MIXED WEAK TYPE INEQUALITIES FOR ONE-SIDED OPERATORS

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 ∫c^x f(y) dy, x > c, is the Hardy averaging operator, u ∈ A1⁻ (one-sided Muckenhoupt A1 class), and v ∈ A1⁺ (another one-sided Muckenhoupt A1 class), then there exists a constant C such that supc∈ℝ ∫{x:|T c f(x)| > v(x)}uv ≤ C ∫ℝ|f|u.