Special Toeplitz operators on strongly pseudoconvex domains
type: Item
instance of: scholarly article
title: Special Toeplitz operators on strongly pseudoconvex domains
zbMATH Open document ID: 1124.32004
DOI: 10.4171/RMI/476
author: Željko Čučković
author: J. D. McNeal
published in: Revista Matemática Iberoamericana
publication date: 2007-05-03T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1169480033
full work available at URL: https://eudml.org/doc/41995
review text: Let \(\Omega\) be a smoothly bounded domain in \(\mathbb C^N\). Denote by \(\delta(z)\) the distance from \(z\in\Omega\) to the boundary of \(\Omega\). By a distance-symbol Toeplitz operator the authors mean a Toeplitz operator, built from the Bergman kernel \(B\), with symbol equal to a positive power of \(\delta\). Thus a distance-symbol Toeplitz operator has the form \(T_{\delta^\eta}(g)(z)= \int_\Omega B(z, w) \delta(w)^\eta g(w)\, dV(w)\) for some power \(\eta>0\), where \(B(z, w)\) denotes the Bergman kernel associated to \(\Omega\) and \(dV\) is the volume measure on \(\Omega\). These operators are naturally expected to have better ``smoothing'' behavior, as \(\eta\) increases. In this paper the authors study how much a distance-symbol Toeplitz operator, depending on the power \(\eta\), improves \(L^p\)-integrability. Their results are obtained on strongly pseudoconvex domains and depend on the dimension. In case where \(N=1\) and \(\eta\) is relatively small, the authors show that a special case of their results is sharp; the sharpness remains open in general. They wonder whether there are similar \(L^p\)-improving estimates which are independent of the dimension. They mention that their proofs can be carried out, with minimal changes, on other classes of domains where good estimates on the Bergman kernel are known, e.g. finite type domains in \(\mathbb C^2\) and convex domains of finite type in \(\mathbb C^N\). They also mention that further results can be obtained, with help of somewhat more complicated functional analysis, on Banach spaces other than Lebesgue spaces, e.g. Hölder spaces and Sobolev spaces .
reviewed by: Boo Rim Choe
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 32T15
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 5149121
zbMATH Keywords: Bergman kernel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5149121