Vector extensions of operators inLpspaces

Abstract

We study 5-valued extensions of operators of weak or strong type (/>, q), where B is a /?-Banach space of a certain type, and present several applications. An old well known result of Marcinkiewicz and Zygmund states that every bounded linear operator T from Lp to Lq has a bounded extension T ® \B from L§ to L%, where B is a Hubert space. The analogous question for certain types of Banach spaces was also considered in [9] and [5] and, for weak type operators, in [13] and [14]. Here we obtain a general result on the extension of bounded linear operators of (weak or strong) type (p,q) to B-valued functions, where B belongs to C. Herz's class of r-spaces (see [5]). Applications are given to pointwise convergence of vector valued functions and to weighted norm inequalities for a wide class of operators. Finally, we prove a mixed norm estimate for translation invariant operators which is a weak type analogue of the main result in [6].