Two weighted inequalities for convolution maximal operators

Abstract

Let ψ: ℝ → [0, ∞) an integrable function such that ψχ(-∞,0) = 0 and ψ is decreasing in (0, ∞). Let τhf(x) = f(x - h), with h ∈ ℝ \ {0} and f R(x) = 1/Rf(x/R), with R > 0. In this paper we characterize the pair of weights (u, v) such that the operators Mτhψf(x) = supR>0 |f| * [τhψ]R(x) are of weak type (p, p) with respect to (u, v), 1 < p < ∞.