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Transactions of the American Mathematical Society

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Duality of holomorphic function spaces and smoothing properties of the Bergman projection

Authors: A.-K. Herbig, J. D. McNeal and E. J. Straube
Journal: Trans. Amer. Math. Soc. 366 (2014), 647-665
MSC (2010): Primary 32A36, 32A25, 32C37
Published electronically: July 3, 2013
MathSciNet review: 3130312
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Abstract: Let $ \Omega \Subset \mathbb{C}^{n}$ be a domain with smooth boundary, whose Bergman projection $ B$ maps the Sobolev space $ H^{k_{1}}(\Omega )$ (continuously) into $ H^{k_{2}}(\Omega )$. We establish two smoothing results: (i) the full Sobolev norm $ \Vert Bf\Vert _{k_{2}}$ is controlled by $ L^2$ derivatives of $ f$ taken along a single, distinguished direction (of order $ \leq k_{1}$), and (ii) the projection of a conjugate holomorphic function in $ L^{2}(\Omega )$ is automatically in $ H^{k_{2}}(\Omega )$. There are obvious corollaries for when $ B$ is globally regular.

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Additional Information

A.-K. Herbig
Affiliation: Department of Mathematics, University of Vienna, Vienna, Austria

J. D. McNeal
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210-1174

E. J. Straube
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Received by editor(s): November 8, 2011
Published electronically: July 3, 2013
Additional Notes: This research was supported in part by Austrian Science Fund FWF grants Y377 and V187N13 (first author), National Science Foundation grants DMS–0752826 (second author) and DMS–0758534 (third author), and by the Erwin Schrödinger International Institute for Mathematical Physics.
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