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In this article we prove weighted norm inequalities and pointwise estimatesbetween the multilinear fractional integral operator and the multilinearfractional maximal. As a consequence of these estimations we obtain weightedweak and strong inequalities for the multilinear fractional integral operator.In particular, we extend some results given in \cite{CPSS} to the multilinearcontext. On the other hand we prove weighted pointwise estimates between themultilinear fractional maximal operator ${\cal M}_{\alpha,B}$ associated to aYoung function $B$ and the multilinear maximal operators ${\cal M}_{\psi}={\calM}_{0,\psi}$, $\psi(t)=B(t^{1-\alpha/(nm)})^{{nm}/{(nm-\alpha)}}$. As anapplication of these estimate we obtain a direct proof of the $L^p-L^q$boundedness results of ${\cal M}_{\alpha,B}$ for the case $B(t)=t$ and$B_k(t)=t(1+\log^+t)^k$ when $1/q=1/p-\alpha/n$. We also give sufficientconditions on the weights involved in the boundedness results of ${\calM}_{\alpha,B}$ that generalizes those given in \cite{M} for $B(t)=t$. Finally,we prove some boundedness results in Banach function spaces for a generalizedversion of the multilinear fractional maximal operator.

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