On Hardy spaces on worm domains

• Autorzy: Alessandro Monguzzi
• Języki publikacji: EN

Abstrakty

EN

In this review article we present the problem of studying Hardy spaces and the related Szeg˝o projection on worm domains. We review the importance of the Diederich–Fornæss worm domain as a smooth bounded pseudoconvex domain whose Bergman projection does not preserve Sobolev spaces of sufficiently high order and we highlight which difficulties arise in studying the same problem for the Szeg˝o projection. Finally, we announce and discuss the results we have obtained so far in the setting of non-smooth worm domains.

Słowa kluczowe

EN

Hardy spaces; Szego projection; Worm domains

• Wydawca: De Gruyter Open
• Czasopismo: Concrete Operators
• Rocznik: 2016
• Tom: 3
• Numer: 1
• Strony: 29-42

Daty

otrzymano

2015-11-09

zaakceptowano

2016-01-27

online

2016-04-13

Twórcy

autor

Alessandro Monguzzi

• Dipartimento di Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133, Milano, Italy

Bibliografia

• [1] Barrett, D. E., Behavior of the Bergman projection on the Diederich-Fornæss worm, Acta Math., 168, 1992, 1-2, 1–10
• [2] Barrett, D. E. and Ehsani, D. and Peloso, M. M., Regularity of projection operators attached to worm domains, ArXiv e-prints, 2014, aug, http://adsabs.harvard.edu/abs/2014arXiv1408.0082B
• [3] Barrett, D. and Lee, L., On the Szeg˝o metric, J. Geom. Anal., 24, 2014, 1, 104–117
• [4] Barrett, D. E. and S¸ ahutog˘ lu, S., Irregularity of the Bergman projection on worm domains in Cn, Michigan Math. J., 61, 2012, 1, 187–198
• [5] Bell, S. R., Biholomorphic mappings and the N@ -problem, Ann. of Math. (2), 114, 1981, 1, 103–113
• [6] Bell, S. R., The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992, x+149
• [7] Bell, S. and Ligocka, E., A simplification and extension of Fefferman’s theorem on biholomorphic mappings, Invent. Math., 57, 1980, 3, 283–289
• [8] Boas, H. P., Regularity of the Szeg˝o projection in weakly pseudoconvex domains, Indiana Univ. Math. J., 34, 1985, 1, 217–223
• [9] Boas, H. P., The Szeg˝o projection: Sobolev estimates in regular domains, Trans. Amer. Math. Soc., 300, 1987, 1, 109–132
• [10] Boas, H. P. and Chen, S.-C. and Straube, E. J., Exact regularity of the Bergman and Szeg˝o projections on domains with partially transverse symmetries, Manuscripta Math., 62, 1988, 4, 467–475
• [11] Boas, H. P. and Straube, E. J., Complete Hartogs domains in C2 have regular Bergman and Szeg˝o projections, Math. Z., 201, 1989, 3, 441–454
• [12] Boas, H. P. and Straube, E. J., Equivalence of regularity for the Bergman projection and the @-Neumann operator, Manuscripta Math., 67, 1990, 1, 25–33
• [13] Boas, H. P. and Straube, E. J., Sobolev estimates for the complex Green operator on a class of weakly pseudoconvex boundaries, Comm. Partial Differential Equations, 16, 1991, 10, 1573–1582
• [14] Butzer, P. L. and Jansche, S., A direct approach to the Mellin transform, J. Fourier Anal. Appl., 3, 1997, 4, 325–376
• [15] Butzer, P. L. and Jansche, S., A self-contained approach to Mellin transform analysis for square integrable functions; applications, Integral Transform. Spec. Funct., 8, 1999, 3-4, 175–198
• [16] Chen, Bo-Yong and Fu, Siqi, Comparison of the Bergman and Szegö kernels, Adv. Math., 228, 2011, 4, 2366–2384
• [17] Chen, So-Chin and Shaw, Mei-Chi, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001, xii+380
• [18] Christ, M., Global C1 irregularity of the @-Neumann problem for worm domains, J. Amer. Math. Soc., 9, 1996, 4, 1171–1185
• [19] Christ, M., The Szeg˝o projection need not preserve global analyticity, Ann. of Math. (2), 143, 1996, 2, 301–330
• [20] Diederich, K. and Fornaess, J. E., Pseudoconvex domains: an example with nontrivial Nebenhülle, Math. Ann., 225, 1977, 3, 275–292
• [21] Fefferman, C., The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math., 26, 1974, 1–65
• [22] Folland, G. B. and Kohn, J. J., The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, No. 75, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972, viii+146
• [23] Grafakos, L., Classical Fourier analysis, Graduate Texts in Mathematics, 249, Second edition, Springer, New York, 2008, xvi+489
• [24] Harrington, P. S. and Peloso, M. M. and Raich, A. S., Regularity equivalence of the Szegö projection and the complex Green operator, Proc. Amer. Math. Soc., 143, 2015, 1, 353–367
• [25] Hartogs, F., Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann., 62, 1906, 1, 1–88
• [26] Kiselman, C. O., A study of the Bergman projection in certain Hartogs domains, Several complex variables and complex geometry, Part 3 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1991, 52, 219–231
• [27] Kohn, J. J., Harmonic integrals on strongly pseudo-convex manifolds. I, Ann. of Math. (2), 78, 1963, 112–148
• [28] Kohn, J. J., Harmonic integrals on strongly pseudo-convex manifolds. II, Ann. of Math. (2), 79, 1964, 450–472
• [29] Krantz, S. G., Function theory of several complex variables, Reprint of the 1992 edition, AMS Chelsea Publishing, Providence, RI, 2001, xvi+564
• [30] Krantz, S. G. and Peloso, M. M., New results on the Bergman kernel of the worm domain in complex space, Electron. Res. Announc. Math. Sci., 14, 2007, 35–41 (electronic)
• [31] Krantz, S. G. and Peloso, M. M., Analysis and geometry on worm domains, J. Geom. Anal., 18, 2008, 2, 478–510
• [32] Krantz, S. G. and Peloso, M. M., The Bergman kernel and projection on non-smooth worm domains, Houston J. Math., 34, 2008, 3, 873–950
• [33] Krantz, S. G. and Peloso, M. M. and Stoppato, C., Bergman kernel and projection on the unbounded worm domain. To appear. doi:10.2422/2036-2145.201503_012
• [34] Lanzani, L. and Stein, E. M., Szegö and Bergman projections on non-smooth planar domains, J. Geom. Anal., 14, 2004, 1, 63–86
• [35] Lanzani, L. and Stein, E. M., Cauchy-type integrals in several complex variables, Bull. Math. Sci., 3, 2013, 2, 241–285
• [36] Lanzani, L. and Stein, E. M., Hardy spaces of holomorphic functions for domains in Cn with minimal smoothness, ArXiv e-prints, 2015, jun, http://adsabs.harvard.edu/abs/2015arXiv150603748L
• [37] Lanzani, L. and Stein, E. M., The Cauchy-Szeg˝o projection for domains in Cn with minimal smoothness, ArXiv e-prints, 2015, jun, http://adsabs.harvard.edu/abs/2015arXiv150603965L
• [38] Ligocka, E., The Sobolev spaces of harmonic functions, Studia Math., 84, 1986, 1, 79–87
• [39] McNeal, J. D. and Stein, E. M., The Szeg˝o projection on convex domains, Math. Z., 224, 1997, 4, 519–553
• [40] Monguzzi, A., A Comparison Between the Bergman and Szegö Kernels of the Non-smooth Worm Domain D'β, Complex Analysis and Operator Theory, 2015, 1–27
• [41] Monguzzi, A., Hardy spaces and the Szeg˝o projection of the non-smooth worm domain D'β, J. Math. Anal. Appl., 436, 2016, 1, 439–466
• [42] Monguzzi, A. and Peloso, M. M, Sharp estimates for the Szeg˝o projection of Hardy spaces on the distinguished boundary of model worm domains. In preparation
• [43] Nagel, A. and Rosay, J.-P. and Stein, E. M. and Wainger, S., Estimates for the Bergman and Szeg˝o kernels in C2, Ann. of Math. (2), 129, 1989, 1, 113–149
• [44] Phong, D. H. and Stein, E. M., Estimates for the Bergman and Szegö projections on strongly pseudo-convex domains, Duke Math. J., 44, 1977, 3, 695–704
• [45] Rooney, P. G., A technique for studying the boundedness and extendability of certain types of operators, Canad. J. Math., 25, 1973, 1090–1102
• [46] Rooney, P. G., Multipliers for the Mellin transformation, Canad. Math. Bull., 25, 1982, 3, 257–262
• [47] Rooney, P. G., A survey of Mellin multipliers, Fractional calculus (Glasgow, 1984), Res. Notes in Math., 138, Pitman, Boston, MA, 1985, 176–187
• [48] Rudin, W., Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969, vii+188
• [49] Stein, E.M., Boundary values of holomorphic functions, Bull. Amer. Math. Soc., 76, 1970, 1292–1296
• [50] Straube, E. J., Exact regularity of Bergman, Szeg˝o and Sobolev space projections in nonpseudoconvex domains, Math. Z., 192, 1986, 1, 117–128

Identyfikatory

DOI

10.1515/conop-2016-0005