Projections onto $L^p$-Bergman spaces of Reinhardt Domains
Debraj Chakrabarti, Luke D. Edholm
TL;DR
The paper develops a Banach-space based framework for projecting onto $A^p(\Omega,\lambda)$ on bounded Reinhardt domains by using Laurent monomials as a natural basis, leading to the Monomial Basis Projection (MBP) and its kernel (MBK). It proves that the MBP is a bounded (and often absolutely bounded) projection in $L^p$ spaces where the classical Bergman projection may fail for $p\neq 2$, and provides an integral kernel representation via $K_{p,\lambda}^{\Omega}$. A key outcome is the absolute boundedness of MBP on monomial polyhedra, together with a detailed transformation theory under monomial maps and a decomposition of the MBK into tensor products of 1D subkernels; these results underpin a duality theory identifying $A^p(\Omega)'$ with weighted Bergman spaces on appropriately transformed domains. The work also clarifies the limitations of the classical Bergman projection in $L^p$ settings (irregularity and non-surjectivity) and shows how MBP yields faithful holomorphic dual spaces, particularly on Reinhardt domains and monomial polyhedra. Overall, the MBP provides a robust, transform-friendly alternative to the Bergman projection for $L^p$-function theory on Reinhardt domains and sharpens understanding of duality in this setting.
Abstract
For $1<p<\infty$, we emulate the Bergman projection on Reinhardt domains by using a Banach-space basis of $L^p$-Bergman space. The construction gives an integral kernel generalizing the ($L^2$) Bergman kernel. The operator defined by the kernel is shown to be absolutely bounded projection on the $L^p$-Bergman space on a class of domains where the $L^p$-boundedness of the Bergman projection fails for certain $p \neq 2$. As an application, we identify the duals of these $L^p$-Bergman spaces with weighted Bergman spaces.