Stability of the Monomial Basis Kernel of Reinhardt domains

TL;DR

This paper studies the stability and continuity of the p-Monomial Basis Kernel on Reinhardt domains, analyzing how it varies with p and domain modifications, with explicit computations for special cases.

Contribution

It establishes the continuous dependence of the p-MBK on p, proves a Ramadanov-type theorem, and explicitly computes index sets for certain monomial polyhedra.

Findings

p-MBK depends continuously on p in [1,∞)

A Ramadanov-type theorem holds for increasing sequences of domains

Explicit index sets and threshold exponents are computed for special classes

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.