arXiv subjects

Fabio Berra

Publications and source records attributed to Fabio Berra.

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]).

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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 $M^{ρ,σ}$, where $ρ$ is a critical radius function and $σ\geq 0$. We also exhibit and prove an extension of Cruz-Uribe, Martell and Pérez extrapolation result in \cite{CruzUribe-Martell-Perez} to the setting of Muckenhoupt weights associated to a critical radius function $ρ$. This theorem allows us to give mixed inequalities for Schrödinger-Calderón-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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arXiv2024

Endpoint estimates for higher order Gaussian Riesz transforms

We will show that, contrary to the behavior of the higher order Riesz transforms studied so far on the atomic Hardy space $\mathcal{H}^1(\mathbb R^n, \gamma)$, associated with the Ornstein-Uhlenbeck operator with respect to the $n$-dimensional Gaussian measure $\gamma$, the new Gaussian Riesz transforms are bounded from $\mathcal{H}^1(\mathbb R^n, \gamma)$ to $L^1(\mathbb R^n, \gamma)$, for any order and dimension $n$. We will also prove that the classical Gaussian Riesz transforms of higher order are bounded from an adequate subspace of $\mathcal{H}^1(\mathbb R^n, \gamma)$ into $L^1(\mathbb R^n, \gamma)$, extending Bruno's result (J. Fourier Anal. Appl. 25, 4 (2019), 1609--1631) for the first order case.

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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}.

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arXiv2023

Smoothness properties related to several commutators of fractional operators for values of $\boldsymbol{p}$ beyond the extreme in the multilinear setting

We prove continuity properties of higher order commutators of fractional operators on the multilinear setting, between a product of weighted Lebesgue spaces into certain weighted Lipschitz spaces. The considered operators include the multilinear fractional integral function and the main results extend previous estimates known for the unweighted case, as well as those established on the linear case. We give a complete study showing the main properties of the related multilinear weights and the optimal region described by the parameters where we can find nontrivial examples, including the restricted case of the one-weight theory. We also exhibit examples of such weights on the whole region.

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arXiv2022

Corrgiendum to "From $A_1$ to $A_\infty$: new mixed inequalities for certain maximal operators''

We devote this note to correct an estimate concerning mixed inequalities for the generalized maximal function $M_Φ$, when certain properties of the associated Young function $Φ$ are assumed. Although the obtained estimates turn out to be slightly different, they are good extensions of mixed inequalities for the classical Hardy-Littlewood maximal functions $M_r$, with $r\geq 1$. They also allow us to obtain mixed estimates for the generalized fractional maximal operator $M_{γ,Φ}$, when $0<γ<n$ and $Φ$ is an $L\log L$ type function.

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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.

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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

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.

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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.

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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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arXiv2021

Better bounds on mixed inequalities involving radial functions and applications

We prove mixed inequalities for the generalized maximal operator $M_Φ$ when the function $v$ is a radial power function that fails to be locally integrable. Concretely, let $u$ be a weight, $v(x)=|x|^β$ with $β<-n$ and $r\geq 1$. If $Φ$ is a Young function with certain properties, then the inequality \[uv^r\left(\left\{x\in\mathbb{R}^n: \frac{M_Φ(fv)(x)}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)v^r(x)Mu(x)\,dx\] holds for every $t>0$ and every bounded function. This improves a similar mixed estimate proved in \cite{BCP-M}. As an application, we give mixed estimates for the generalized fractional maximal operator $M_{γ,Φ}$, where $0<γ<n$ and $Φ$ is of $L\log L$ type. A special case involving the fractional maximal operator $M_γ$ allows to obtain a similar estimate for the fractional integral operator $I_γ$ through an extrapolation result. Furthermore, we also give mixed estimates for commutators of singular integral Calderón-Zygmund operators and of $I_γ$, both with Lipschitz symbol.

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arXiv2021

Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operators in higher dimensions

We give a quantitative characterization of the pairs of weights $(w,v)$ for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak $(p,p)$ type inequality, for $1\leq p<\infty$. More precisely, given any measurable set $E_0$ the estimate \[w(\{x\in \mathbb{R}^n: M^{+,d}(\mathcal{X}_{E_0})(x)>t\})\leq \frac{C[(w,v)]_{A_p^{+,d}(\mathcal{R})}^p}{t^p}v(E_0)\] holds if and only if the pair $(w,v)$ belongs to $A_p^{+,d}(\mathcal{R})$, that is \[\frac{|E|}{|Q|}\leq [(w,v)]_{A_p^{+,d}(\mathcal{R})}\left(\frac{v(E)}{w(Q)}\right)^{1/p}\] for every dyadic cube $Q$ and every measurable set $E\subset Q^+$. The proof follows some ideas appearing in [Sheldy Ombrosi, \emph{Weak weighted inequalities for a dyadic one-sided maximal function in {$\Bbb R^n$}}, Proc. Amer. Math. Soc. \textbf{133} (2005), no.~6, 1769--1775]. We also obtain a similar quantitative characterization for the non-dydadic case in $\mathbb{R}^2$ by following the main ideas in [L.~Forzani, F.~J. Mart\'ın-Reyes, and S.~Ombrosi, \emph{Weighted inequalities for the two-dimensional one-sided {H}ardy-{L}ittlewood maximal function}, Trans. Amer. Math. Soc. \textbf{363} (2011), no.~4, 1699--1719].

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