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The method of variation of the domain for poly‐Bergman spaces
Author(s) -
Pessoa Luís V.
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201010057
Subject(s) - mathematics , bergman space , bergman kernel , sequence (biology) , pure mathematics , domain (mathematical analysis) , corollary , projection (relational algebra) , boundary (topology) , mathematical analysis , bounded function , algorithm , biology , genetics
IfU n , n ∈ N , is an increasing sequence (well ordered by inclusion) of domains then the sequence of poly‐Bergman projections on the domains U n strongly converges to the poly‐Bergman projection on the limit domain. As a corollary some properties of the poly‐Bergman spaces on the half‐planes are deduced from the corresponding ones in the unit disk. We obtain explicit representation of the poly‐Bergman projections in terms of the two‐dimensional singular integral operatorsS Π , j , j ∈ Z ± , likewise explicit formulas for the poly‐Bergman kernels. We prove that the poly‐Bergman projections on the sectors with a non‐smooth boundary do not admit the usual representations by the two‐dimensional singular integral operators. The variation of the domain and the latter peculiarity of the poly‐Bergman projections allow us to furnish a larger class of domains not admitting Dzhuraev's formulas.