Abstract
We set a framework for the study of Hardy spaces inherited by complements of analytic hypersurfaces in domains with a prior Hardy space structure. The inherited structure is a filtration, various aspects of which are studied in specific settings. For punctured planar domains, we prove a generalization of a famous rigidity lemma of Kerzman and Stein. A stabilization phenomenon is observed for egg domains. Finally, using proper holomorphic maps, we derive a filtration of Hardy spaces for certain power-generalized Hartogs triangles, although these domains fall outside the scope of the original framework.
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References
Aronszajn, N.: Theory of reproducing kernels. Trans. AMS 68(3), 337–404 (1950)
Baratchart, L., Fischer, Y., Leblond, J.: Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation. Complex Var. Elliptic Equ. 59(4), 504–538 (2014)
Barrett, D.E., Lanzani, L.: The spectrum of the Leray transform for convex Reinhardt domains in \(\mathbb{C}^2\). J. Funct. Anal. 257(9), 2780–2819 (2009)
Bell, S.R.: The Cauchy transform, potential theory and conformal mapping. CRC Press, Boca Raton (1992)
Chakrabarti, D., Shaw, M.-C.: Sobolev regularity of the \(\overline{\partial }\)-equation on the Hartogs triangle. Math. Ann. 356(1), 241–258 (2013)
Chakrabarti, D., Zeytuncu, Y.: \({L}^p\) mapping properties of the Bergman projection on the Hartogs triangle. Proc. Am. Math. Soc. 144(4), 1643–1653 (2016)
Chakrabarti, D., Edholm, L., McNeal, J.: Duality and approximation of Bergman spaces. Adv. Math. 341, 616–656 (2019)
Chaumat, J., Chollet, A.-M.: Régularité höldérienne de l’opérateur \(\overline{\partial }\) sur le triangle de Hartogs. Ann. Inst. Fourier (Grenoble) 41(4), 867–882 (1991)
L. Chen and J. D. McNeal: A solution operator for \(\overline{\partial }\) on the Hartogs triangle and \({L}^p\) estimates. Math. Ann. (to appear)
Chirka, E.M.: Complex analytic sets, vol. 46. Springer Science & Business Media, Berlin (2012)
Dales, H.G.: The ring of holomorphic functions on a Stein compact set as a unique factorization domain. Proc. Am. Math. Soc. 44(1), 88–92 (1974)
Duren, P.L.: Theory of \(H^p\) Spaces. Academic press, Cambridge (1970)
Edholm, L.: Bergman theory of certain generalized Hartogs triangles. Pac. J. Math. 284(2), 327–342 (2016)
Edholm, L., McNeal, J.: The Bergman projection on fat Hartogs triangles: \({L}^p\) boundedness. Proc. Am. Math. Soc. 144(5), 2185–2196 (2016)
Edholm, L., McNeal, J.: Bergman subspaces and subkernels: Degenerate \({L}^p\) mapping and zeroes. J. Geom. Anal. 27(4), 2658–2683 (2017)
Hansson, T.: On Hardy spaces in complex ellipsoids. Ann. Inst. Fourier (Grenoble) 49(5), 1477–1501 (1999)
Huo, Z., Wick, B.D.: Weighted estimates for the Bergman projection on the Hartogs triangle. J. Funct. Anal. 279(9), 108727 (2020)
Kerzman, N., Stein, E.M.: The Cauchy kernel, the Szegő kernel, and the Riemann mapping function. Math. Ann. 236(1), 85–93 (1978)
Lanzani, L.: Szegő projection vs. potential theory for non-smooth planar domains. Indiana U. Math. J. 48(2), 537–555 (1998)
Laurent-Thiébaut, C., Shaw, M.-C.: Solving \(\overline{\partial }\) with prescribed support on Hartogs triangles in \(\mathbb{C}^2\) and \(\mathbb{CP}^2\). Trans. Am. Math. Soc. 371(9), 6531–6546 (2019)
Ma, L., Michel, J.: \(\cal{C}^{k,\alpha }\)-estimates for the \(\overline{\partial }\)-equation on the Hartogs triangle. Math. Ann. 294(1), 661–675 (1992)
A. Monguzzi: Holomorphic function spaces on the Hartogs triangle. Math. Nachr. (to appear)
A. Nagel and M. Pramanik. Bergman spaces under maps of monomial type. arXiv preprint arXiv:2002.02915, 2020
Ohsawa, T.: Analysis of several complex variables, 211th edn. American Mathematical Society, Providence (2002)
Poletsky, E.A., Stessin, M.I.: Hardy and Bergman spaces on hyperconvex domains and their composition operators. Indiana Univ. Math. J. 57(5), 2153–2201 (2008)
Pommerenke, C.: Boundary behaviour of conformal maps, vol. 299. Springer Science & Business Media, Berlin (2013)
Rademacher, H., Toeplitz, O.: The enjoyment of mathematics. Princeton U. press, Princeton (1957)
Sahin, S.: Poletsky-Stessin Hardy spaces on Complex Ellipsoids in \(\mathbb{C}^n\). Complex Anal. Oper. Theory 10(2), 295–309 (2016)
Shaw, M.-C.: The Hartogs triangle in complex analysis. Contemp. Math. 646, (2015)
Stein, E.M.: Boundary behavior of holomorphic functions of several complex variables.(MN-11). Princeton University Press, Princeton (2015)
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L. Lanzani and L. Vivas were supported in part by the National Science Foundation (DMS-1901978 and DMS-1800777). P. Gupta was supported in part by a UGC CAS-II grant (Grant No. F.510/25/CAS-II/ 2018(SAP-I)). Part of this work took place at the Banff International Research station during a workshop of the Women in Analysis (WoAn), an AWM Research Network. We are grateful to the Institute for its kind hospitality and to the Association of Women in Mathematics for its generous support. We also wish to thank Mei-Chi Shaw for providing the inspiration for this work, Björn Gustafsson for offering helpful feedback on an earlier version of this manuscript, and the anonymous referee for their useful comments.
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Gallagher, AK., Gupta, P., Lanzani, L. et al. Hardy spaces for a class of singular domains. Math. Z. 299, 2171–2197 (2021). https://doi.org/10.1007/s00209-021-02755-1
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DOI: https://doi.org/10.1007/s00209-021-02755-1