Multilinear version of reversed Hölder inequality and its applications to multilinear Calderón-Zygmund operators

Abstract

In this paper, we give a natural, and generalized reverse Hölder inequality, which says that if $\omega_{i}\in A_\infty$, then for every cube $Q$, \begin{equation} \int_Q\prod_{i =1}^m{\omega_i}^{\theta_i} \geq \prod_{i = 1}^m \left( \frac{\int_Q{\omega_i}}{[\omega_i]_{A_\infty}} \right)^{\theta_i}, \end{equation} where $\sum_{i=1}^m\theta_{i}=1$, $0\leq\theta_i\leq1$.

As a consequence, we get a more general inequality, which can be viewed as an extension of the reverse Jensen inequality in the theory of weighted inequalities. Based on this inequality (0.1), we then give some results concerning multilinear Calderón-Zygmund operators and maximal operators on weighted Hardy spaces, which improve some known results significantly.