Restricted Weak Type Inequalities for Convolution Maximal Operators in Weighted LP Spaces

Abstract

Let ϕ : R →[ 0, ∞) be an integrable function such that ϕχ(−∞,0) = 0 and ϕ is decreasing in (0, ∞). Let τh f (x) = f (x − h), with h ∈ R \{ 0} and ϕR(x) = (1/R)ϕ(x/R) ,w ith R > 0. In this paper we study the pair of weights (u ,v) such that the operators Mτhϕ f (x) = sup R>0 | f |∗[ τhϕ]R(x) are of restricted weak type ( p, p) with respect to (u ,v) ,1 p < ∞ .A sparticular cases, these operators include some maximal operators related to Ces ` aro convergence. We characterize those functions ϕ for which Mτhϕ is of (restricted) weak type