L^p REGULARITY OF THE WEIGHTED BERGMAN PROJECTION ON THE FOCK–BARGMANN–HARTOGS DOMAIN. Issue 2 (8th October 2020) - HE, LE; TANG, YANYAN; TU, ZHENHAN
Abstract : The Fock–Bargmann–Hartogs domain D_n, m( [STIX]x1D707):={(z, w)C^n× C^m:w^2<e^-[STIX]x1D707z^2}, where [STIX]x1D707>0, is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. We compute the weighted Bergman kernel of D_n, m( [STIX]x1D707) with respect to the weight (-[STIX]x1D70C)^[STIX]x1D6FC, where [STIX]x1D70C(z, w):=w^2-e^-[STIX]x1D707z^2 and [STIX]x1D6FC>-1 . Then, for p[1, ∞ ), we show that the corresponding weighted Bergman projection P_D_n, m( [STIX]x1D707), (-[STIX]x1D70C)^[STIX]x1D6FC is unbounded on L^p(D_n, m( [STIX]x1D707), (-[STIX]x1D70C)^[STIX]x1D6FC), except for the trivial case p=2 . This gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is L^p irregular when p[1, ∞ ){2}, in contrast to the well-known positive L^p regularity result on a bounded strongly pseudoconvex domain.