Criteria for Bochner's extension problem

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–category. The solution of the L–extension problem by Bochner [1] giving the relation between the order of the operator, the dimension, and index p, for which the L–extension property holds, can be viewed as a subcritical case of the general L–extension problem. In general, this property fails in some critical and in all supercritical cases. In this paper, the L–extension problem is investigated for operators of all orders and for all 1 ≤ p ≤ ∞. Necessary and sufficient conditions on the subset of L are given for which the L–extension property still holds, in the critical and supercritical cases.