Abstract
For bounded domains in \({\mathbb {R}}^n\) 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 non-smooth domains.
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Notes
In this context, reflexivity means that \((L_h^p (\Omega ))^{\star } = L_h^{p^{\prime }} (\Omega )\) if and only if \((L_h^{p^{\prime }} (\Omega ))^{\star } = L_h^p (\Omega )\).
The Bergman theory for punctured domains in \({\mathbb {C}}^n\) is the same as for smooth domains, since any holomorphic function on a punctured domain extends holomorphically to the unpunctured domain.
The same interval of p also applies to any simply connected local graph domain in \({\mathbb {C}}\) [12, §3].
For the punctured unit ball in \({\mathbb {R}}^3\), this function is the constant 3/2.
Alternatively, if the duality property held for some \(p<\frac{n}{n-2}\), then necessarily \(L_h^p(\Omega _n^*)=L_h^p(\Omega _n)\) for that p which contradicts Proposition 5.4.
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Koenig, K.D., Wang, Y. Harmonic Bergman Theory on Punctured Domains. J Geom Anal 31, 7410–7435 (2021). https://doi.org/10.1007/s12220-020-00542-8
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DOI: https://doi.org/10.1007/s12220-020-00542-8