Duality of holomorphic function spaces and smoothing properties of the Bergman projection
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- by A.-K. Herbig, J. D. McNeal and E. J. Straube PDF
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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 $\|Bf\|_{k_{2}}$ 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.References
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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
- MR Author ID: 267191
- Email: mcneal@math.ohio-state.edu
- E. J. Straube
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 168030
- Email: straube@math.tamu.edu
- 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.
- © Copyright 2013 American Mathematical Society
- 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
- MathSciNet review: 3130312