arXiv subjects

Gladis Pradolini

Publications and source records attributed to Gladis Pradolini.

At least 19 recordsLinked to original sources

arXiv2026

Mixed Weak type inequalities for pairs of weights related to the Hardy-Littlewood maximal funcion, Calder\'on-Zygmund operators and their commutators

We study two-weight weak-type estimates for the operator $S_v f = \mathcal{T}(fv)/v$, where $\mathcal{T}$ is the Hardy-Littlewood maximal operator or a Calder\'on-Zygmund operator (CZO) and $v$ is a weight. Concretely, under certain conditions on the weights involved, we prove that $S_v$ is bounded from $L^{1}(wv)$ to $L^{1,\infty} (uv)$. We also consider the corresponding inequalities when $\mathcal{T}$ is a higher-order commutator of a CZO. These types of results are inspired by the article of Sawyer in [21], (see also [17]).

math.CA↗

arXiv2026

Continuity properties of strongly singular integral operators for extreme values of $p$

In this work, we establish continuity properties of strongly singular integral operators for extreme values of $p$. Particularly, weighted $L^\infty$-$BMO$ boundedness is obtained, generalizing Miyachi's result to the context of Muckenhoupt weights. As an application, we get an alternative proof of Chanillo's weighted $L^p$ estimates via extrapolation techniques.

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arXiv2025

Characterization of the continuity properties of maximal operators associated to critical radius functions via Dini type conditions

We give a characterization of the continuity properties of a Luxemburg maximal type operator associated to a critical radius function $\rho$ between Orlicz spaces. This goal is achieved by means of a Dini type condition that includes certain Young functions related to the maximal operator and the spaces involved. Our results provide not only weak Fefferman-Stein type inequalities but also a weak weighted estimate of modular type for the considered operators, which is interesting in its own right. On the other hand, we prove the boundedness of the Hardy-Littlewood maximal function associated to $\rho$ between Zygmund spaces of $L\,\log\,L$ type with $A_p$ weights.

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arXiv2025

Revisiting mixed weak inequalities of Fefferman-Stein type for commutators of Calder\'on-Zygmund operators: an improvement

In this paper we establish mixed weak inequalities of Fefferman-Stein type for Calder\'on-Zygmund operators and their commutators, improving some previous results known in the literature. The main estimates also generalize the classical weighted weak Fefferman-Stein inequalities proved in [19] and [22]. In order to obtain the main results, our approach is to give a strong Fefferman-Stein type inequality for the operators involved with respect to an adequate measure.

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arXiv2024

Weighted mixed inequalities for commutators of Schr\"odinger type operators

We obtain weighted mixed inequalities for the first order commutator of singular integral operators in the Schr\"odinger setting. Concretely, for $0<\delta\leq 1$ we give estimates of commutators of Schr\"odinger-Calder\'on-Zygmund operators of $(s,\delta)$ type with $1<s\leq \infty$, and $\text{BMO}(\rho)$ symbols associated to a critical radious function $\rho$. Our results generalizes some previous estimates about mixed inequalities for Schr\"odinger type operators. We also deal with $A_p^\rho$ weights, which can be understood as a perturbation of the $A_p$ Muckenhoupt classes by means of function $\rho$.

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arXiv2024

Sawyer estimates of mixed type for operators associated to a critical radius function

We prove mixed inequalities for the Hardy-Littlewood maximal function $M^{\rho,\sigma}$, where $\rho$ is a critical radius function and $\sigma\geq 0$. We also exhibit and prove an extension of Cruz-Uribe, Martell and P\'erez extrapolation result in \cite{CruzUribe-Martell-Perez} to the setting of Muckenhoupt weights associated to a critical radius function $\rho$. This theorem allows us to give mixed inequalities for Schr\"odinger-Calder\'on-Zygmund operators, extending some previous estimates that we have already proved in \cite{BPQ}. Since we are dealing with unrelated weights, the proof involves a quite subtle argument related with the original ideas from Sawyer in \cite{Sawyer}.

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arXiv2023

Weighted mixed endpoint estimates of Fefferman-Stein type for commutators of singular integral operators

We deal with mixed weak estimates of Fefferman-Stein type for higher order commutators of Calderón-Zygmund operators with BMO symbol. The results obtained are Fefferman-Stein inequalities that include the estimates proved in \cite{BCP22(JMS)} for the case of singular integral operators, as well as the classical weak endpoint estimate for commutators given in \cite{PP01}. We also consider commutators of operators involving less regular kernels satisfying an $L^Φ$--Hörmander condition. Particularly, the obtained results contain some previous estimates proved in \cite{BCP22(JMS)} and \cite{Lorente-Martell-Perez-Riveros}.

math.CA↗

arXiv2022

Some extensions of classes involving pair of weights related to the boundedness of multilinear commutators associated to generalized fractional integral operators

We deal with the boundedness properties of higher order commutators related to some generalizations of the multilinear fractional integral operator of order $m$, $I_α^m$, from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces, extending some previous estimates for the linear case. Our study includes two different types of commutators and sufficient conditions on the weights in order to guarantee the continuity properties described above. We also exhibit the optimal range of the parameters involved. The optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region, being the weights trivial outside of it. We further show examples of weights for the class which cover the mentioned area.

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arXiv2022

Mixed inequalities for operators associated to critical radius functions with applications to Schrödinger type operators

We obtain weighted mixed inequalities for operators associated to a critical radius function. We consider Schrödinger Calderón-Zygmund operators of $(s,δ)$ type, for $1<s\leq \infty$ and $0<δ\leq 1$. We also give estimates of the same type for the associated maximal operators. As an application, we obtain a wide variety of mixed inequalities for Schrödinger type singular integrals. As far as we know, these results are a first approach of mixed inequalities in the Schrödinger setting.

math.AP↗

arXiv2022

Two-weighted estimates of the multilinear fractional integral operator between weighted Lebesgue and Lipschitz spaces with optimal parameters

Given an $m$-tuple of weights $\vec{v}=(v_1,\dots,v_m)$, we characterize the classes of pairs $(w,\vec{v})$ involved with the boundedness properties of the multilinear fractional integral operator from $\prod_{i=1}^mL^{p_i}\left(v_i^{p_i}\right)$ into suitable Lipschitz spaces associated to a parameter $δ$, $\mathcal{L}_w(δ)$. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. We emphasize the study related to the range of the parameters involved with the problem described above, which is optimal in the sense that they become trivial outside of the region obtained. We also exhibit nontrivial examples of pairs of weights in this region.

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arXiv2022

On optimal parameters involved with two-weighted estimates of commutators of singular and fractional integral operators

In this paper we prove two-weighted norm estimates for higher order commutator of singular integral and fractional type operators between weighted $L^p$ and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces belong to certain region out of which the classes of weights are satisfied by trivial weights. We also exhibit pairs of non-trivial weights in the optimal region satisfying the conditions required.

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arXiv2022

Optimal parameters related with continuity properties of the multilinear fractional integral operator between Lebesgue and Lipschitz spaces

We deal with the boundedness of the multilinear fractional integral operator $I_{γ,m}$ from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. We characterize the classes of weights for which the problem described above holds and show the optimal range of the parameters involved. The optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region. We further exhibit examples of weights for the class which cover the mentioned area.

math.CA↗

arXiv2022

Mixed inequalities of Fefferman-Stein type for singular integral operators

We give Feffermain-Stein type inequalities related to mixed estimates for Calderón-Zygmund operators. More precisely, given $δ>0$, $q>1$, $φ(z)=z(1+\log^+z)^δ$, a nonnegative and locally integrable function $u$ and $v\in \mathrm{RH}_\infty\cap A_q$, we prove that the inequality \[uv\left(\left\{x\in \mathbb{R}^n: \frac{|T(fv)(x)|}{v(x)}>t\right\}\right)\leq \frac{C}{t}\int_{\mathbb{R}^n}|f|\left(M_{φ, v^{1-q'}}u\right)M(Ψ(v))\] holds with $Ψ(z)=z^{p'+1-q'}\mathcal{X}_{[0,1]}(z)+z^{p'}\mathcal{X}_{[1,\infty)}(z)$, for every $t>0$ and every $p>\max\{q,1+1/δ\}$. This inequality provides a more general version of mixed estimates for Calderón-Zygmund operators proved in \cite{CruzUribe-Martell-Perez}. It also generalizes the Fefferman-Stein estimates given in \cite{P94} for the same operators. We further get similar estimates for operators of convolution type with kernels satisfying an $L^Φ-$Hörmander condition, generalizing some previously known results which involve mixed estimates and Fefferman-Stein inequalities for these operators.

math.CA↗

arXiv2021

Mixed inequalities for commutators with multilinear symbol

We prove mixed inequalities for commutators of Calderón-Zygmund operators (CZO) with multilinear symbols. Concretely, let $m\in\mathbb{N}$ and $\mathbf{b}=(b_1,b_2,\dots, b_m)$ be a vectorial symbol such that each component $b_i\in \mathrm{Osc}_{\mathrm{exp}\, L^{r_i}}$, with $r_i\geq 1$. If $u\in A_1$ and $v\in A_\infty(u)$ we prove that the inequality \[uv\left(\left\{x\in \mathbb{R}^n: \frac{|T_\mathbf{b}(fv)(x)|}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\|\mathbf{b}\|\frac{|f(x)|}{t}\right)u(x)v(x)\,dx\] holds for every $t>0$, where $Φ(t)=t(1+\log^+t)^r$, with $1/r=\sum_{i=1}^m 1/r_i$. We also consider operators of convolution type with kernels satisfying less regularity properties than CZO. In this setting, we give a Coifman type inequality for the associated commutators with multilinear symbol. This result allows us to deduce the $L^p(w)$-boundedness of these operators when $1<p<\infty$ and $w\in A_p$. As a consequence, we can obtain the desired mixed inequality in this context.

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arXiv2020

On two weighted problems for commutators of classical operators with optimal behaviour on the parameters involved and extrapolation results

We give two weighted norm estimates for higher order commutator of classical operators such as singular integral and fractional type operators, between weighted $L^p$ and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces belong to certain region out of which the classes of weights are satisfied by trivial weights. We also exhibit pairs of non-trivial weights in the optimal region satisfying the conditions required. Finally, we exhibit an extrapolation result that allows us to obtain boundedness results of the type described above in the variable setting and for a great variety of operators, by starting from analogous inequalities in the classical context. In order to get this result we prove a Calderón-Scott type inequality with weights that connects adequately the spaces involved.

math.AP↗

arXiv2019

Musielak Orlicz bumps and Bloom type estimates for commutators of Calderón Zygmund and fractional integral operators on variable Lebesgue spaces via sparse operators

We obtain Musielak Orlicz bumps conditions on a pair of weights for the boundedness of Calderón Zygmund operators and their commutators between variable Lebesgue spaces with different weights. The symbols of the commutators belong to a wider class of functions. We also give Bloom type estimates for commutators of Calderón Zygmund and fractional integral operators in the variable Lebesgue context. The techniques involved in both type of results are related with the theory of sparse domination.

math.AP↗

arXiv2019

Commutators of potential type operators with Lipschitz symbols on variable Lebesgue spaces with different weights

We prove that a generalized Fefferman-Phong type condition on a pair of weights $u$ and $v$ is sufficient for the boundedness of the commutators of potential type operators from $L^{p(\cdot)}_v$ into $L^{q(\cdot)}_u$. We also give an improvement of this result in the sense that we not only consider a variable version of power bump conditions, but also weaker norms related to Musielak-Orlicz functions. We consider a wider class of symbols including Lipschitz symbols and some generalizations.

math.CA↗

arXiv2019

Improvements on Sawyer type estimates for generalized maximal functions

In this paper we prove mixed inequalities for the maximal operator $M_Φ$, for general Young functions $Φ$ with certain additional properties, improving and generalizing some previous estimates for the Hardy-Littlewood maximal operator proved by E. Sawyer. We show that given $r\geq 1$, if $u,v^r$ are weights belonging to the $A_1$-Muckenhoupt class and $Φ$ is a Young function as above, then the inequality \[uv^r\left(\left\{x\in \mathbb{R}^n: \frac{M_Φ(fv)(x)}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx\] holds for every positive $t$. A motivation for studying these type of estimates is to find an alternative way to prove the boundedness properties of $M_Φ$. Moreover, it is well-known that for the particular case $Φ(t)=t(1+\log^+t)^m$ with $m\in\mathbb{N}$ these maximal functions control, in some sense, certain operatos in Harmonic Analysis.

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