Criteria for Bochner's extension problem

Details Criteria for Bochner's extension problem Criteria for Bochner's extension problem

2008-02-05

arxiv.org/abs/0802.0565v1

Asymptotic Analysis, 66 (2010), 125-138.

Mathematics

Analysis of PDEs

12 pages

Scientific paper

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.

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