Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators

GLADIS PRADOLINI

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

Elsevier

Lugar: San Diego; Año: 2009

0022-247X

In this article we prove weighted norm inequalities and pointwiseestimates between the multilinear fractional integral operator andthe multilinear fractional maximal. As a consequence of theseestimations we obtain weighted weak and strong inequalities forthe multilinear fractional maximal operator or function. Inparticular, we extend some results given in cite{CPSS} to themultilinear context. On the other hand we prove weighted pointwiseestimates between the multilinear fractional maximal operator${cal M}_{alpha,B}$ associated to a Young function $B$ and themultilinear maximal operators ${cal M}_{psi}={cal M}_{0,psi}$,$psi(t)=B(t^{1-alpha/(nm)})^{{nm}/{(nm-alpha)}}$. As anapplication of these estimate we obtain a direct proof of the $L^p-L^q$ boundedness results of ${cal M}_{alpha,B}$ for thecase $B(t)=t$ and $B_k(t)=t(1+log^+t)^k$ when $1/q=1/p-alpha/n$.We also give sufficient conditions on the weights involved in theboundedness results of ${cal M}_{alpha,B}$ that generalizesthose given in cite{M} for $B(t)=t$. Finally, we prove someboundedness results in Banach function spaces for a generalizedversion of the multilinear fractional maximal operator.