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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
DOI: https://doi.org/10.1090/S0002-9947-2013-05827-8
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
Email: anne-katrin.herbig@univie.ac.at

J. D. McNeal
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210-1174
Email: mcneal@math.ohio-state.edu

E. J. Straube
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: straube@math.tamu.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05827-8
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.
Article copyright: © Copyright 2013 American Mathematical Society

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