Weighted Lebesgue and $BMO^\gamma $; norm inequalities for the Calderón and Hilbert operators

Abstract

Necessary and sufficient conditions are given for generalized Calderón and Hilbert operators to be bounded from weighted Lebesgue spaces into suitable weighted BMO and Lipschitz spaces. Moreover, we have obtained new results on the boundedness of these operators from L∞ into BMO, even in the unweighted case for the Hilbert operator. The class of weights involved are close to the doubling and reverseHölder conditions related to the Muckenhoupt?s classes.Keywords Calderón operator · BMO spaces · Weighted inequalities · Integral operators