Linear isometric invariants of bounded domains

TL;DR

This paper establishes conditions under which linear isometries between spaces of $L^p$ holomorphic functions imply the domains are holomorphically equivalent, introducing new domain invariants for such function spaces.

Contribution

It introduces $A^p$-completeness and boundary blow down type conditions, and proves that under these, linear isometries imply domain equivalence for certain $p$.

Findings

Linear isometries imply domain equivalence under new conditions.

Introduces $A^p$-completeness and boundary blow down type as invariants.

Results hold for $p>0$, $p eq$ even integers.

Abstract

We introduce two new conditions for bounded domains, namely $A^p$-completeness and boundary blow down type, and show that, for two bounded domains $D_1$ and $D_2$ that are $A^p$-complete and not of boundary blow down type, if there exists a linear isometry from $A^p(D_1)$ to $A^{p}(D_2)$ for some real number $p>0$ with $p\neq $ even integers, then $D_1$ and $D_2$ must be holomorphically equivalent, where for a domain $D$, $A^p(D)$ denotes the space of $L^p$ holomorphic functions on $D$.