$L^p$ -regularity of the Bergman projection on quotient domains

We obtain sharp ranges of $L^p$ -boundedness for domains in a wide class of Reinhardt domains representable as sublevel sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating $L^p$ -boundedness on a domain and its quotient by a finite group. The range of p for which the Bergman projection is $L^p$ -bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases.

[1]  N. Sibony.  Prolongement des fonctions holomorphes bornées et métrique de Carathéodory , 1975 .

[2]  H. Cartan.  Quotient d'un Espace Analytique par un Groupe d'Automorphismes , 1957 .

[3]  S. Bell.  The Bergman kernel function and proper holomorphic mappings , 1982 .

[4]  G. Misra,et al.  REPRODUCING KERNEL FOR A CLASS OF WEIGHTED BERGMAN SPACES ON THE SYMMETRIZED POLYDISC , 2011, 1106.3868.

[5]  Jong-Do Park.  The explicit forms and zeros of the Bergman kernel for 3-dimensional Hartogs triangles , 2017 .

[6]  L. Collatz.  Functional analysis and numerical mathematics , 1968 .

[7]  B. M. Fulk.  MATH , 1992 .

[8]  J. McNeal,et al.  The Bergman projection on fat Hartogs triangles: L^p boundedness , 2015, 1502.07302.

[9]  $L^p$ estimates for the Bergman projection on some Reinhardt domains , 2017, 1710.02449.

[10]  Yunus E. Zeytuncu,et al.  mapping properties of the Bergman projection on the Hartogs triangle , 2014, 1410.1105.

[11]  Liwei Chen.  The $L^p$ boundedness of the Bergman projection for a class of bounded Hartogs domains , 2013, 1304.7898.

[12]  Bergman projection on the symmetrized bidisk , 2020, 2004.02785.

[13]  S. Krantz,et al.  Lp regularity of the Bergman projection on domains covered by the polydisc , 2020 .

[14]  D. Barrett.  Irregularity of the Bergman Projection on a Smooth Bounded Domain in C 2 , 1984 .

[15]  H. O. Foulkes.  Abstract Algebra , 1967, Nature.

[16]  Meera Mainkar,et al.  Bergman kernels of elementary Reinhardt domains , 2019, Pacific Journal of Mathematics.

[17]  S. Bergmann.  Über die Entwicklung der harmonischen Funktionen der Ebene und des Raumes nach Orthogonalfunktionen , 1922 .

[18]  H. Hedenmalm.  The dual of a bergman space on simply connected domains , 2002 .

[19]  W. Rudin,et al.  Projections on Spaces of Holomorphic Functions in Balls , 1974 .

[20]  P. Galanopoulos.  Theory of Bergman spaces (I) , 2008, 0801.1512.

[21]  Saulo Alves de Araujo,et al.  Identification of novel keloid biomarkers through Profiling of Tissue Biopsies versus Cell Cultures in Keloid Margin specimens Compared to adjacent Normal Skin , 2010, Eplasty.

[22]  Kehe Zhu,et al.  The Bergman Spaces , 2000 .

[23]  M. Ziegler.  Volume 152 of Graduate Texts in Mathematics , 1995 .

[24]  K. Diederich,et al.  Boundary regularity of proper holomorphic mappings , 1982 .

[25]  E. Stein,et al.  Estimates for the Bergman and Szegö projections on strongly pseudo-convex domains , 1977 .

[26]  S. Krantz,et al.  $L^p$ regularity of the Bergman Projection on domains covered by the polydisk , 2019, 1903.10497.

[27]  S. Bell.  Proper holomorphic mappings and the Bergman projection , 1981 .

[28]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

[29]  Bergman theory of certain generalized Hartogs triangles , 2015, 1504.07914.

[30]  W. Massey.  A basic course in algebraic topology , 1991 .

[31]  E. Stein,et al.  Mapping properties of the Bergman projection on convex domains of finite type , 1994 .

[32]  Shôshichi Kobayashi.  Geometry of bounded domains , 1959 .

[33]  S. Krantz,et al.  Analysis and Geometry on Worm Domains , 2007, 0710.4087.

[34]  J. McNeal,et al.  Duality and approximation of Bergman spaces , 2018, Advances in Mathematics.

[35]  H. Behnke.  Zur Theorie der Singularitäten der Funktionen mehrerer komplexen Veränderlichen , 1933 .

[36]  P. Steerenberg,et al.  Targeting pathophysiological rhythms: prednisone chronotherapy shows sustained efficacy in rheumatoid arthritis. , 2010, Annals of the rheumatic diseases.

[37]  David Békollé.  Inégalités à poids pour le projecteur de Bergman dans la boule unité de $C^{n}$ , 1982 .

[38]  J. McNeal,et al.  Bergman Subspaces and Subkernels: Degenerate $$L^p$$Lp Mapping and Zeroes , 2016, 1605.06223.

[39]  S. Bell,et al.  Boundary regularity of proper holomorphic mappings , 1982 .

[40]  Shuo Zhang.  L p boundedness for the Bergman projections over n-dimensional generalized Hartogs triangles , 2020 .

[41]  Einzelwerken Muster,et al.  Invent , 2021, Encyclopedic Dictionary of Archaeology.

[42]  D. Catlin.  Boundary behavior of holomorphic functions on pseudoconvex domains , 1980 .

[43]  N. Sibony,et al.  Spectre de A(\̄gW) pour des domaines bornés faiblement pseudoconvexes réguliers , 1980 .

[44]  W. Zwonek.  Completeness, Reinhardt domains and the method of complex geodesics in the theory of invariant functions , 2000 .

[45]  D. Chakrabarti.  On an observation of Sibony , 2018, Proceedings of the American Mathematical Society.

[46]  W. Zwonek.  On hyperbolicity of pseudoconvex Reinhardt domains , 1999 .