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

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 integral operator. In particular, we extend some results given in \cite{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 \cite{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

Boundedness of fractional operators in weighted variable exponent spaces with non doubling measures

In the context of variable exponent Lebesgue spaces equipped with a lower Ahlfors measure we obtain weighted norm inequalities over bounded domains for the centered fractional maximal function and the fractional integral operator

Two-weighted estimates for generalized fractional maximal operators on non-homogeneous spaces

summary:Let $\mu$ be a nonnegative Borel measure on $\mathbb R^d$ satisfying that $\mu (Q)\le l(Q)^n$ for every cube $Q\subset \mathbb R^n$, where $l(Q)$ is the side length of the cube $Q$ and $0<n\leq d$. \endgraf We study the class of pairs of weights related to the boundedness of radial maximal operators of fractional type associated to a Young function $B$ in the context of non-homogeneous spaces related to the measure $\mu$. Our results include two-weighted norm and weak type inequalities and pointwise estimates. Particularly, we give an improvement of a two-weighted result for certain fractional maximal operator proved in W. Wang, C. Tan, Z. Lou (2012)

End-point estimates for iterated commutators of multilinear singular integrals

Iterated commutators of multilinear Calderon-Zygmund operators and pointwise multiplication with functions in $BMO$ are studied in products of Lebesgue spaces. Both strong type and weak end-point estimates are obtained, including weighted results involving the vectors weights of the multilinear Calderon-Zygmund theory recently introduced in the literature. Some better than expected estimates for certain multilinear operators are presented too.Comment: A typo in the original manuscript lead to overlook a gap in one of our arguments which has been fixed. The new arguments are provided in the proof of Theorem 3.1 in Section 3. With the exception of some new notation introduced and some minor changes in wording in a few places, those new details are the only modifications to the original manuscrip