Bochner延拓问题的判定
Criteria for Bochner's extension problem
摘要：
给出了弱扩张问题可解的一个充要条件。这个准则也被应用于给出经典Bochner扩张问题在$L^p$ -范畴中可解性的一个准则。解决方案的$L^p$ -扩展问题的Bochner给之间的关系的秩序的运营商，尺寸，和指数$p$，其中$L^p$ -扩展属性举行，可以被视为一个亚临界情况下的一般$L^p$ -扩展问题。一般来说，这种性质在某些临界和所有超临界情况下都失效。本文研究了所有阶算子和所有$1\leq p\leq\infty$的$L^p$ -扩张问题.给出了$L^p$的子集在临界和超临界情况下仍具有$L^p$ -扩张性质的充要条件.
英文摘要：
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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