aa r X i v : . [ m a t h . C V ] F e b BERGMAN METRIC AND CAPACITY DENSITIES ON PLANAR DOMAINS
BO-YONG CHENA
BSTRACT . We give quantitative estimates of the Bergman distance through positivity of capac-ity densities.
1. I
NTRODUCTION
Let Ω be a bounded domain in C with ∈ ∂ Ω . We set D r := { z : | z | < r } and K r := D r − Ω . The famous Wiener’s criterion states that is a regular point if and only if ∞ X k =1 k log[1 /C l ( K − k )] = ∞ , where C l ( K − k ) denotes the logarithmic capacity of K − k . In a similar manner, Carleson-Totik[4] characterized the Hölder continuity of the Green function g Ω ( · , w ) at through positivityof certain capacity density, under some mild restrictions on ∂ Ω near . It is also known thatthe regularity of implies that the Bergman kernel is exhaustive at (cf. [12]) and the Bergmanmetric is complete at (cf. [2], [11]). Zwonek [16] showed that the Bergman kernel is exhaustiveat if and only if ∞ X k =1 k log[1 /C l ( K − k )] = ∞ . On the other hand, a similar characterization for the Bergman completeness at is still missing,although some partial results exist (cf. [14]).The goal of this paper is to get quantitative estimates of the Bergman distance through posi-tivity of certain capacity densities.Motivated by the work of Carleson-Totik [4], we give the following Definition 1.1.
Let ε > and < λ < be fixed. For every a ∈ ∂ Ω we set K t ( a ) := D t ( a ) − Ω; D t ( a ) := { z : | z − a | < t }N a ( ε, λ ) := (cid:8) n ∈ Z + : C l ( K λ n ( a )) ≥ ελ n (cid:9) N na ( ε, λ ) := N a ( ε, λ ) ∩ { , , · · · , n } . We define the ( ε, λ ) − capacity density of ∂ Ω at a by D a ( ε, λ ) := lim inf n →∞ |N na ( ε, λ ) | n . Supported by National Natural Science Foundation of China, No.11771089.
We define the weak and strong ( ε, λ ) − capacity density of ∂ Ω by D W ( ε, λ ) := lim inf n →∞ inf a ∈ ∂ Ω |N na ( ε, λ ) | n and D S ( ε, λ ) := lim inf n →∞ | T a ∈ ∂ Ω N na ( ε, λ ) | n respectively. It is easy to see that D W ( ε, λ ) ≥ D S ( ε, λ ) , and C l ( K t ( a )) ≥ εt implies D a ( ε, λ ) = 1 . Recallthat ∂ Ω is said to be uniformly perfect if inf a ∈ ∂ Ω C l ( K t ( a )) ≥ εt (cf. [15]). Thus if ∂ Ω isuniformly perfect then D W ( ε, λ ) = D S ( ε, λ ) = 1 for some ε > . On the other hand, it waspointed out in [4] that the domain Ω = D − { } − ∞ [ k =1 h − k +1 , − k i satisfies D W ( ε, / > for some ε > while ∂ Ω is non-uniformly perfect. Actually, one mayverify that D S ( ε, / > .Let δ ( z ) denote the euclidean distance from z to ∂ Ω and d B ( z , z ) the Bergman distance froma fixed point z to z . We have Theorem 1.1. (1) If D S ( ε, λ ) > for some ε, λ , then (1.1) d B ( z , z ) & | log δ ( z ) | , ∀ z ∈ Ω . (2) If D W ( ε, λ ) > for some ε, λ , then (1.2) d B ( z , z ) & | log δ ( z ) | log | log δ ( z ) | for all z sufficiently close to ∂ Ω . In [5], (1.1) was verified by a different method in case ∂ Ω is uniformly perfect. Estimate oftype (1.2) was first obtained by Blocki [1] for bounded pseudoconvex domains with Lipschitzboundaries in C n (see also [6] and [8] for related results). Definition 1.2.
For ε > , < λ < and γ > we set N a ( ε, λ, γ ) := (cid:8) n ∈ Z + : C l ( K λ n ( a )) ≥ ελ γn (cid:9) N na ( ε, λ, γ ) := N a ( ε, λ, γ ) ∩ { , , · · · , n } . We define the ( ε, λ, γ ) − capacity density of ∂ Ω at a by D a ( ε, λ, γ ) := lim inf n →∞ P k ∈N na ( ε,λ,γ ) k − log n and the weak and strong ( ε, λ, γ ) − capacity densities of ∂ Ω by D W ( ε, λ, γ ) := lim inf n →∞ inf a ∈ ∂ Ω |N na ( ε, λ, γ ) | log n ERGMAN METRIC AND CAPACITY DENSITIES ON PLANAR DOMAINS 3 and D S ( ε, λ, γ ) := lim inf n →∞ | T a ∈ ∂ Ω N na ( ε, λ, γ ) | log n respectively. Note that D W ( ε, λ, γ ) ≥ D S ( ε, λ, γ ) . If C l ( K t ( a )) ≥ εt γ for some ε > and γ > , then D a ( ε, λ, γ ) = 1 in view of the following well-known formula lim n →∞ (cid:18) · · · + 1 n − log n (cid:19) = Euler constant . Theorem 1.2. If D W ( ε, λ, γ ) > for some ε, λ, γ , then (1.3) d B ( z , z ) & log log | log δ ( z ) | for all z sufficiently close to ∂ Ω . The proofs of the theorems depend on precise estimates of the Green function. We also developa general method to obtain quantitative results on hyperconvexity of bounded planar domainsthrough the logarithmic capacity, which is of independent interest.Some interesting questions arise.
Problem 1.
Does there exist an example with D W ( ε, λ ) > resp. D W ( ε, λ, γ ) > for some ε, λ ( resp. ε, λ, γ ) while D S ( ε, λ ) = 0 ( resp. D S ( ε, λ, γ ) = 0) ? Problem 2.
Suppose D S ( ε, λ, γ ) > for some ε, λ, γ . Does one have (1.4) d B ( z , z ) & log | log δ ( z ) | for z sufficiently close to ∂ Ω ?
2. C
APACITIES
In this section we shall review different notions of capacities and present some basic propertiesof them. Let Ω be a bounded domain in C and K ⊂ Ω a compact (non-polor) set in Ω . We definethe Dirichlet capacity C d ( K, Ω) of K relative to Ω by(2.1) C d ( K, Ω) = inf φ ∈L ( K, Ω) Z Ω |∇ φ | where L ( K, Ω) is the set of all locally Lipschitz functions φ on Ω with a compact support in Ω such that ≤ φ ≤ and φ | K = 1 . If Ω = C then we write C d ( K ) for C d ( K, Ω) . By thedefinition we have(2.2) K ⊆ K and Ω ⊇ Ω ⇒ C d ( K , Ω ) ≤ C d ( K , Ω ) . In view of Dirichlet’s principle, the infimum in (2.1) is attained at the function φ min which isexactly the Perron solution to the following (generalized) Dirichlet problem in Ω \ K :(2.3) ∆ u = 0; u = 0 n.e. on ∂ Ω; u = 1 n.e. on ∂K. BO-YONG CHEN
We call φ min the capacity potential of K relative to Ω . In case ∂ Ω and ∂K are both C − smooth,integration by parts gives C d ( K, Ω) = Z Ω |∇ φ min | = Z Ω \ K |∇ φ min | = − Z Ω \ K φ min ∆ φ min + Z ∂ (Ω \ K ) φ min ∂φ min ∂ν dσ = Z ∂K ∂φ min ∂ν dσ =: − flux ∂K φ min (2.4)where ν is the outword unit normal vector fields on ∂ (Ω \ K ) . By the maximum principle weconclude that ∂φ min /∂ν ≥ holds on ∂K .Let g Ω ( z, w ) be the (negative) Green function on Ω . Let z ∈ Ω \ K be given. Since ∆ g Ω ( · , z ) =2 πδ z , where δ z stands for the Dirac measure at z , we infer from Green’s formula that πφ min ( z ) = Z Ω \ K φ min ∆ g Ω ( · , z ) = Z Ω \ K g Ω ( · , z ) ∆ φ min + Z ∂ (Ω \ K ) φ min ∂g Ω ( · , z ) ∂ν dσ − Z ∂ (Ω \ K ) g Ω ( · , z ) ∂φ min ∂ν dσ = Z ∂K ∂g Ω ( · , z ) ∂ν dσ − Z ∂K g Ω ( · , z ) ∂φ min ∂ν dσ = − Z ∂K g Ω ( · , z ) ∂φ min ∂ν dσ (2.5)because g Ω ( · , z ) is harmonic on K . This equality combined with (2.4) gives the following fun-damental inequality which connects the capacity, Green’s function and the capacity potential:(2.6) C d ( K, Ω)2 π inf ∂K ( − g Ω ( · , z )) ≤ φ min ( z ) ≤ C d ( K, Ω)2 π sup ∂K ( − g Ω ( · , z )) , z ∈ Ω \ K. Since Ω \ K can be exhausted by bounded domains with smooth boundaries, we conclude bypassing to a standard limit process that the same inequality holds for every compact set K .For a finite Borel measure µ on C whose support is contained in K we define its Green poten-tial relative to Ω by p µ ( z ) = Z Ω g Ω ( z, w ) dµ ( w ) , z ∈ Ω . ERGMAN METRIC AND CAPACITY DENSITIES ON PLANAR DOMAINS 5
Clearly, p µ is negative, subharmonic on Ω , harmonic on Ω \ K , and p µ ( z ) → as z → ∂ Ω . Given φ ∈ C ∞ (Ω) we have Z Ω p µ ∆ φdV = Z Ω (cid:20)Z Ω g Ω ( z, w ) dµ ( w ) (cid:21) ∆ φ ( z ) dV ( z )= Z Ω (cid:20)Z Ω g Ω ( z, w )∆ φ ( z ) dV ( z ) (cid:21) dµ ( w ) ( Fubini’s theorem )= Z Ω (cid:20)Z Ω ∆ g Ω ( z, w ) φ ( z ) dV ( z ) (cid:21) dµ ( w ) ( Green’s formula )= Z Ω πφ ( w ) dµ ( w ) . Thus we obtain(2.7) ∆ p µ = 2 πµ in the sense of distributions. The Green energy I ( µ ) of µ is given by I ( µ ) := Z Ω p µ dµ = Z Ω Z Ω g Ω ( z, w ) dµ ( z ) dµ ( w ) . By (2.7) we have(2.8) I ( µ ) = 12 π Z Ω p µ ∆ p µ = − π Z Ω |∇ p µ | . Every compact set K has an equilibrium measure µ max , which maximizes I ( µ ) among all Borelprobability measures µ on K . A fundamental theorem of Frostman states that(1) p µ max ≥ I ( µ max ) on Ω ;(2) p µ max = I ( µ max ) on K \ E for some F σ polor set E ⊂ ∂K .By the uniqueness of the solution of the (generalized) Dirichlet problem we have(2.9) φ min = p µ max /I ( µ max ) . We define the Green capacity C g ( K, Ω) of K relative to Ω by C g ( K, Ω) := e I ( µ max ) . It follows from (2.8) and (2.9) that(2.10) C d ( K, Ω)2 π = − C g ( K, Ω) . Analogously, we may define the logarithmic capacity C l ( K ) of K by log C l ( K ) := sup µ Z C Z C log | z − w | dµ ( z ) dµ ( w ) where the supremum is taken over all Borel propability measures µ on C whose support is con-tained in K . Let R be the diameter of Ω and set d = d ( K, Ω) . Since log | z − w | /R ≤ g Ω ( z, w ) ≤ log | z − w | /d, z, w ∈ K, BO-YONG CHEN we have(2.11) log C l ( K ) − log R ≤ log C g ( K, Ω) ≤ log C l ( K ) − log d.
3. E
STIMATES OF THE CAPACITY POTENTIAL
We first give a basic lemma as follows.
Lemma 3.1.
Let Ω be a bounded domain in C with ∈ ∂ Ω . Let ≤ h ≤ be a harmonicfunction on Ω such that h = 0 n.e. on ∂ Ω ∩ D r for some r < . For all < α < / we have (3.1) sup Ω ∩ D r h ≤ exp " − log 1 / (16 α )log 1 /α Z αr r (cid:18) t log t/α C l ( K t ) (cid:19) − dt where K t := D t − Ω .Proof. The idea of the proof comes from [10] (see also [7]). Let D be the unit disc. For t < r and | z | = t we have(3.2) sup ∂K αt ( − g D ( · , z )) ≤ log 2 + sup ∂K αt ( − log | · − z | ) ≤ log 2 − log | t − αt | ≤ log 4 /t. Let φ αt be the capacity potential of K αt relative to D . By (2.6) and (3.2) we have φ αt ( z ) ≤ (log 4 /t ) · C d ( K αt , D )2 π for | z | = t. It follows that for z ∈ Ω ∩ ∂ D t (3.3) (1 − φ αt ( z )) sup Ω ∩ D t h ≥ (cid:20) − log 4 t · C d ( K αt , D )2 π (cid:21) h ( z ) , while the same inequality holds for z ∈ ∂ Ω ∩ D t , because lim z → ζ h = 0 for n.e. ζ ∈ ∂ Ω ∩ D t .By the (generalized) maximum principle, (3.3) holds on Ω ∩ D t . On the other hand, since for | z | = αt we have(3.4) inf ∂K αt ( − g D ( · , z )) ≥ log 1 / ∂K αt ( − log | · − z | ) ≥ log 1 / − log(2 αt ) = log 14 αt , it follows from (2.6) that(3.5) φ αt ( z ) ≥ log 14 αt · C d ( K αt , D )2 π for | z | = αt. Substituting (3.5) into (3.3) we have h ( z ) ≤ sup Ω ∩ D t h · − log αt · C d ( K αt , D )2 π − log t · C d ( K αt , D )2 π ≤ sup Ω ∩ D t h α ) · C d ( K αt , D )2 π − log t · C d ( K αt , D )2 π ! (3.6) ERGMAN METRIC AND CAPACITY DENSITIES ON PLANAR DOMAINS 7 for z ∈ Ω ∩ D αt . Set M ( t ) := sup Ω ∩ D t h . It follows from (3.6) and (2.10) that log M ( t ) t − log M ( αt ) t ≥ log 116 α (cid:18) t log t C g ( K αt , D ) (cid:19) − ≥ log 116 α (cid:18) t log t C l ( K αt ) (cid:19) − ( by (2 . . Integration from r/α to r gives log 116 α Z r r/α (cid:20) t log t C l ( K αt ) (cid:21) − dt ≤ Z r r/α log M ( t ) t dt − Z r r/α log M ( αt ) t dt = Z r r/α log M ( t ) t dt − Z αr r log M ( t ) t dt ≤ Z r αr log M ( t ) t dt − Z r/αr log M ( t ) t dt ≤ (log M ( r ) − log M ( r )) log 1 /α, because M ( t ) is nondecreasing. Thus (3.1) holds because M ( r ) ≤ . (cid:3) Theorem 3.2.
Fix a compact set E in Ω with C l ( E ) > . Let φ E be the capacity potential of E relative to Ω . Set d = d ( E, ∂ Ω) . Then for all < α < / (3.7) sup Ω ∩ D r φ E ≤ exp " − log 1 / (16 α )log 1 /α Z αdr (cid:18) t log t/α C l ( K t ) (cid:19) − dt . Proof.
The solution of the (generalized) Dirichlet problem gives lim z → ζ φ E ( z ) = 0 for n.e. ζ ∈ ∂ Ω , and the (generalized) maximum principle gives ≤ φ E ≤ . Thus Lemma 3.1 applies. (cid:3) We shall give a few interesting consequences of Theorem 3.2. Let < λ < . For N ≫ wehave Z αd (cid:18) t log t/α C l ( K t ) (cid:19) − dt ≥ Z αd (cid:18) t log d C l ( K t ) (cid:19) − dt = log 1 λ · Z ∞ log αd log λ (cid:18) log d C l ( K λ s ) (cid:19) − ds & ∞ X n = N Z λ − n λ − n +1 (cid:18) log 12 C l ( K λ s ) (cid:19) − ds & ∞ X n = N λ − n log[1 /C l ( K λ λ − n )] . (3.8) BO-YONG CHEN
On the other hand, we have ∞ X k = λ − N k log[1 /C l ( K λ k )] = ∞ X n = N λ − n − X k = λ − n k log[1 /C l ( K λ k )] ≤ λ · ∞ X n = N λ − n log[1 /C l ( K λ λ − n )] . (3.9)By (3 . ∼ (3 . we see that if ∞ X k =1 k log[1 /C l ( K λ k )] = ∞ , then is a regular point for Ω , which is exactly the sufficient part of Wiener’s criteria.Theorem 3.2 also yields a new proof of the following result due to Carleson-Totik [4]. Corollary 3.3. If D W ( ε, λ ) > for some ε, λ , then there exists β > such that (3.10) φ E ( z ) ≤ δ ( z ) β where δ denotes the boundary distance of Ω .Proof. Since D W ( ε, λ ) > , there exist c > and n ∈ Z + such that |N na ( ε, λ ) | ≥ cn, ∀ n ≥ n and a ∈ ∂ Ω . Since N na ( ε, λ ) is decreasing in ε , we may assume that ε is as small as we want. Note that for n ≫ N ≫ Z λ N λ n (cid:18) t log t/α C l ( K t ( a )) (cid:19) − dt ≥ X k ∈N na ( ε,λ ) \N Na ( ε,λ ) Z λ k − λ k (cid:18) t log t/α C l ( K t ( a )) (cid:19) − dt ≥ X k ∈N na ( ε,λ ) \N Na ( ε,λ ) (cid:18) log 12 λεα (cid:19) − Z λ k − λ k dtt = log 1 /λ · (cid:18) log 12 λεα (cid:19) − |N na ( ε, λ ) \N Na ( ε, λ ) |≥ log 1 /λ · (cid:18) log 12 λεα (cid:19) − · cn c · (cid:18) log 12 λεα (cid:19) − · log 1 /λ n . (3.11)Since for every z there exists n ∈ Z + such that λ n ≤ | z − a | ≤ λ n − , it follows from (3.7) and(3.11) that φ E ( z ) ≤ | z − a | β for suitable constant β > which is independent of a , so that (3.10) holds. (cid:3) ERGMAN METRIC AND CAPACITY DENSITIES ON PLANAR DOMAINS 9
Remark.
It is remarkable that the converse of Corollary 3.3 holds under the additional conditionthat Ω contains a fixed size cone with vertex at any a ∈ ∂ Ω (cf. [4]). Corollary 3.4. If D W ( ε, λ, γ ) > for some ε, λ, γ , then there exists β > such that (3.12) φ E ( z ) ≤ ( − log δ ( z )) − β for all z sufficiently close to ∂ Ω .Proof. Since D W ( ε, λ, γ ) > , there exist c > and n ∈ Z + such that |N na ( ε, λ, γ ) | ≥ c log n, ∀ n ≥ n and a ∈ ∂ Ω . Note that for n ≫ N ≫ Z λ N λ n (cid:18) t log t/α C l ( K t ( a )) (cid:19) − dt ≥ X k ∈N na ( ε,λ,γ ) \N Na ( ε,λ,γ ) Z λ k − λ k (cid:18) t log t/α C l ( K t ( a )) (cid:19) − dt ≥ X k ∈N na ( ε,λ,γ ) \N Na ( ε,λ,γ ) (cid:18) log λ (1 − γ ) k − εα (cid:19) − Z λ k − λ k dtt & X k ∈N na ( ε,λ,γ ) \N Na ( ε,λ,γ ) k − & log n where the implicit constants are independent of a . This combined with (3.7) gives φ E ( z ) ≤ ( − log | z − a | ) − β for some constant β > independent of a , which in turn implies (3.12). (cid:3) Corollary 3.5.
Suppose inf a ∈ ∂ Ω C l ( K t ( a )) ≥ εt γ for some ε > and γ > . For every τ < γ − we have φ E ( z ) ≤ const τ ( − log δ ( z )) − τ for all z sufficiently close to ∂ Ω .Proof. By (3.7) we have for every a ∈ ∂ Ωsup Ω ∩ D r ( a ) φ E ≤ exp " − log 1 / (16 α )log 1 /α Z αdr (cid:18) t log t/α C l ( K t ( a )) (cid:19) − dt ≤ exp (cid:20) − log 1 / (16 α )log 1 /α Z αdr dtt (( γ −
1) log 1 /t + log 1 / (2 αε )) (cid:21) ≤ const τ ( − log r ) τ provided α sufficiently small, from which the assertion follows. (cid:3)
4. E
STIMATES OF THE G REEN FUNCTION
Proposition 4.1. If D W ( ε, λ, γ ) > for some ε, λ, γ , then there exists c ≫ such that (4.1) { g Ω ( · , w ) ≤ − } ⊂ n c − φ E ( w ) ββ < φ E < c φ E ( w ) β β o where β is given as (3 . .Proof. We shall first adopt a trick from [3]. Consider two points z, w with | z − w | , δ ( z ) and δ ( w ) are sufficiently small. We want to show(4.2) | φ E ( z ) − φ E ( w ) | ≤ c ( − log | z − w | ) − β for some numerical constant c > . Without loss of generality, we assume δ ( w ) ≥ δ ( z ) . If | z − w | ≥ δ ( w ) / , this follows directly from (3.12). Since φ E is a positive harmonic function on D ( w, δ ( w )) , we see that if | z − w | ≤ δ ( w ) / then | φ E ( z ) − φ E ( w ) | ≤ sup D ( w,δ ( w ) / |∇ φ E | | z − w |≤ c δ ( w ) − ( − log δ ( w )) − β | z − w | ( by (3 . ≤ c (2 | z − w | ) − ( − log(2 | z − w | )) − β | z − w |≤ c ( − log | z − w | ) − β . The remaining argument is standard. Let R be the diameter of Ω . By (4.2) we conclude that if φ E ( z ) = φ E ( w ) / then log | z − w | R ≥ − (cid:18) c φ E ( w ) (cid:19) /β − log R ≥ − c φ E ( w ) − /β Since − φ E is subharmonic on Ω , it follows that ψ ( z ) := ( log | z − w | R if φ E ( z ) ≥ φ E ( w ) / n log | z − w | R , − c φ E ( w ) − − /β φ E ( z ) o otherwise . is a well-defined negative subharmonic function on Ω with a logarithmic pole at w , and if φ E ( z ) ≤ φ E ( w ) / then we have g Ω ( z, w ) ≥ ψ ( z ) ≥ − c φ E ( w ) − − /β φ E ( z ) , so that { g Ω ( · , w ) ≤ − } ∩ { φ E ≤ φ E ( w ) / } ⊂ (cid:8) φ E ≥ (2 c ) − φ E ( w ) /β (cid:9) . Since { φ E ≥ φ E ( w ) / } ⊂ { φ E > c − φ E ( w ) /β } if c ≫ , we have { g Ω ( · , w ) ≤ − } ⊂ (cid:8) φ E > c − φ E ( w ) /β (cid:9) . By the symmetry of g Ω , we immediately get { g Ω ( · , w ) ≤ − } ⊂ n φ E < ( cφ E ( w )) β β o . (cid:3) ERGMAN METRIC AND CAPACITY DENSITIES ON PLANAR DOMAINS 11
Lemma 4.2 (cf. [9]) . Let Ω be a bounded domain in C and U a relatively compact open set in Ω . For every w ∈ U we have (4.3) min ∂U ( − g Ω ( · , w )) ≤ πC d (cid:0) U , Ω (cid:1) ≤ max ∂U ( − g Ω ( · , w )) . Proof.
Since g Ω ( · , w ) is harmonic on Ω \ U and vanishes n.e on ∂ Ω , it follows from the maximumprinciple that sup Ω \ U ( − g Ω ( · , w )) = max ∂U ( − g Ω ( · , w )) and inf U ( − g Ω ( · , w )) = min ∂U ( − g Ω ( · , w )) . Then we have n − g Ω ( · , w ) ≥ max ∂U ( − g Ω ( · , w )) o ⊂ U ⊂ n − g Ω ( · , w ) ≥ min ∂U ( − g Ω ( · , w )) o . Set F c = {− g Ω ( · , w ) ≥ c } . It suffices to show C d ( F c , Ω) = 2 π/c.
Indeed, the function φ c := − c − g Ω ( · , w ) is the capacity potential of F c relative to Ω . Thus wehave C d ( F c , Ω) = − flux ∂F c φ c = − flux ∂ Ω φ c = c − flux ∂ Ω g Ω ( · , w ) = 2 π/c. where the second and last equalities follow from Green’s formula. (cid:3) Lemma 4.3.
Let Ω be a bounded domain in C with ∈ ∂ Ω . Let β > α > . Suppose C l ( K r ) ≥ εr for some ε, r > . There exists a positive number c depending only on α, β, ε suchthat for every point w with | w | = βr and D αr ( w ) ⊂ Ω we have (4.4) { g Ω ( · , w ) ≤ − c } ⊂ D αr ( w ) . Proof.
By (4.3) and Harnack’s inequality it suffices to show(4.5) C d (cid:16) D αr ( w ) , Ω (cid:17) ≥ c ′ for some positive constant c ′ depending only on α, β, ε . By the definition we see that C d (cid:16) D αr ( w ) , Ω (cid:17) = C d (cid:16) Ω c , C ∞ − D αr ( w ) (cid:17) ≥ C d (cid:16) K r , C ∞ − D αr ( w ) (cid:17) . (4.6)We consider the conformal map T : C ∞ − D αr ( w ) → D , z αrz − w . Since the Dirichlet energy is invariant under conformal maps, it follows that(4.7) C d (cid:16) K r , C ∞ − D αr ( w ) (cid:17) = C d ( T ( K r ) , D ) . Since K r ⊂ D (1+ β ) r ( w ) − D αr ( w ) , we have T ( K r ) ⊂ D / − D α/ (1+ β ) , so that C d ( T ( K r ) , D ) = − π log C g ( T ( K r ) , D ) ≥ π log 2 − log C l ( T ( K r )) . (4.8)Since | T − ( z ) − T − ( z ) | = αr | z z | · | z − z | ≤ (1 + β ) rα · | z − z | for all z , z ∈ T ( K r ) , we have C l ( T ( K r )) ≥ α (1 + β ) r · C l ( K r ) ≥ αε (1 + β ) . This combined with (4 . ∼ (4 . gives (4 . . (cid:3) Proposition 4.4.
Suppose D S ( ε, λ ) > for some ε, λ . There exists c ≫ such that for every k ∈ T a ∈ ∂ Ω N na ( ε, λ ) and every w with λ k − / ≤ δ ( w ) ≤ λ k − / , (4.9) { g Ω ( · , w ) ≤ − c } ⊂ (cid:8) λ k < δ < λ k − (cid:9) . Proof.
Take a ( w ) ∈ ∂ Ω such that | w − a ( w ) | = δ ( w ) . Note that C l ( K λ k ( a ( w ))) ≥ ελ k . Thus Lemma 4.3 applies. (cid:3)
5. P
ROOFS OF T HEOREM
AND T HEOREM
Proof of Theorem 1.1. (1)
Let c be as Proposition 4.4. Let z be sufficiently close to ∂ Ω . Take n ∈ Z + such that λ n ≤ δ ( z ) ≤ λ n − . Write \ a ∈ ∂ Ω N na ( ε, λ ) = { k < k < · · · < k m n } . We may choose a Bergman geodesic jointing z to z , and a finite number of points on thisgeodesic with the following order z → z k → z k → · · · → z, such that λ k j − / ≤ δ ( z k j ) ≤ λ k j − / . By Proposition 4.4 we have { g Ω ( · , z k j ) ≤ − c } ∩ { g Ω ( · , z k j +1 ) ≤ − c } = ∅ so that d B ( z k j , z k j +1 ) ≥ c > for all j , in view of Theorem 1.1 in [1]. Since m n & n , we have d B ( z , z ) ≥ X j d B ( z k j , z k j +1 ) & n & | log δ ( z ) | . (2) The assertion follows directly from Corollary 3.3 and Corollary 1.8 in [6]. (cid:3)
ERGMAN METRIC AND CAPACITY DENSITIES ON PLANAR DOMAINS 13
Proof of Theorem 1.2.
Let c be as Proposition 4.1. Let z be sufficiently close to ∂ Ω . We maychoose a Bergman geodesic jointing z to z , and a finite number of points { z k } mk =1 on this geo-desic with the following order z → z → z → · · · → z m → z, where c φ E ( z k +1 ) β β = c − φ E ( z k ) ββ and c − φ E ( z m ) ββ ≤ φ E ( z ) ≤ cφ E ( z m ) β β . By Proposition 4.1 we have { g Ω ( · , z k ) ≤ − } ∩ { g Ω ( · , z k +1 ) ≤ − } = ∅ so that d B ( z k , z k +1 ) ≥ c > for all k .Note that log φ E ( z ) = (cid:18) β β (cid:19) log φ E ( z ) + β β log c = · · · = (cid:18) β β (cid:19) m log φ E ( z m ) + β β − (cid:16) β β (cid:17) m − (cid:16) β β (cid:17) log c . Thus we have m ≍ log | log φ E ( z m ) | ≍ log | log φ E ( z ) | & log log | log δ ( z ) | , so that d B ( z , z ) ≥ m − X k =1 d B ( z k , z k +1 ) ≥ c ( m − & log log | log δ ( z ) | . (cid:3) R EFERENCES [1] Z. Blocki,
The Bergman metric and the pluricomplex Green function , Trans. Amer. Math. Soc. (2004),2613–2625.[2] Z. Blocki and P. Pflug,
Hyperconvexity and Bergman completeness , Nagoya Math. J. (1998), 221–225.[3] L. Carleson and T. W. Gamelin, Complex Dynamics, Springer, 1993.[4] L. Carleson and V. Totik,
Hölder continuity of Green’s functions , Acta Sci. Math. (Szeged) (2004), 558–608.[5] B.-Y. Chen, An essay on Bergman completeness , Ark. Mat. (2013), 269–291.[6] ———-, Bergman kernel and hyperconvexity index , Analysis & PDE (2017), 1429–1454.[7] ———-, Every bounded pseudoconvex domain with Hölder boundary is hyperconvex , arXiv:2004.09696v1.[8] K. Diederich and T. Ohsawa,
An estimate for the Bergman distance on pseudoconvex domains , Ann. of Math. (1995), 181–190.[9] A. Grigor’yan,
Analytic and geometric background for recurrence and non-explosion of the Brownian motionon Riemannian manifolds , Bull. Amer. Math. Soc. (1999), 135–249.[10] M. Grüter and K.-O. Widman, The Green function for uniformly elliptic equations , Manuscripta Math. (1982), 303–342.[11] G. Herbort, The Bergman metric on hyperconvex domains , Math. Z. (1999), 183–196. [12] T. Ohsawa,
On the Bergman kernel of hyperconvex domains , Nagoya Math. J. (1993), 43–52.[13] T. Ransford, Potential Theory in the Complex Plane, Cambridge Univ. Press, Cambridge, 1995.[14] P. Pflug and W. Zwonek,
Logarithmic capacity and Bergman functions , Arch. Math. (2003), 536–552.[15] H. Pommerenke, Uniformly perfect sets and the Poincaré metric , Arch. Math. (1979), 192–199.[16] W. Zwonek, Wiener’s type criterion for Bergman exhaustiveness , Bull. Pol. Acad. Sci. Math. (2002), 297–311.D EPARTMENT OF M ATHEMATICAL S CIENCES , F
UDAN U NIVERSITY , S
HANGHAI , 20043, C
HINA
Email address ::