On Hardy spaces on worm domains

Alessandro Monguzzi 1
  • 1 Dipartimento di Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133, Milano, Italy

Abstract

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.

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  • [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

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