New Proofs of Two-Weight Norm Inequalities for the Maximal Operator

Abstract

We give a new and simpler proof of Sawyer's theorem characterizing the weights governing the two-weight, strong-type norm inequality for the Hardy-Littlewood maximal operator and the fractional maximal operator. As a further application of our techniques, we give new proofs of two sufficient conditions for such weights due to Wheeden and Sawyer. § 1. Introduction 2000 Mathematics Subject Classification. 42B25.

Classical Fourier Analysis

“…Sawyer [312] obtained that the condition sup Q Q v 1−p dx −1 Q M(v 1−p χ Q ) p wdx < ∞ provides a characterization of all pairs of weights (v, w) for which the Hardy-Littlewood maximal operator M maps L p (v) to L p (w). Simpler proofs of this result were obtained by Cruz-Uribe [77] and Verbitsky [367]. The fact that Sawyer's condition reduces to the usual A p condition when v = w was shown by Hunt, Kurtz, and Neugebauer [166].…”

Fractional maximal operators with weighted Hausdorff content

“…In Theorem 1.2, if we take p = q and γ = 0 and we assume w ∈ A 1 (R n ), then we obtain the Tang theorem. Finally, let p = q, γ = α, and d = n, then δ becomes n and we obtain the Fefferman-Stein inequality for the fractional maximal operator due to Sawyer [7] (see also [3]).…”

Two-weight inequalities for multilinear maximal operators

“…The two-weight norm inequalities of the maximal operator M˛were studied extensively in [2,3,7,8,10,11] and the many references given there. In [10], Sawyer showed that in order for M˛to be bounded from L p .w/ to L q .v/, 1 < p Ä q < 1, This research was supported by the National Natural Science Foundation of China (10771049) and the Natural Science Foundation of Hebei Province (08M001).…”

“…In this paper, using the method in Cruz-Uribe [2], we give some results. The following theorems are our main results.…”