Affiliations: Department of Mathematics, Imperial College London, London, UK. E-mail: m.ruzhansky@imperial.ac.uk | Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Japan. E-mail: sugimoto@math.nagoya-u.ac.jp
Note: [] Corresponding author: Michael Ruzhansky, Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK. E-mail: m.ruzhansky@imperial.ac.uk
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 Lp-category. The solution of the Lp-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 Lp-extension property holds, can be viewed as a subcritical case of the general Lp-extension problem. In general, this property fails in some critical and in all supercritical cases. In this paper, the Lp-extension problem is investigated for operators of all orders and for all 1≤p≤∞. Necessary and sufficient conditions on the subset of Lp are given for which the Lp-extension property still holds, in the critical and supercritical cases.
