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Proceedings of the American Mathematical Society

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Convex domains, Hankel operators, and maximal estimates

Authors: Mehmet Çelik, Sönmez Şahutoğlu and Emil J. Straube
Journal: Proc. Amer. Math. Soc. 148 (2020), 751-764
MSC (2010): Primary 32W05; Secondary 47B35
Published electronically: August 28, 2019
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Abstract: Let $ 1\leq q\leq (n-1)$. We first show that a necessary condition for a Hankel operator on $ (0,q-1)$-forms on a convex domain to be compact is that its symbol is holomorphic along $ q$-dimensional analytic varieties in the boundary. Because maximal estimates (equivalently, a comparable eigenvalues condition on the Levi form of the boundary) turn out to be favorable for compactness of Hankel operators, this result then implies that on a convex domain, maximal estimates exclude analytic varieties from the boundary, except ones of top dimension $ (n-1)$ (and their subvarieties). Some of our techniques apply to general pseudoconvex domains to show that if the Levi form has comparable eigenvalues, or equivalently, if the domain admits maximal estimates, then compactness and subellipticity hold for forms at some level $ q$ if and only if they hold at all levels.

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Additional Information

Mehmet Çelik
Affiliation: Department of Mathematics, Texas A&M University-Commerce, Commerce, Texas 75429

Sönmez Şahutoğlu
Affiliation: Department of Mathematics and Statistics, Toledo, University of Toledo, Ohio 43606

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

Keywords: $\dj$-Neumann problem, Hankel operators, convex domains, maximal estimates, compactness
Received by editor(s): March 1, 2019
Received by editor(s) in revised form: June 3, 2019, and June 5, 2019
Published electronically: August 28, 2019
Additional Notes: This work was supported in part by Qatar National Research Fund Grant NPRP 7-511-1-98 and by the Erwin Schrödinger International Institute for Mathematics and Physics, workshop Analysis and CR Geometry, Dec. 2018.
Communicated by: Harold P. Boas
Article copyright: © Copyright 2019 American Mathematical Society