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: 2010 vol. 367 p. 640 - 656

0022-247X

In this article we prove weighted norm inequalities and pointwise estimates between the multilinear fractional integral operator and the multilinear fractional maximal. As a consequence of these estimations we obtain weighted weak and strong inequalities for the multilinear fractional maximal operator or function. In particular, we extend some results given in [CPSS] to the multilinear context. On the other hand we prove weighted pointwise estimates between the multilinear fractional maximal operator ${cal M}_{alpha,B}$ associated to a Young function $B$ and the multilinear maximal operators ${cal M}_{psi}={cal M}_{0,psi}$, $psi(t)=B(t^{1-alpha/(nm)})^{{nm}/{(nm-alpha)}}$. As an application 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 sufficient conditions on the weights involved in the boundedness results of ${cal M}_{alpha,B}$ that generalizes those given in [M] for $B(t)=t$. Finally, we prove some boundedness results in Banach function spaces for a generalized version of the multilinear fractional maximal operator.