Harmonic Bergman Theory on Punctured Domains
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Harmonic Bergman Theory on Punctured Domains Kenneth D. Koenig1 · Yuda Wang1 Received: 25 July 2020 / Accepted: 9 October 2020 © Mathematica Josephina, Inc. 2020
Abstract For bounded domains in Rn with smooth boundary that are punctured by removing a point, we give a complete description of when basic duality and approximation properties hold for harmonic Bergman spaces and determine the L p mapping properties of the harmonic Bergman projection. Our findings reveal some unexpected dimension-dependent behavior of harmonic Bergman spaces that can occur for nonsmooth domains. Keywords Harmonic Bergman · Bergman spaces · Duality · Lp mapping · Weak type estimate · Sobolev irregularity Mathematics Subject Classification 32A36 · 32A25 · 42B35 · 46B10
1 Introduction This paper examines some previously unobserved dimension-dependent regularity phenomena exhibited by harmonic Bergman spaces of punctured domains in Rn . In particular, our results provide a complete characterization of those values of p for which the harmonic Bergman projection is L p bounded and for which the spaces of L p harmonic functions satisfy duality and L 2 approximation properties. Our work also clarifies how these three regularity conditions are related to each other in general. p Let ⊂ Rn be a domain. For 1 ≤ p < ∞, the harmonic Bergman space L h () is the closed subspace of L p (, dx) consisting of harmonic functions on , where dx denotes Lebesgue measure. The orthogonal projection Bh from L 2 (, dx) onto L 2h () is the harmonic Bergman projection. It can be expressed as an integral operator
B
Yuda Wang [email protected] Kenneth D. Koenig [email protected]
1
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
123
K. D. Koenig, Y. Wang
Bh ( f )(x)
=
R (x, y) f (y)dy,
where R (x, y) is the harmonic Bergman kernel, i.e., the reproducing kernel of L 2h (). The harmonic Bergman kernel is real-valued, symmetric in x and y, and represented by any orthonormal basis {φk } of L 2h () via the formula R (x, y) =
∞
φk (x)φk (y).
(1.1)
k=1
The sum converges absolutely and uniformly on compact subsets of ×. For details and additional background on harmonic Bergman spaces, see [1, Chapter 8]. The preceding discussion with “holomorphic” in place of “harmonic” describes the usual Bergman spaces, Bergman projection, and (conjugate-symmetric) Bergman kernel for domains in Cn . The harmonic Bergman kernel is difficult to compute explicitly from (1.1)—even more so than the ordinary Bergman kernel—and is not known except for basic examples such as a half-space ([1, Theorem 8.24]), a ball ([1, Theorem 8.13], [15]), or a punctured ball ([16] and Sect. 3 of this paper). For bounded domains with C ∞ smooth boundary, it is still possible to obtain size estimates for the harmonic Bergman kernel and its derivatives (in terms of an isotropic Euclidean distance function) by a scaling argument; these kernel estimates are sufficient to establish L p -Sobolev and Lipschitz mapping properties of the harmonic Be
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