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Gladis Pradolini

Publications and source records attributed to Gladis Pradolini.

27 records · Page 2Linked to original sources
arXiv2017

Mixed weak estimates of Sawyer type for fractional integrals and some related operators

We prove mixed weak estimates of Sawyer type for fractional operators. More precisely, let $\mathcal{T}$ be either the maximal fractional function $M_γ$ or the fractional integral operator $I_γ$, $0<γ t\right\}\right)^{1/q}\leq \frac{C}{t}\left(\int_{\R^n}|f(x)|^pu(x)^{p/q}v(x)\,dx\right)^{1/p}, \end{equation*} holds for every positive $t$ and every bounded function with compact support. As an important application of the results above we further more exhibe mixed weak estimates for commutators of Calderón-Zygmund singular integral and fractional integral operators when the symbol $b$ is in the class Lipschitz-$δ$, $0<δ\leq 1$.

math.AP↗
arXiv2017

Mixed weak estimates of Sawyer type for commutators of singular integrals and related operators

We study mixed weak type inequalities for the commutator $[b,T]$, where $b$ is a BMO function and $T$ is a Calderón-Zygmund operator. More precisely, we prove that for every $t>0$ \begin{equation*}%\label{tesis_teo2.2} uv(\{x\in\R^n: |\frac{[b,T](fv)(x)}{v(x)}|>t\})\leq C\int_{\R^n}ϕ(\frac{|f(x)|}{t})u(x)v(x)\,dx, \end{equation*} where $ϕ(t)=t(1+\log^{+}{t})$, $u\in A_1$ and $v\in A_{\infty}(u)$. Our technique involves the classical Calderón-Zygmund decomposition, which allow us to give a direct proof. We use this result to prove an analogous inequality for higher order commutators. We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of $L\log L$ type which are in intimate relation with the commutators. This last estimate involves an arbitrary weight $u$ and a radial function $v$ which is not even locally integrable.

math.CA↗
arXiv2017

The effect of the smoothness of fractional type operators over their commutators with Lipschitz symbols on weighted spaces

We prove boundedness results for integral operators of fractional type and their higher order commutators between weighted spaces, including $L^p$-$L^q$, $L^p$-$BMO$ and $L^p$-Lipschitz estimates. The kernels of such operators satisfy certain size condition and a Lipschitz type regularity, and the symbol of the commutator belongs to a Lipschitz class. We also deal with commutators of fractional type operators with less regular kernels satisfying a Hörmander's type inequality. As far as we know, these last results are new even in the unweighted case. Moreover, we give a characterization result involving symbols of the commutators and continuity results for extreme values of $p$.

math.AP↗
arXiv2016

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

Let $μ$ be a non-negative Borel measure on $R^d$ satisfying that the measure of a cube in $R^d$ is smaller than the length of its side raised to the $n$-th power, $0<n\leq d$. In this article we study the class of weights related to the boundedness of radial fractional type maximal operator associated to a Young function $B$ in the context of non-homogeneous spaces related with the measure $μ$. This type of maximal operators are the adequate operators related with commutators of singular and fractional operators. Particularly, we give an improvement of a two weighted result for certain fractional maximal operator proved in [26].

math.AP↗
arXiv2011

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.

math.CA↗
arXiv2010

Weighted inequalities for multilinear potential operators and its commutators

We prove weighted strong inequalities for the multilinear potential operator ${\cal T}_ϕ$ and its commutator, where the kernel $ϕ$ satisfies certain growth condition. For these operators we also obtain Fefferman-Stein type inequalities and Coifman type estimates. On the other hand we prove weighted weak type inequalities for the multilinear maximal operator $\mathcal{M}_{φ,B}$ associated to a essentially nondecreasing function $φ$ and to a submultiplicative Young function $B$. This result allows us to obtain a weighted weak type inequality for the operator ${\cal T}_ϕ$.

math.CA↗
arXiv2009

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 ${\cal M}_{α,B}$ associated to a Young function $B$ and the multilinear maximal operators ${\cal M}_ψ={\cal M}_{0,ψ}$, $ψ(t)=B(t^{1-α/(nm)})^{{nm}/{(nm-α)}}$. As an application of these estimate we obtain a direct proof of the $L^p-L^q$ boundedness results of ${\cal M}_{α,B}$ for the case $B(t)=t$ and $B_k(t)=t(1+\log^+t)^k$ when $1/q=1/p-α/n$. We also give sufficient conditions on the weights involved in the boundedness results of ${\cal M}_{α,B}$ 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.

math.AP↗