Berezin regularity of domains in ℂⁿ and the essential norms of Toeplitz operators

Čučković, Željko; Şahutoğlu, Sönmez

doi:10.1090/tran/8201

Berezin regularity of domains in $\mathbb {C}^n$ and the essential norms of Toeplitz operators
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by Željko Čučković and Sönmez Şahutoğlu PDF

Trans. Amer. Math. Soc. 374 (2021), 2521-2540 Request permission

Abstract:

For the open unit disc $\mathbb {D}$ in the complex plane, it is well known that if $\phi \in C(\overline {\mathbb {D}})$ then its Berezin transform $\widetilde {\phi }$ also belongs to $C(\overline {\mathbb {D}})$. We say that $\mathbb {D}$ is BC-regular. In this paper we study BC-regularity of some pseudoconvex domains in $\mathbb {C}^n$ and show that the boundary geometry plays an important role. We also establish a relationship between the essential norm of an operator in a natural Toeplitz subalgebra and its Berezin transform.

References

J. Arazy and M. Engliš, Iterates and the boundary behavior of the Berezin transform, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 4, 1101–1133 (English, with English and French summaries). MR 1849217
Sheldon Axler, John B. Conway, and Gerard McDonald, Toeplitz operators on Bergman spaces, Canadian J. Math. 34 (1982), no. 2, 466–483. MR 658979, DOI 10.4153/CJM-1982-031-1
Sheldon Axler and Dechao Zheng, Compact operators via the Berezin transform, Indiana Univ. Math. J. 47 (1998), no. 2, 387–400. MR 1647896, DOI 10.1512/iumj.1998.47.1407
D. Békollé, C. A. Berger, L. A. Coburn, and K. H. Zhu, BMO in the Bergman metric on bounded symmetric domains, J. Funct. Anal. 93 (1990), no. 2, 310–350. MR 1073289, DOI 10.1016/0022-1236(90)90131-4
Steve Bell, Differentiability of the Bergman kernel and pseudolocal estimates, Math. Z. 192 (1986), no. 3, 467–472. MR 845219, DOI 10.1007/BF01164021
Harold P. Boas, Extension of Kerzman’s theorem on differentiability of the Bergman kernel function, Indiana Univ. Math. J. 36 (1987), no. 3, 495–499. MR 905607, DOI 10.1512/iumj.1987.36.36027
Harold P. Boas, Small sets of infinite type are benign for the $\overline \partial$-Neumann problem, Proc. Amer. Math. Soc. 103 (1988), no. 2, 569–578. MR 943086, DOI 10.1090/S0002-9939-1988-0943086-2
Harold P. Boas and Emil J. Straube, Sobolev estimates for the $\overline \partial$-Neumann operator on domains in $\textbf {C}^n$ admitting a defining function that is plurisubharmonic on the boundary, Math. Z. 206 (1991), no. 1, 81–88. MR 1086815, DOI 10.1007/BF02571327
David W. Catlin, Global regularity of the $\bar \partial$-Neumann problem, Complex analysis of several variables (Madison, Wis., 1982) Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984, pp. 39–49. MR 740870, DOI 10.1090/pspum/041/740870
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
L. A. Coburn, A Lipschitz estimate for Berezin’s operator calculus, Proc. Amer. Math. Soc. 133 (2005), no. 1, 127–131. MR 2085161, DOI 10.1090/S0002-9939-04-07476-3
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
So-Chin Chen and Mei-Chi Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR 1800297, DOI 10.1090/amsip/019
Mehmet Çelik and Sönmez Şahutoğlu, On compactness of the $\overline \partial$-Neumann problem and Hankel operators, Proc. Amer. Math. Soc. 140 (2012), no. 1, 153–159. MR 2833527, DOI 10.1090/S0002-9939-2011-11350-9
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
eljko C̆u ković and Sönmez Şahutoğlu, Axler-Zheng type theorem on a class of domains in $\Bbb {C}^n$, Integral Equations Operator Theory 77 (2013), no. 3, 397–405. MR 3116666, DOI 10.1007/s00020-013-2088-7
eljko C̆u ković and Sönmez Şahutoğlu, Erratum to: Axler-Zheng type theorem on a class of domains in $\Bbb {C}^n$ [MR3116666], Integral Equations Operator Theory 79 (2014), no. 3, 449–450. MR 3216813, DOI 10.1007/s00020-014-2138-9
Željko Čučković and Sönmez Şahutoğlu, Essential norm estimates for Hankel operators on convex domains in $\Bbb C^2$, Math. Scand. 120 (2017), no. 2, 305–316. MR 3657418, DOI 10.7146/math.scand.a-25793
Željko Čučković and Sönmez Şahutoğlu, Essential norm estimates for the $\overline \partial$-Neumann operator on convex domains and worm domains, Indiana Univ. Math. J. 67 (2018), no. 1, 267–292. MR 3776022, DOI 10.1512/iumj.2018.67.6252
Željko Čučković, Sönmez Şahutoğlu, and Yunus E. Zeytuncu, A local weighted Axler-Zheng theorem in $\Bbb C^n$, Pacific J. Math. 294 (2018), no. 1, 89–106. MR 3743367, DOI 10.2140/pjm.2018.294.89
John P. D’Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2) 115 (1982), no. 3, 615–637. MR 657241, DOI 10.2307/2007015
John P. D’Angelo, Several complex variables and the geometry of real hypersurfaces, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993. MR 1224231
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
D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications. MR 929030
Miroslav Engliš, Singular Berezin transforms, Complex Anal. Oper. Theory 1 (2007), no. 4, 533–548. MR 2358010, DOI 10.1007/s11785-007-0023-0
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
Siqi Fu and Emil J. Straube, Compactness in the $\overline \partial$-Neumann problem, Complex analysis and geometry (Columbus, OH, 1999), Ohio State Univ. Math. Res. Inst. Publ., vol. 9, de Gruyter, Berlin, 2001, pp. 141–160.
Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152. MR 179443, DOI 10.1007/BF02391775
Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. MR 1242120, DOI 10.1515/9783110870312
J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. I, Ann. of Math. (2) 78 (1963), 112–148. MR 153030, DOI 10.2307/1970506
J. J. Kohn and L. Nirenberg, A pseudo-convex domain not admitting a holomorphic support function, Math. Ann. 201 (1973), 265–268. MR 330513, DOI 10.1007/BF01428194
Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625, DOI 10.1090/chel/340
R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR 847923, DOI 10.1007/978-1-4757-1918-5
Sönmez Şahutoğlu, Localization of compactness of Hankel operators on pseudoconvex domains, Illinois J. Math. 56 (2012), no. 3, 795–804. MR 3161351
Nessim Sibony, Une classe de domaines pseudoconvexes, Duke Math. J. 55 (1987), no. 2, 299–319 (French). MR 894582, DOI 10.1215/S0012-7094-87-05516-5
Sönmez Şahutoğlu and Emil J. Straube, Analytic discs, plurisubharmonic hulls, and non-compactness of the $\overline \partial$-Neumann operator, Math. Ann. 334 (2006), no. 4, 809–820. MR 2209258, DOI 10.1007/s00208-005-0737-0
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
Daniel Suárez, The essential norm of operators in the Toeplitz algebra on $A^p(\Bbb B_n)$, Indiana Univ. Math. J. 56 (2007), no. 5, 2185–2232. MR 2360608, DOI 10.1512/iumj.2007.56.3095