We study the Bergman kernel and projection on the worm domains for β > π. We calculate the kernels explicitly, up to an error term that can be controlled. Denote by P the Bergman projection on Dβ and by P′ the one on D′β. We show that is bounded when 1 < p < ∞, while if and only if 2/(1 + vβ) < p < 2/(1 - vβ), where vβ = π/(2β - π). Along the way, we give a new proof of the failure of Condition R on these worms. Finally, we are able to show that the singularities of the Bergman kernel on the boundary are not contained in the boundary diagonal.
New results on the Bergman kernel of the worm domain in complex space / S. G. Krantz, M. M. Peloso. - In: ELECTRONIC RESEARCH ANNOUNCEMENTS IN MATHEMATICAL SCIENCES. - ISSN 1935-9179. - 14(2007), pp. 35-41.
New results on the Bergman kernel of the worm domain in complex space
M. M. PelosoUltimo
2007
Abstract
We study the Bergman kernel and projection on the worm domains for β > π. We calculate the kernels explicitly, up to an error term that can be controlled. Denote by P the Bergman projection on Dβ and by P′ the one on D′β. We show that is bounded when 1 < p < ∞, while if and only if 2/(1 + vβ) < p < 2/(1 - vβ), where vβ = π/(2β - π). Along the way, we give a new proof of the failure of Condition R on these worms. Finally, we are able to show that the singularities of the Bergman kernel on the boundary are not contained in the boundary diagonal.Pubblicazioni consigliate
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