Convex domains, Hankel operators, and maximal estimates
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- by Mehmet Çeli̇k, Sönmez Şahutoğlu and Emil J. Straube PDF
- Proc. Amer. Math. Soc. 148 (2020), 751-764 Request permission
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.References
- Benoît Ben Moussa, Analyticité semi-globale pour le $\overline \partial$-Neumann dans des domaines pseudoconvexes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 1, 51–100 (French, with English summary). MR 1765538
- Thomas Bloom and Ian Graham, A geometric characterization of points of type $m$ on real submanifolds of $\textbf {C}^{n}$, J. Differential Geometry 12 (1977), no. 2, 171–182. MR 492369
- David Catlin, Necessary conditions for subellipticity and hypoellipticity for the $\bar \partial$-Neumann problem on pseudoconvex domains, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 93–100. MR 627751
- David Catlin, Necessary conditions for subellipticity of the $\bar \partial$-Neumann problem, Ann. of Math. (2) 117 (1983), no. 1, 147–171. MR 683805, DOI 10.2307/2006974
- Mehmet Çelik and Emil J. Straube, Observations regarding compactness in the $\overline {\partial }$-Neumann problem, Complex Var. Elliptic Equ. 54 (2009), no. 3-4, 173–186. MR 2513533, DOI 10.1080/17476930902759478
- Timothy G. Clos, Mehmet Çelik, and Sönmez Şahutoğlu, Compactness of Hankel operators with symbols continuous on the closure of pseudoconvex domains, Integral Equations Operator Theory 90 (2018), no. 6, Paper No. 71, 14. MR 3877477, DOI 10.1007/s00020-018-2497-8
- Željko Čučković and Sönmez Şahutoğlu, Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains, J. Funct. Anal. 256 (2009), no. 11, 3730–3742. MR 2514058, DOI 10.1016/j.jfa.2009.02.018
- John P. D’Angelo and Joseph J. Kohn, Subelliptic estimates and finite type, Several complex variables (Berkeley, CA, 1995–1996) Math. Sci. Res. Inst. Publ., vol. 37, Cambridge Univ. Press, Cambridge, 1999, pp. 199–232. MR 1748604
- M. Derridj, Regularité pour $\bar \partial$ dans quelques domaines faiblement pseudo-convexes, J. Differential Geometry 13 (1978), no. 4, 559–576 (1979) (French). MR 570218
- Klas Diederich and Peter Pflug, Necessary conditions for hypoellipticity of the $\bar \partial$-problem, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 151–154. MR 627755
- Siqi Fu and Emil J. Straube, Compactness of the $\overline \partial$-Neumann problem on convex domains, J. Funct. Anal. 159 (1998), no. 2, 629–641. MR 1659575, DOI 10.1006/jfan.1998.3317
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
- Kenneth D. Koenig, Maximal hypoellipticity for the $\overline \partial$-Neumann problem, Adv. Math. 282 (2015), 128–219. MR 3374525, DOI 10.1016/j.aim.2015.06.013
- J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492. MR 181815, DOI 10.1002/cpa.3160180305
- Jeffery D. McNeal, $L^2$ estimates on twisted Cauchy-Riemann complexes, 150 years of mathematics at Washington University in St. Louis, Contemp. Math., vol. 395, Amer. Math. Soc., Providence, RI, 2006, pp. 83–103. MR 2206894, DOI 10.1090/conm/395/07419
- Shigeyuki Morita, Geometry of differential forms, Translations of Mathematical Monographs, vol. 201, American Mathematical Society, Providence, RI, 2001. Translated from the two-volume Japanese original (1997, 1998) by Teruko Nagase and Katsumi Nomizu; Iwanami Series in Modern Mathematics. MR 1851352, DOI 10.1090/mmono/201
- Emil J. Straube, Lectures on the $\scr L^2$-Sobolev theory of the $\overline {\partial }$-Neumann problem, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2010. MR 2603659, DOI 10.4171/076
Additional Information
- Mehmet Çeli̇k
- Affiliation: Department of Mathematics, Texas A&M University-Commerce, Commerce, Texas 75429
- MR Author ID: 869210
- Email: mehmet.celik@tamuc.edu
- Sönmez Şahutoğlu
- Affiliation: Department of Mathematics and Statistics, Toledo, University of Toledo, Ohio 43606
- ORCID: 0000-0003-0490-0113
- Email: sonmez.sahutoglu@utoledo.edu
- Emil J. Straube
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- MR Author ID: 168030
- Email: straube@math.tamu.edu
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 751-764
- MSC (2010): Primary 32W05; Secondary 47B35
- DOI: https://doi.org/10.1090/proc/14729
- MathSciNet review: 4052212