ONE DIMENSIONAL ESTIMATES FOR THE BERGMAN KERNEL AND LOGARITHMIC CAPACITY ZBIGNIEW BLOCKI AND WLODZIMIERZ ZWONEK (Communicated by Filippo Bracci) Abstract

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(1)PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 146, Number 6, June 2018, Pages 2489–2495 http://dx.doi.org/10.1090/proc/13916 Article electronically published on February 16, 2018. ONE DIMENSIONAL ESTIMATES FOR THE BERGMAN KERNEL AND LOGARITHMIC CAPACITY ZBIGNIEW BLOCKI AND WLODZIMIERZ ZWONEK (Communicated by Filippo Bracci) Abstract. Carleson showed that the Bergman space for a domain on the plane is trivial if and only if its complement is polar. Here we give a quantitative version of this result. One is the Suita conjecture, established by the first-named author in 2012, and the other is an upper bound for the Bergman kernel in terms of logarithmic capacity. We give some other estimates for those quantities as well. We also show that the volume of sublevel sets for the Green function is not convex for all regular non-simply connected domains, generalizing a recent example of Fornæss.. 1. Introduction For w ∈ Ω, where Ω is a domain in C, Carleson [7] (see also [8]) showed the Bergman space A2 (Ω) of square integrable holomorphic functions in Ω is trivial if and only if the complement C \ Ω is polar. The estimate conjectured by Suita [12] and proved in [4] (1). cΩ (w)2 ≤ πKΩ (w),. w ∈ Ω,. gives a quantitative version of one of the implications. Here (2) cΩ (w) = exp lim GΩ (z, w) − log |z − w| z→w. is the logarithmic capacity of C \ Ω with respect to w, GΩ (z, w) = sup{u(z) : u ∈ SH(Ω), u < 0, lim sup (u(ζ) − log |ζ − w|) < ∞} ζ→w. is the (negative) Green function, KΩ (w) = sup{|f (w)|2 : f ∈ A2 (Ω), ||f || ≤ 1} is the Bergman kernel on the diagonal, and ||f || = ||f ||L2 (Ω) . Received by the editors March 27, 2017, and, in revised form, July 19, 2017. 2010 Mathematics Subject Classification. Primary 30H20, 30C85, 32A36. Key words and phrases. Bergman kernel, logarithmic capacity, Suita conjecture. The first-named author was supported by the Ideas Plus grant no. 0001/ID3/2014/63 of the Polish Ministry of Science and Higher Education and the second-named author by the Polish National Science Centre (NCN) Opus grant no. 2015/17/B/ST1/00996. c 2018 American Mathematical Society. 2489. Licensed to Jagiellonian University. Prepared on Sat Aug 25 05:13:17 EDT 2018 for download from IP 149.156.75.40. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use.

(2) 2490. ZBIGNIEW BLOCKI AND WLODZIMIERZ ZWONEK. Our first result is the following upper bound for the Bergman kernel: Theorem 1. Let Ω be a domain in C and w ∈ Ω. Assume that 0 < r ≤ δΩ (w) := dist (w, ∂Ω). Then 1 KΩ (w) ≤ . −2πr 2 max GΩ (z, w) z∈Δ(w,r). As a consequence we will obtain the following quantitative version of the other implication in the Carleson characterization: Theorem 2. There exists a uniform constant C > 0 such that for w ∈ Ω, where Ω is a domain in C, we have KΩ (w) ≤. C δΩ. (w)2 log (1/(δ. Ω (w)cΩ (w))). .. We will also consider the following counterparts of the Bergman kernel for higher derivatives for j = 0, 1, . . .: (j). KΩ (w) := sup{|f (j) (w)|2 : f ∈ A2 (Ω), ||f || ≤ 1, f (0) = f (0) = · · · = f (j−1) (0) = 0}. We will prove the following generalization of (1): Theorem 3. For w ∈ Ω ⊂ C and j = 0, 1, 2, . . . we have j!(j + 1)! (cΩ (w))2j+2 . π The inequality is optimal; one can easily check that the equality holds for Ω = Δ, the unit disc, and w = 0. It is clear that the dimension of A2 (Ω) is infinite if and only if, for a given w, (j) there exists infinitely many j’s such that KΩ (w) > 0. Therefore, Theorem 3 gives a quantitative version of a result of Wiegerinck [13] who showed that if C \ Ω is not polar, then A2 (Ω) is infinitely dimensional. Since the proof of Theorem 2 also easily gives the upper bound (j). KΩ (w) ≥. (j). KΩ (w) ≤. Cj , δΩ (w)2+j log (1/(δΩ (w)cΩ (w))). w ∈ Ω,. and by Proposition 6 below we have the following characterization of domains in dimension one: Theorem 4. For w ∈ Ω ⊂ C and j = 0, 1, 2, . . . the following are equivalent: i) C \ Ω is not polar; (j) ii) KΩ (w) > 0; (j) iii) log KΩ is smooth and strongly subharmonic; 2 iv) A (Ω) = {0}; v) dim A2 (Ω) = ∞. A different proof of the Suita conjecture (1) was given in [5]. It follows from the lower bound 1 (3) KΩ (w) ≥ −2t e λ({GΩ (·, w) < t}). Licensed to Jagiellonian University. Prepared on Sat Aug 25 05:13:17 EDT 2018 for download from IP 149.156.75.40. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use.

(3) BERGMAN KERNEL AND LOGARITHMIC CAPACITY. 2491. for t < 0. This inequality was proved in [5] using the tensor power trick which requires a corresponding inequality for pseudoconvex domains in Cn for arbitrary n. As noticed by Lempert (see also [3]), (3) can also be proved using the variational formula for the Bergman kernel in C2 of Maitani-Yamaguchi [11] (generalized by Berndtsson [2] to higher dimensions). Both proofs therefore make crucial use of several complex variables. It would be interesting to find a purely one dimensional proof of (3). It was shown in [6] that the right-hand side of (3) is non-increasing in t (it is an open problem in higher dimensions). Also, a more general conjecture was given, namely that the function (−∞, 0) t −→ log λ({GΩ (·, w) < t}). (4). is convex. A counterexample was found by Fornæss [10]. It was also shown numerically in [1] that the conjecture does not hold in an annulus. Here we will generalize and simplify both results proving the following: Theorem 5. Assume that w ∈ Ω, where Ω is a domain in C, are such that ∇G(z0 ) = 0 for some z0 ∈ Ω \ {w}, where G = GΩ (·, w). Then the function (4) is not convex near t0 = G(z0 ). Note that for example any regular domain Ω which is not simply connected satisfies the assumption of Theorem 5 for any w: it is enough to take maximal t0 such that {G < t0 } is simply connected. Then there exists z0 such that ∇G(z0 ) = 0. 2. Upper bounds for the Bergman kernel In this section we will prove Theorems 1 and 2. Proof of Theorem 1. We may assume that Ω is bounded and smooth, w = 0, and r < δΩ (0). Take f ∈ A2 (Ω). Without loss of generality we may take such an f which is defined in a neighborhood of Ω. Let u ∈ C ∞ (Ω \ Δr ), where Δr := Δ(0, r), be harmonic in Ω \ Δr and such that u = 1 on ∂Ω and u = 0 on ∂Δr . Then 1 1 1 f (z) fu f uz¯ f (0) = dz = dz = dλ. 2πi ∂Ω z 2πi ∂(Ω\Δr ) z π Ω\Δr z Therefore, ||f ||2 |f (0)| ≤ 2 2 π r 2. . ||f ||2 |uz¯| dλ = 4π 2 r 2 Ω\Δr. . 2. un dσ, ∂Ω. where un denotes the outer normal derivative of u at ∂Ω. Denoting G = GΩ (·, 0), we have G + 1 ≤ u, − max G Δr. and therefore on ∂Ω un ≤. Gn . − max G Δr. The required estimate now follows from the fact that Gn dσ = 2π. ∂Ω. . Licensed to Jagiellonian University. Prepared on Sat Aug 25 05:13:17 EDT 2018 for download from IP 149.156.75.40. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use.

(4) 2492. ZBIGNIEW BLOCKI AND WLODZIMIERZ ZWONEK. Proof of Theorem 2. Denote G = GΩ (·, w), R = δΩ (w) and assume that 0 < r < R. Then by the Poisson formula 2π R−r R−r log(RcΩ (w)). max G ≤ G(w + Reit )dt = 2π(R + r) 0 R+r Δ(w,r) By Theorem 1 R+r . 2π(R − r)r 2 log(1/(RcΩ (w))) We can now take for example r = R/2 and the estimate follows. KΩ (w) ≤. √ The smallest constant the above proof gives will be obtained for r = ( 5−1)R/2; then √ 11 + 5 5 . C= 4π (j). 3. Proof of the lower bound for KΩ. In this section we prove Theorem 3. We follow the method from [4]. We could have also used another method from [5] but this would require us to go to several complex variables and we prefer to have a purely one dimensional argument. Proof of Theorem 3. We assume that w = 0, Ω is bounded and smooth, and denote G = GΩ (·, 0). Set z j+1 χ (|z|) ∂ j z χ(|z|) = α := ∂ z¯ 2|z| and ϕ := (2j + 2)G + η ◦ G, ψ := γ ◦ G, 0,1 where χ ∈ C ((0, ∞)), η ∈ C 1,1 ((−∞, 0)), γ ∈ C 0,1 ((−∞, 0)) will be defined later. We assume that η is convex and non-decreasing (so that ϕ is subharmonic), (γ )2 < η , and that (γ ◦ G)2 ≤ δη ◦ G on the support of α for some constant δ with 0 < δ < 1. Then by Theorem 2 in [4] one can find u ∈ L2loc (Ω) such that F := z j χ(|z|) − u is holomorphic and √ |α|2 1+ δ 2 √ |u| Γ ◦ G dλ ≤ e2ψ−ϕ dλ, (5) 1 − δ Ω η ◦ G |Gz |2 Ω (γ )2 Γ := 1 − e2γ−η−(2j+2)t . η Take ε > 0 and assume that χ(|z|) ⊂ {|z| ≤ ε}. We choose T = T (ε) < 0 such that {|z| ≤ ε} ⊂ {G ≤ T }. Since |G − log |z|| ≤ C0 near 0, we may take T := log ε + C0 . Similarly as in [4] for s < 0 we define where. η0 (s) := − log(−s + es − 1), γ0 (s) := − log(−s + es − 1) + log(1 − es ), so that. (γ )2 e2γ0 −η0 −s = 1 1 − 0 η0. and (6). lim (2γ0 (s) − η0 (s) − log η0 (s)) = 0.. s→−∞. Licensed to Jagiellonian University. Prepared on Sat Aug 25 05:13:17 EDT 2018 for download from IP 149.156.75.40. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use.

(5) BERGMAN KERNEL AND LOGARITHMIC CAPACITY. We now set. and. η0 (2j + 2)t , η(t) := −δ log(T − t + a) + b,. t ≥ T, t < T,. γ0 (2j + 2)t , γ(t) := −δ log(T − t + a) + b,. t ≥ T, t < T,. 2493. where δ = δ(ε) > 0 will be determined later and a, b, b are uniquely determined by the conditions η ∈ C 1,1 , γ ∈ C 0,1 : δ , (2j + 2)T (2j + b = b(ε) = η0 (2j + 2)T + δ log a, . b = b(ε) = γ0 (2j + 2)T + δ log a. √ We see that if we choose δ = −T , then δ(ε) → 0 and a(ε) → ∞ as ε → 0. On (−∞, T ) we have (γ )2 = δη and a = a(ε) =. 2)η0. . Γ = (1 − δ)e2b−b. e−(2j+2)t , (T − t + a)δ. so that |z|2j Γ ◦ G is not locally integrable near 0. By (5) it implies that F (0) = F (0) = · · · = F (j−1) (0) = 0 and F (j) (0) = j! χ(0). One can also check that Γ ≥ 1 on (−∞, T ). Since |2Gz − 1/z| ≤ C1 near 0, we have 2|z||Gz | ≥ 1 − C1 ε on {|z| ≤ ε}. There we also have . e2ψ−ϕ e2b−b = (T − G + a)2−δ e−(2j+2)G η ◦G δ . e2b−b (log ε − log |z| + 2C0 + a)2−δ e−(2j+2)G . δ. ≤. Therefore, the right-hand side of (5) can be bounded from above by √ (1 + δ)e2b−b A(ε) √ (χ (|z|))2 (log ε − log |z| + 2C0 + a)2−δ dλ, (7) δ(1 − δ)(cΩ (0))2j+2 {|z|≤ε} where A(ε) → 1 as ε → 0. The optimal choice for χ is χ(r) = (2C0 + a)δ−1 − (log ε − log r + 2C0 + a)δ−1 ; then (7) takes the form 2π(1 +. √. Note that . e2b−b (2C0 + a)1−δ = δ. . δ)2 e2b−b (2C0 + a)δ−1 A(ε) . δ(cΩ (0))2j+2 . 2C0 + a a. 1−δ. . e2γ0 (S)−η0 (S)−log η0 (S) , 2j + 2. where S = (2j +2)T . Combining this with (6) and the fact that χ(0) = (2C0 +a)δ−1. Licensed to Jagiellonian University. Prepared on Sat Aug 25 05:13:17 EDT 2018 for download from IP 149.156.75.40. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use.

(6) 2494. ZBIGNIEW BLOCKI AND WLODZIMIERZ ZWONEK. we will obtain lim inf ε→0. |F (j) (0)|2 j!(j + 1)! = (cΩ (w))2j+2 . ||F ||2 π. . Using standard methods we will also prove the following formula for the Lapla(j) cian of log KΩ . It is of course well known for j = 0 and the proof is essentially the same in general. Proposition 6. For a domain Ω in C such that C \ Ω is not polar and j = 0, 1, . . . we have (j+1) K ∂2 (log KΩ ) = Ω(j) . ∂z∂ z¯ K Ω. Proof. Denote H0 = A (Ω) and for a fixed w ∈ Ω and k = 1, 2, . . . set 2. Hk := {f ∈ A2 (Ω) : f (w) = f (w) = · · · = f (k−1) (w) = 0}. Since Hk is of codimension at most 1 in Hk−1 , we can find an orthonormal system (j) ϕ0 , ϕ1 , . . . in A2 (Ω) such that ϕk ∈ Hk for all k. This means that ϕl (w) = 0 for l > j. For f ∈ Hj we have. f, ϕl ϕl . f= l≥j. Therefore,. (j). KΩ (z) =. (j). |ϕl (z)|2. l≥j. and (j). (j). KΩ (w) = |ϕj (w)|2 . Since. KKzz¯ − |Kz |2 K2. (log K)zz¯ = and (j). (j+1). (KΩ )zz¯(w) = |ϕj (j). (j+1). (w)|2 + |ϕj+1 (w)|2 ,. (j+1). (KΩ )z (w) = ϕj. (j). (w) ϕj (w),. we will obtain. (j+1). (j). (log KΩ )zz¯(w) =. |ϕj+1 (w)|2 (j). |ϕj (w)|2 . and the proposition follows. 4. Proof of Theorem 5. Let tj → t0 be a sequence of regular values for G. It will be enough to show that γ (tj ) → ∞, where γ(t) = λ({GΩ (·, w) < t}). By the co-area formula we have t dσ ds, γ(t) = |∇G| −∞ {G=s} and, therefore, . γ (tj ) =. {G=tj }. dσ . |∇G|. Licensed to Jagiellonian University. Prepared on Sat Aug 25 05:13:17 EDT 2018 for download from IP 149.156.75.40. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use.

(7) BERGMAN KERNEL AND LOGARITHMIC CAPACITY. 2495. It is convenient to assume that z0 = 0. Since G is harmonic in Ω \ {w}, it follows that there exists a holomorphic h near 0 such that h(0) = 0 and for some n ≥ 2 we have G(z) = t0 + Re (z n h(z)) . It follows that near 0 we have |∇G(z)| ≤ C1 |z|n−1 . We can also find a biholomorphic F near 0 such that G(F (ζ)) = t0 + Re (ζ n ). We then have |∇G(F (ζ))| ≤ C2 |ζ|n−1 and for some r > 0 1 dσ dσ ≥ −→ ∞ |∇G| C3 |ζ|n−1 {G=tj }. {ζ∈Δ(0,r) : Re (ζ n )=tj −t0 }. as j → ∞.. References. [1] [2]. [3] [4] [5]. [6] [7] [8] [9] [10] [11] [12] [13]. P. ˚ Ahag, R. Czy˙z, and P. H. Lundow, A Counterexample to a Conjecture by Blocki–Zwonek, Exp. Math. 27 (2018), no. 1, 119–124. MR3750933 Bo Berndtsson, Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains (English, with English and French summaries), Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, 1633–1662. MR2282671 Bo Berndtsson and L´ aszl´ o Lempert, A proof of the Ohsawa-Takegoshi theorem with sharp estimates, J. Math. Soc. Japan 68 (2016), no. 4, 1461–1472. MR3564439 Zbigniew Blocki, Suita conjecture and the Ohsawa-Takegoshi extension theorem, Invent. Math. 193 (2013), no. 1, 149–158. MR3069114 Zbigniew Blocki, A lower bound for the Bergman kernel and the Bourgain-Milman inequality, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2116, Springer, Cham, 2014, pp. 53–63. MR3364678 Zbigniew Blocki and Wlodzimierz Zwonek, Estimates for the Bergman kernel and the multidimensional Suita conjecture, New York J. Math. 21 (2015), 151–161. MR3318425 Lennart Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR0225986 John B. Conway, Functions of one complex variable. II, Graduate Texts in Mathematics, vol. 159, Springer-Verlag, New York, 1995. MR1344449 Harold Donnelly and Charles Fefferman, L2 -cohomology and index theorem for the Bergman metric, Ann. of Math. (2) 118 (1983), no. 3, 593–618. MR727705 John Erik Fornæss, On a conjecture by Blocki and Zwonek, Sci. China Math. 60 (2017), no. 6, 963–966. MR3647125 Fumio Maitani and Hiroshi Yamaguchi, Variation of Bergman metrics on Riemann surfaces, Math. Ann. 330 (2004), no. 3, 477–489. MR2099190 Nobuyuki Suita, Capacities and kernels on Riemann surfaces, Arch. Rational Mech. Anal. 46 (1972), 212–217. MR0367181 Jan J. O. O. Wiegerinck, Domains with finite-dimensional Bergman space, Math. Z. 187 (1984), no. 4, 559–562. MR760055. ´ski, L ´ w, Instytut Matematyki, Uniwersytet Jagiellon ojasiewicza 6, 30-348 Krako Poland Email address: zbigniew.blocki@im.uj.edu.pl ´ski, L ´ w, Instytut Matematyki, Uniwersytet Jagiellon ojasiewicza 6, 30-348 Krawo Poland Email address: wlodzimierz.zwonek@im.uj.edu.pl. Licensed to Jagiellonian University. Prepared on Sat Aug 25 05:13:17 EDT 2018 for download from IP 149.156.75.40. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use.

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