  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.
