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Weighted norm inequalities for averaging operators of monotone functions

Publicacions Matemàtiques
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Publicacions Matemátiques, Vol 35 (1991), 429-447 . Abstract WEIGHTED NORM INEQUALITIES FOR AVERAGING OPERATORS OF MONOTONE FUNCTIONS C .J . NEUGEBAUER We prove weighted norm inequalities for the averaging operator Af(x) = l fs J f of monotone functions . x o 1 . Introduction This paper is concerned with weighted Hardy type inequalities of the form f°°(1 fx f)Pw(x)dx < c J oo f(x)Pv(x)dx . o x o 0 Muckenhoupt [6] has given necessary and sufficient conditions for (*) to hold for arbitrary f. In their paper [1] Ariño and Muckenhoupt studied the problem when the Hardy-Littlewood maximal operator is bounded on Lorentz spaces and observed that this leads to the study of (*) for non-increasing f . There are more weights in this case than for general f [1] . They solved the problem for w = v by the 00 r P r condition BP , Le ., w E BP if and only if -) w(x)dx <_ cf w(x)dx, fr x o r > 0. The proof is rather lengthy and first establishes that BP implies BP_E (Lemma 2.1 of [1]) . The purpose of this paper is (i) to give a much shorter proof of a somewhat more general version of (*) without BP implies BP_E , (ii) to prove then BP implies BP_E using an iterated version of (*), (iii) to investigate the reverse inequalities 00 00 1 x f (x)Pw(x)dx < cf (-1 f f)Pv(x)dx, o o x o (iv) to study the same questions for non-decreasing functions, and finally (v) to present some properties of BPweights suggested by the analogous properties of Ap-weights as, e.g . the A1 A1-P factorization of an AP weight [3] . 430 C.J . NEUGEBAUER We point out that the double weight inequality (*) has been characterized in a recent paper by E . Sawyer [7] for non-increasing functions with the q-norm of the averaging operator on the left and the p-norm on the right . It is also possible to prove some of our results by the methods developed in the paper by D.W. Boyd [2] . Throughout the paper we shall use the following notation . The sy

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