Criteria for Bochner's extension problem

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 [J. Math. Pures Appl. 35 (1956), 193–202] 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≤p≤∞. 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.