O’Neil Inequality for Multilinear Convolutions and Some Applications

In this paper we prove the O’Neil inequality for the k-linear convolution f ⊗ g. By using the O’Neil inequality for rearrangements we obtain a pointwise rearrangement estimate of the k-linear convolution. As an application, we obtain necessary and sufficient conditions on the parameters for the boundedness of the k-sublinear fractional maximal operator MΩ, α and k-linear fractional integral operator IΩ, α with rough kernels from the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{p1} \times L_{p2} \times . . . \times L_{pk} ({\mathbb{R}}^{n}) {\rm to} L_{q}({\mathbb{R}}^{n}).$$\end{document}