Two-weighted estimations for the Hardy-Littlewood maximal function in ideal Banach spaces

Abstract

We give conditions on a couple of ideal Banach spaces with weights which are both necessary and sufficient for the Hardy-Littlewood maximal function to satisfy the two-weighted estimations of weak type, and we consider a modification of the Hardy-Littlewood maximal function. We also give some conditions on weights in order for the Hardy-Littlewood maximal function and the modification under consideration to fulfil the two-weighted estimations of strong type. 0. INTRODUCTION The general problem of two-weighted estimations of weak type in LP-spaces for the Hardy-Littlewood maximal function has been solved in [9]. In the case of Lorentz spaces Lpq a corresponding result has been obtained in [2, 7] (see also [3, 4]). In order to obtain a positive answer the AP-condition for weights is required in both cases mentioned. The problem of two-weighted estimations of strong type for the Hardy-Littlewood maximal function in LP-spaces has been considered in [12]. The solution is boundedness of the operator of the Hardy-Littlewood maximal function on the set of the characteristic functions of cubes. Simpler conditions, necessary and sufficient on a certain class of weights, have been considered by many authors (see for example [3] or [4]). Among recent papers we should mention [11], where, for a large class of weights, there are given necessary and sufficient conditions for boundedness of the weighted fractional maximal operator. In the present paper we give necessary and sufficient conditions for two-weighed estimations of weak type for the Hardy-Littlewood maximal function and a modification in ideal Banach spaces. We also give some conditions on weights assuring two-weighted estimations of strong type for the Hardy-Littlewood maximal function and a modification. The obtained results are new even in the case of Orlicz spaces. 1. PRELIMINARIES Let S = S(R', ,u) be the space of all Lebesgue measurable real-valued functions on R' endowed with the topology of convergence in measure on each set of finite measure. Recall that a Banach subspace X in S is said to be ideal [8] if x E X, Received by the editors December 11, 1991 and, in revised form, September 19, 1996. 1991 Mathematics Subject Classification. Primary 42B20, 42B25. ?)1999 American Mathematical Society 79 This content downloaded from 157.55.39.144 on Mon, 25 Jul 2016 03:48:01 UTC All use subject to http://about.jstor.org/terms