On the equivalence of weighted inequalities for a class of operators

Abstract

A characterization is obtained on weight function u so that $\smash{T\colon L_{p}^+\mapsto L_{q,u}^+}$ is bounded for 1 < p < ∞ and 0 < q < ∞, where T are integral operators and related maximal operators, and for 0 < p, q < ∞, where T are geometric mean operators and related geometric maximal operators. The equivalence of such weighted inequalities for these operators are established.