Stability of the Monomial Basis Kernel of Reinhardt domains

Abstract

On a pseudoconvex Reinhardt domain $\Omega\subset\mathbb{C}^n$ the $p$-Bergman space $A^p(\Omega)$ admits a canonical basis of monomials indexed by a subset $S_p(\Omega)\subset\mathbb{Z}^n$. The corresponding $p$-Monomial Basis Kernel (or $p$-MBK) is defined by a series involving these monomials and their norms. This article records stability properties of the $p$-MBK and of the index set $S_p(\Omega)$ with respect to the parameter $p$. First, under mild hypotheses, the $p$-MBK depends continuously on $p\in[1,\infty)$, and a Ramadanov-type theorem holds for $p$-MBK for an increasing sequence of pseudoconvex Reinhardt domains. Second, for certain special classes of monomial polyhedra, we explicitly compute the index set and the associated Threshold exponents. Finally, these explicit models are used to illustrate structural properties of the index sets under finite unions, intersections, and products.