Restricted weak type inequalities for convolution maximal operators in weighted Lp spaces

Abstract

Let φ: ℝ → [0, ∞) be an integrable function such that φ(-∞,0) = 0 and φ is decreasing in (0, ∞). Let τh f (x) = f (x - h), with h ∈ ℝ/{0} and φ R(x) = (1/R)φ(x/R), with R > 0. In this paper we study the pair of weights (u, v) such that the operators Mτhφ f (x) = supR>0 |f| * [tau;hφ]R (x) are of restricted weak type (p, p) with respect to (u, v), 1 ≤ p < ∞. As particular cases, these operators include some maximal operators related to Cesàro convergence. We characterize those functions φ for which Mτhφ is of (restricted) weak type (p, p) with respect to the Lebesgue measure. Unlike the case of the Cesàro maximal operators, it follows from the characterization that the interval of those p such that M τhφ is of weak type (p, p) can be left-closed, [p 0, ∞], or left-open, (p0, ∞], without having restricted weak type (p0, p0).