Mixed estimates for singular integrals on weighted Hardy spaces

Abstract

In this paper we give quantitative bounds for the norms of different kinds of singular integral operators on weighted Hardy spaces $$H_w^p$$ , where $$0<p\le 1$$ and w is a weight in the Muckenhoupt $$A_{\infty }$$ class. We deal with Fourier multiplier operators, Calderon–Zygmund operators of homogeneous type which are particular cases of the first ones, and, more generally, we study singular integrals of convolution type. In order to prove mixed estimates in the setting of weighted Hardy spaces, we need to introduce several characterizations of weighted Hardy spaces by means of square functions, Littlewood–Paley functions and the grand maximal function. We also establish explicit quantitative bounds depending on the weight w when switching between the $$H^p_w$$ -norms defined by the Littlewood–Paley–Stein square function and its discrete version, and also by applying the mixed bound $$A_q-A_\infty $$ result for the vector-valued extension of the Hardy–Littlewood maximal operator given in Buckley (Trans Am Math Soc 340(1):253–272, 1993).