R h 0 (x) A ) h i (y i R ) h i(x) for x D. In particular, dim P ξ N. i=1 i=1 Contents 17
3. Domar s argument Lemma (Domar [Dom57]) Let u 0 be subharmonic in Ω s.t. I = (log + u) n 1+ε dx < for Ω ε > 0. Then u(x) exp(ai 1/ε dist(x, Ω) n/ε ). Lemma 4 Let L n = (e 2 / B(0, 1) ) 1/n. Let u 0 be subharmonic in B(x, R). If u(x) t > 0 and (2) R L n {y B(x, R) : e 1 t < u(y) et} 1/n, then x B(x, R) s.t. u(x ) > et. Contents 18
Proof. Observe that (2) is equivalent to {y B(x, R) : e 1 t < u(y) et} B(x, R) 1 e 2. If u et on B(x, R), then the mean value property yields 1 t u(x) u(y)dy B(x, R) B(x,R) ( ) 1 = udy + udy B(x, R) B(x,R) {u e 1 t} B(x,R) {u>e 1 t} e 1 t + 1 et < t. e2 This is a contradiction. Contents 19
Proof of Domar s Lemma. It is sufficient to show that (3) δ Ω (x) AI 1/n (log u(x)) ε/n, whenever u(x) > e 2. Fix x 1 Ω with u(x 1 ) > e 2 and let us prove (3) with x = x 1. Let R j = L n {y Ω : e j 2 u(x 1 ) < u(y) e j u(x 1 )} 1/n for j 1. Choose {x j } as follows: If δ Ω (x 1 ) < R 1, then we stop. If δ Ω (x 1 ) R 1, then B(x 1, R 1 ) Ω, so that there exists x 2 B(x 1, R 1 ) such that u(x 2 ) > eu(x 1 ) by Lemma 4. Next we consider δ Ω (x 2 ). If δ Ω (x 2 ) < R 2, then we stop. If δ Ω (x 2 ) R 2, then B(x 2, R 2 ) Ω, so that there exists x 3 B(x 2, R 2 ) such that u(x 3 ) > e 2 u(x 1 ) by Lemma 4. Repeat this procedure to obtain a finite or infinite sequence {x j }. Contents 20
We claim (4) δ Ω (x 1 ) 2 R j. Suppose first {x j } is finite. If δ Ω (x 1 ) < R 1, then (4) trivially holds. If δ Ω (x 1 ) R 1, then we have an integer J 2 such that δ Ω (x 1 ) R 1,..., δ Ω (x J 1 ) R J 1, δ Ω (x J ) < R J, x 2 B(x 1, R 1 ), x 3 B(x 2, R 2 ),..., x J B(x J 1, R J 1 ). Hence we have (4) as δ Ω (x 1 ) x 1 x 2 + + x J 1 x J + δ Ω (x J ) < R 1 + + R J 1 + R J. Suppose next {x j } is infinite. Since u(x j ) > e j u(x 1 ), the local boundedness of a subharmonic function shows that x j Ω. Hence, J 2 s.t. δω (x J ) 1 2 δ Ω(x 1 ). j=1 Contents 21
Then δ Ω (x 1 ) x 1 x 2 + + x J 1 x J + δ Ω (x J ) R 1 + + R J 1 + 1 2 δ Ω(x 1 ), so that (4) follows. In view of (4) we observe that (3) follows from (5) j=1 R j AI 1/n (log u(x 1 )) ε/n. To show (5), let j 1 be the integer such that e j 1 j 1 2 and < u(x 1 ) e j 1+1. Then R j L n {y Ω : e j 1+ j 2 < u(y) e j 1+ j+1 } 1/n. Contents 22
Since the family of intervals {(e j 1+ j 2, e j 1+ j+1 ]} j overlaps at most 3 times, it follows from Hölder s inequality that R j 3L n j=1 3L n A j ε/n 1 j= j 1 {y Ω : e j 1 < u(y) e j } 1/n 1 (n 1)/n j (n 1+ε)/(n 1) j= j 1 ( (log + u) n 1+ε dy Ω A(log u(x 1 )) ε/n I 1/n. ) 1/n Thus (5) follows. The lemma is proved. j n 1+ε {y Ω : e j 1 < u(y) e j } j= j 1 1/n Contents 23
4. Union of convex sets John const c J is close to 1 = D is better. Yet two minimal Marin boundary points. Condition for 1 minimal Marin boundary point? Ancona [Anc79, Théorème]: D is admissible: (A1) D = λ B(x λ, ρ 0 ). (A2) Let ξ D. If D B 1, B 2 with radius ρ 0 tangential at ξ, then D Γ θ (ξ, y) B(ξ, r), a truncated circular cone with aperture θ > 0, radius r > 0 and axis on the tangent hyperplane. B 1 ξ B 2 000 111 000 111 00000 11111 00000 11111 000000 111111 Γ θ (ξ,y) B(ξ,r) Contents 24
Theorem A (Ancona) If D is a bounded admissible domain, then D = D. Generalize both (A1) and (A2). (I) D = λ C λ ; C λ are open convex sets s.t. B(z λ, ρ 0 ) C λ B(z λ, A 1 ρ 0 ). 00000 11111 000000000000 111111111111 000000000000000 111111111111111 000000000000000 111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 C 0000000000000000 1111111111111111 λ ρ 0 0000000000000000 1111111111111111 0000000000000000 1111111111111111 000000000000000 111111111111111 z λ 0000000000000 1111111111111 000000000000 111111111111 0000000000 1111111111 0000000 1111111 A 1 ρ 0 Contents 25
(II) For ξ D θ 1 sin 1 (1/A 1 ), ρ 1 ρ 0 cos θ 1 s.t. C (ξ) = Γ θ1 (ξ, y) B(ξ, 2ρ 1 ) is connected. y D, Γ θ1 (ξ,y) B(ξ,2ρ 1 ) D Ω 00000000 11111111 000000000000 111111111111 0000000000000 1111111111111 000000000000 111111111111 C (ξ) 0000000000 1111111111 000000000 111111111 θ 1 00000000 11111111 000000 111111 0000 1111 000 111 01 ξ Theorem 2 Let D satisfy (I) and (II). Then D = D. Contents 26
Remark 2 Denjoy domain = D = λ B(x λ, ρ 0 ). Lipschitz Denjoy domains sectorial domain = D = λ C λ with (I). Remark 3 The bounds θ 1 sin 1 (1/A 1 ) and ρ 1 ρ 0 cos θ 1 are sharp. Contents 27
References [AHL06] [Aik01] [Anc79] [Anc84] [Anc90] H. Aikawa, K. Hirata, and T. Lundh, Martin boundary points of a John domain and unions of convex sets, J. Math. Soc. Japan 58 (2006), no. 1, 247 274. [3] H. Aikawa, Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan 53 (2001), no. 1, 119 145. [16] A. Ancona, Une propriété de la compactification de Martin d un domaine euclidien, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, 71 90. [24], Régularité d accès des bouts et frontière de Martin d un domaine euclidien, J. Math. Pures Appl. (9) 63 (1984), no. 2, 215 260. [5, 6, 16], Sur la frontière de Martin des domaines de Denjoy, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), no. 2, 259 271. [5] [Anc07], Sur la théorie du potentiel dans les domaines de John, Publ. Mat. 51 (2007), no. 2, 345 396. [16] [BB91] R. F. Bass and K. Burdzy, A boundary Harnack principle in twisted Hölder domains, Ann. of Math. (2) 134 (1991), no. 2, 253 276. [16] Contents 28
[Ben80] [Che89] [CS93] [Dom57] [FH76] [Gar89] M. Benedicks, Positive harmonic functions vanishing on the boundary of certain domains in R n, Ark. Mat. 18 (1980), no. 1, 53 72. [3, 5] N. Chevallier, Frontière de Martin d un domaine de R n dont le bord est inclus dans une hypersurface lipschitzienne, Ark. Mat. 27 (1989), no. 1, 29 48. [5] M. C. Cranston and T. S. Salisbury, Martin boundaries of sectorial domains, Ark. Mat. 31 (1993), no. 1, 27 49. [5, 8] Y. Domar, On the existence of a largest subharmonic minorant of a given function, Ark. Mat. 3 (1957), 429 440. [12, 18] S. Friedland and W. K. Hayman, Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions, Comment. Math. Helv. 51 (1976), no. 2, 133 161. [13] S. J. Gardiner, Minimal harmonic functions on Denjoy domains, Proc. Amer. Math. Soc. 107 (1989), no. 4, 963 970. [5] [Löm00] A. Lömker, Martin boundaries of quasi-sectorial domains, Potential Anal. 13 (2000), no. 1, 11 67. [5, 8] Contents 29
[Seg88] S. Segawa, Martin boundaries of Denjoy domains, Proc. Amer. Math. Soc. 103 (1988), no. 1, 177 183. [5] [Seg90], Martin boundaries of Denjoy domains and quasiconformal mappings, J. Math. Kyoto Univ. 30 (1990), no. 2, 297 316. [5] Contents 30