Criteria for Bochner's extension problem

TL;DR

This paper establishes a comprehensive criterion for solving Bochner's extension problem across various operator orders and p-values, clarifying when the extension property holds or fails in critical and supercritical cases.

Contribution

It provides necessary and sufficient conditions for the $L^p$-extension property in all cases, extending Bochner's classical results to a broader setting.

Findings

Criteria for $L^p$-extension property in critical and supercritical cases

Relation between operator order, dimension, and $p$ for extension solvability

Identification of cases where extension property fails

Abstract

A necessary and sufficient condition for the resolution of the weak extension problem is given. This criterion is applied to also give a criterion for the solvability of the classical Bochner's extension problem in the $L^p$-category. The solution of the $L^p$-extension problem by Bochner giving the relation between the order of the operator, the dimension, and index $p$, for which the $L^p$-extension property holds, can be viewed as a subcritical case of the general $L^p$-extension problem. In general, this property fails in some critical and in all supercritical cases. In this paper, the $L^p$-extension problem is investigated for operators of all orders and for all $1\leq p\leq\infty$. Necessary and sufficient conditions on the subset of $L^p$ are given for which the $L^p$-extension property still holds, in the critical and supercritical cases.