Mixed weak type inequalities for one-sided operators

Abstract

We discuss mixed weak type inequalities in weighted spaces for one-sided operators. In particular, we prove that if Tc f(x) = (x - c)-1 f x c f(y) dy, x > c, is the Hardy averaging operator, u ∈ A- 1 (one-sided Muckenhoupt A1 class), and v ∈ A- 1 (another one-sided Muckenhoupt A1 class), then there exists a constant C such that supc∈ ∫x:Tcf(x) > v(x)uv ≤ C ∫fu.