Abstract
AbstractIn this review article we present the problem of studying Hardy spaces and the related Szeg˝o projection on worm domains. We review the importance of the Diederich–Fornæss worm domain as a smooth bounded pseudoconvex domain whose Bergman projection does not preserve Sobolev spaces of sufficiently high order and we highlight which difficulties arise in studying the same problem for the Szeg˝o projection. Finally, we announce and discuss the results we have obtained so far in the setting of non-smooth worm domains.
Highlights
The smooth bounded pseudoconvex domain introduced by Diederich and Fornæss in [20] has a central role in complex analysis in several variables
This domain, known in the literature as the worm domain, provided counterexamples to many important conjectures for the last 40 years. The goal of this primarily expository paper is to show that the worm domain is, once again, a good starting point to study some problems; namely, the worm domain is a good candidate to be a smooth bounded pseudoconvex domain whose Szegoprojection is unbounded with respect to Lp and W k;p norms for some values of p and k
Our goal is to prove an analogous of Barrett’s theorem, i.e., to show that the Szegoprojection of the worm does not preserve Sobolev spaces of high order
Summary
The smooth bounded pseudoconvex domain introduced by Diederich and Fornæss in [20] has a central role in complex analysis in several variables. This domain, known in the literature as the worm domain, provided counterexamples to many important conjectures for the last 40 years. The goal of this primarily expository paper is to show that the worm domain is, once again, a good starting point to study some problems; namely, the worm domain is a good candidate to be a smooth bounded pseudoconvex domain whose Szegoprojection is unbounded with respect to Lp and W k;p norms for some values of p and k. A pseudoconvex domain D is said to have a Stein neighborhood basis if it exists a family of smooth bounded pseudoconvex domains fBj g such that B1 B2 does not exist, D is said to have nontrivial Nebenhülle
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