Research Projects

Scientific Objective I: We denote the Bergman space of D by A²(D). The orthogonal projection B: L²(D) to A²(D) is said to be the Bergman projection. We say B is L^p regular for some real number p in (1, infinity), if B: L^p(D) to L^p(D) is bounded. Let D₁ and D₂ be two bounded domains in C^n. Suppose that f from D₁ to D₂ is a proper holomorphic mapping which is factored by G where G is a subgroup of the automorphism group of D₁. Here one can show that D₂ is biholomorphic to D₁/G, so we call D₂ by quotient domain. We note that B_ D₁ is L^p regular for p in (1, infinity). We want to formularize a range for p for which B_ D₂ is L^p regular using the invariant theory of the group G (here D₂ is a quotient domain of the form D₁/G). The motivation for such a possibility arises from a very recent paper where the PI with her collaborators has noted such a phenomena for the Szeg\"o projections on certain quotient domains. Methodology: PI has planned to start with a bounded symmetric domain D₁. She wants to classify Bekolle-Bonami weight class for D₁ and for certain Siegel domains. A Lanzani-Stein type characterization is then required associating Bekolle-Bonami weight classes of such Siegel domains and boundedness of corresponding weighted Bergman projections. Next she plans to use it and the transformation rule for Bergman projections under the proper holomorphic map, to conclude about the problem stated above. This method will not provide a sharp range for p. To surpass the shortcomings, her plan is to use Forelli-Rudin type theorem. Scientific Objective II: If D₁ is a bounded symmetric domain and D₂ is as defined above, a notion of the Hardy space H²(D₂) can be given on D₂. H²(D₂) is a closed subspace of a L² space with respect to some measure supported on the Shilov boundary dD₂ of D₂. Let S:L²(dD₂) to H²(D₂) be the orthogonal projection. We aim to establish a formula for the range of p for which S is L^p regular using invariant theoretic information of the group G. Altogether this study will help us to develop a better understanding regarding the geometry of these quotient domains.