Weighted mixed weak-type inequalities for multilinear fractional operators

Abstract

The aim of this paper is to obtain mixed weak-type inequalities for multilinear fractional operators, extending results by F. Berra, M. Carena and G. Pradolini \cite{BCP}. We prove that, under certain conditions on the weights, there exists a constant $C$ such that $$\Vert \frac{{\mathcal{G}}_{\alpha }(\vec{f})}{v}{\Vert }_{L^{q,\infty }(\nu v^q)}\le C\prod _{i=1}^m\Vert f_i{\Vert }_{L^1(u_i)},$$ where ${\mathcal{G}}_{\alpha }(\vec{f})$ is the multilinear maximal function ${\mathcal{M}}_{\alpha }(\vec{f})$ that was introduced by K. Moen in \cite{M} or the multilineal fractional integral ${\mathcal{I}}_{\alpha }(\vec{f})$. As an application a vector-valued weighted mixed inequality for ${\mathcal{I}}_{\alpha }(\vec{f})$ will be provided as well.

References in corpus (2)

• Extrapolation for multilinear Muckenhoupt classes and applications to the bilinear Hilbert transform
• Weighted mixed weak-type inequalities for multilinear operators