Mixed Inequalities of Fefferman-Stein Type for Singular Integral Operators

Abstract

We give Feffermain-Stein type inequalities related to mixed estimates for Calderón-Zygmund operators. More precisely, given 𝛿 > 0 , q > 1 , (z) = z(1 + log + z) , a nonnegative and locally integrable function u and v ∈ RH ∞ ∩ A q , we prove that the inequality holds with, for every t > 0 and every p > max{q, 1 + 1∕𝛿} . This inequality provides a more general version of mixed estimates for Calderón-Zygmund operators proved in [6]. It also generalizes the Fefferman-Stein estimates given in [17] for the same operators. We further get similar estimates for operators of convolution type with kernels satisfying an L Φ −Hörmander condition, generalizing some previously known results which involve mixed estimates and Fefferman-Stein inequalities for these operators.