Abstract
Sobolev irregularity of the Bergman projection on a family of domains containing the Hartogs triangle is shown. On the Hartogs triangle itself, a sub-Bergman projection is shown to satisfy better Sobolev norm estimates than its Bergman projection.
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References
Barrett, D., Şahutoğlu, S.: Irregularity of the Bergman projection on worm domains in \({\mathbb{C}}^n\). Michigan Math. J. 61(1), 187–198 (2012)
Barrett, D.E.: Irregularity of the Bergman projection on a smooth bounded domain in \({ C}^{2}\). Ann. Math. 119(2), 431–436 (1984)
Barrett, D.E.: Regularity of the Bergman projection and local geometry of domains. Duke Math. J. 53(2), 333–343 (1986)
Barrett, D.E.: Behavior of the Bergman projection on the Diederich-Fornæss worm. Acta Math. 168(1–2), 1–10 (1992)
Bell, S., Ligocka, E.: A simplification and extension of Fefferman’s theorem on biholomorphic mappings. Invent. Math. 57, 283–289 (1980)
Bell, S.R.: Biholomorphic mappings and the \(\bar{\partial }\)-problem. Ann. Math. 114(1), 103–113 (1981)
Boas, H.P.: The Szegő projection: Sobolev estimates in regular domains. Trans. Am. Math. Soc. 300(1), 109–132 (1987)
Boas, H.P., Straube, E.J.: Sobolev estimates for the \(\overline{\partial }\)-Neumann operator on domains in \({ C}^n\) admitting a defining function that is plurisubharmonic on the boundary. Math. Z. 206(1), 81–88 (1991)
Boas, H.P., Straube, E.J.: Global regularity of the \(\overline{\partial }\)-Neumann problem: a survey of the \(L^2\)-Sobolev theory. In: Several complex variables (Berkeley, CA, 1995–1996), vol. 37 of Math. Sci. Res. Inst. Publ. Cambridge Univ. Press, Cambridge, pp. 79–111 (1999)
Chakrabarti, D., Edholm, L.D., McNeal, J.D.: Duality and approximation of Bergman spaces. Adv. Math. 341, 616–656 (2019)
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)
Charpentier, P., Dupain, Y.: Estimates for the Bergman and Szegö projections on pseudoconvex domains of finite type with locally diagonalizable Levi form. Publ. Math. 50(2), 413–446 (2006)
Chen, L.: The \(L^p\) boundedness of the Bergman projection for a class of bounded Hartogs domains. J. Math. Anal. Appl. 448(1), 598–610 (2017)
Chen, L.: Weighted Sobolev regularity of the Bergman projection on the Hartogs triangle. Pac. J. Math. 288, 307–318 (2017)
Detraz, J.: Classes de Bergman de fonctions harmoniques. Bull. Soc. Math. France 109, 259–268 (1981)
Diederich, K., Fornæss, J.E.: Pseudoconvex domains: an example with nontrivial Nebenhülle. Math. Ann. 225(3), 275–292 (1977)
Duren, P., Schuster, A.: BergmanSpaces. Mathematical Surveys and Monographs. American Mathematical Society, Providence (2004)
Edholm, L.D.: Bergman theory of certain generalized Hartogs triangles. Pac. J. Math. 284(2), 327–342 (2016)
Edholm, L.D., McNeal, J.D.: The Bergman projection on fat Hartogs triangles: \({L}^p\) boundedness. Proc. Am. Math. Soc. 144(5), 2185–2196 (2016)
Edholm, L.D., McNeal, J.D.: Bergman subspaces and subkernels: degenerate \({L}^p\) mapping and zeroes. J. Geom. Anal. 27(4), 2658–2683 (2017)
Forelli, F., Rudin, W.: Projections on spaces of holomorphic functions in balls. Ind. Univ. Math. J. 24, 593–602 (1974)
Herbig, A.-K., McNeal, J.: A smoothing property of the Bergman projection. Math. Ann. 354(2), 427–449 (2012)
Koenig, K.D.: On maximal Sobolev and Hölder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian. Am. J. Math. 124(1), 129–197 (2002)
Kohn, J.J.: Quantitative estimates for global regularity. In: Analysis and geometry in several complex variables (Katata, 1997), Trends Math. Birkhäuser Boston, Boston, MA, pp. 97–128 (1999)
McNeal, J.D.: Boundary behavior of the Bergman kernel function in \({ C}^2\). Duke Math. J. 58(2), 499–512 (1989)
McNeal, J.D.: Local geometry of decoupled pseudoconvex domains. In: Complex Analysis (Wuppertal, 1991), Aspects Math., E17. Vieweg, Braunschweig, pp. 223–230 (1991)
McNeal, J.D.: Estimates on the Bergman kernels of convex domains. Adv. Math. 109(1), 108–139 (1994)
McNeal, J.D., Stein, E.M.: Mapping properties of the Bergman projection on convex domains of finite type. Duke Math. J. 73(1), 177–199 (1994)
McNeal, J.D., Stein, E.M.: The Szegö projection on convex domains. Math. Z. 224(4), 519–553 (1997)
Nagel, A., Rosay, J.-P., Stein, E.M., Wainger, S.: Estimates for the Bergman and Szego kernels in \({ C}^2\). Ann. Math. 129(1), 113–149 (1989)
Phong, D.H., Stein, E.M.: Estimates for the Bergman and Szegö projections on strongly pseudo-convex domains. Duke Math. J. 44(3), 695–704 (1977)
Rudin, W.: Function Theory in the Unit Ball in \({\mathbb{C}}^n\). Grundlehren der Mathematischen Wissenschaften, vol. 241. Springer, New York (1980)
Zaharjuta, V., Judovic, V.: The general form of a linear functional on \({H}_p^{\prime }\). Uspekhi Mat. Nauk. 19(2), 139–142 (1964)
Zeytuncu, Y.: \({L}^p\) regularity of weighted Bergman projections. Trans. Am. Math. Soc. 365(6), 2959–2976 (2013)
Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics, vol. 226. Springer, New York (2005)
Acknowledgements
The authors thank the Erwin Schrödinger Institute, Vienna for providing them a fruitful environment for collaboration during a December 2018 workshop. The first author also thanks Texas A&M at Qatar for hosting the stimulating workshop Analysis and Geometry in Several Complex Variables III in January 2019. The authors are also grateful to the two anonymous referees, whose comments improved the mathematical and expository content of the paper.
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Edholm, L.D., McNeal, J.D. Sobolev Mapping of Some Holomorphic Projections. J Geom Anal 30, 1293–1311 (2020). https://doi.org/10.1007/s12220-019-00345-6
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DOI: https://doi.org/10.1007/s12220-019-00345-6