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Ezequiel Rela

Publications and source records attributed to Ezequiel Rela.

At least 19 recordsLinked to original sources
arXiv2026

Asymptotically sharp embedding of $A_\infty$ into $A_p$ for flat weights and applications to Poincar\'e-Sobolev inequalities

We provide new quantitative results on the embedding of the Muckenhoupt class $A_\infty$ into $A_p$ with the correct asymptotic behavior when the Fujii--Wilson constant $[w]_{A_\infty}$ is close to 1, namely that the parameter $p$ goes to 1 when the weight is nearly constant. As intermediate steps towards the result, we obtain quantitative estimates on the weighted and unweighted BMO norms of $\log w$ for an $A_\infty$ weight $w$. As a consequence, we show that a precise quantitative weighted Poincar\'e-Sobolev inequality can be proved for weights with small $[w]_{A_\infty}$ that recovers the classical Sobolev exponent $p^*=\frac{np}{n-p}$ when $[w]_{A_\infty}\to 1^+$.

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arXiv2026

Weighted Sobolev Inequalities via the Meyers--Ziemer Framework: Measures, Isoperimetric Inequalities, and Endpoint Estimates

We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This result has several significant consequences. It extends naturally to functions of weighted bounded variation and yields corresponding capacity and isoperimetric inequalities. The inequality is also closely connected to endpoint estimates for fractional operators, including bounds for fractional maximal functions and Hardy space endpoint estimates for the Riesz potential. Our main inequality yields a family of endpoint inequalities, characterized in terms of subrepresentation formulas, Lorentz space improvements, and isoperimetric inequalities for measures and bounded open sets. When one moves away from the endpoint to $p>1$, the analogous inequalities no longer hold in general; however, we identify a sharp bumped maximal function for which the corresponding non-endpoint inequality is valid. Finally, we show that this framework yields new $(p,p)$ two-weight Sobolev inequalities.

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arXiv2026

Slowly decaying Rajchman measures and a restriction theorem for the Fourier transform at the limit case of zero Fourier dimension

In this article we prove the existence of sets $E \subseteq \mathbb{R}$ of zero Fourier dimension such that it is possible to restrict the Fourier transform to $E$ on a certain non-trivial range $[1,\tilde{p})$ with $1<\tilde{p}<2$. This builds upon Mockenhaupt's Restriction Theorem; while this theorem could only be applied to sets of positive Fourier dimension, we show that the existence of a measure with polylogarithmic Fourier decay combined with full Hausdorff dimension 1 on the real line is enough to guarantee restriction. In order to achieve this, we combine two different tools: a modification of a construction from a recent work of Li and Liu to produce a set with specific Hausdorff and Fourier dimensions, and a generalization of the Stein-Tomas-Mockenhaupt Restriction Theorem.

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arXiv2024

The exact dimension of Liouville numbers: The Fourier side

In this article we study the generalized Fourier dimension of the set of Liouville numbers $\mathbb{L}$. Being a set of zero Hausdorff dimension, the analysis has to be done at the level of functions with a slow decay at infinity acting as control for the Fourier transform of (Rajchman) measures supported on $\mathbb{L}$. We give an almost complete characterization of admissible decays for this set in terms of comparison to power-like functions. This work can be seen as the ``Fourier side'' of the analysis made by Olsen and Renfro regarding the generalized Hausdorff dimension using gauge functions. We also provide an approach to deal with the problem of classifying oscillating candidates for a Fourier decay for $\mathbb{L}$ relying on its translation invariance property.

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arXiv2022

Self-improving Poincaré-Sobolev type functionals in product spaces

In this paper we give a geometric condition which ensures that $(q,p)$-Poincaré-Sobolev inequalities are implied from generalized $(1,1)$-Poincaré inequalities related to $L^1$ norms in the context of product spaces. The concept of eccentricity plays a central role in the paper. We provide several $(1,1)$-Poincaré type inequalities adapted to different geometries and then show that our selfimproving method can be applied to obtain special interesting Poincaré-Sobolev estimates. Among other results, we prove that for each rectangle $R$ of the form $R=I_1\times I_2 \subset \mathbb{R}^{n}$ where $I_1\subset \mathbb{R}^{n_1}$ and $I_2\subset \mathbb{R}^{n_2}$ are cubes with sides parallel to the coordinate axes, we have that % \begin{equation*} \left( \frac{1}{w(R)}\int_{ R } |f -f_{R}|^{p_{δ,w}^*} \,wdx\right)^{\frac{1}{p_{δ,w}^*}} \leq c\,(1-δ)^{\frac1p}\,[w]_{A_{1,\mathfrak{R}}}^{\frac1p}\, \Big(a_1(R)+a_2(R)\Big), \end{equation*} % where $δ\in (0,1)$, $w \in A_{1,\mathfrak{R}}$, $\frac{1}{p} -\frac{1}{ p_{δ,w}^* }= \fracδ{n} \, \frac{1}{1+\log [w]_{A_{1,\mathfrak{R}}}}$ and $a_i(R)$ are bilinear analog of the fractional Sobolev seminorms $[u]_{W^{δ,p}(Q)}$ (See Theorem 2.18). This is a biparameter weighted version of the celebrated fractional Poincaré-Sobolev estimates with the gain $(1-δ)^{\frac1p}$ due to Bourgain-Brezis-Minorescu.

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arXiv2021

Maximal operators on the infinite-dimensional torus

We study maximal operators related to bases on the infinite-dimensional torus $\mathbb{T}^ω$. {For the normalized Haar measure $dx$ on $\mathbb{T}^ω$ it is known that $M^{\mathcal{R}_0}$, the maximal operator associated with the dyadic basis $\mathcal{R}_0$, is of weak type $(1,1)$, but $M^{\mathcal{R}}$, the operator associated with the natural general basis $\mathcal{R}$, is not. We extend the latter result to all $q \in [1,\infty)$. Then we find a wide class of intermediate bases $\mathcal{R}_0 \subset \mathcal{R}' \subset \mathcal{R}$, for which maximal functions have controlled, but sometimes very peculiar behavior.} Precisely, for given $q_0 \in [1, \infty)$ we construct $\mathcal{R}'$ such that $M^{\mathcal{R}'}$ is of restricted weak type $(q,q)$ if and only if $q$ belongs to a predetermined range of the form $(q_0, \infty]$ or $[q_0, \infty]$. Finally, we study the weighted setting, considering the Muckenhoupt $A_p^\mathcal{R}(\mathbb{T}^ω)$ and reverse Hölder $\mathrm{RH}_r^\mathcal{R}(\mathbb{T}^ω)$ classes of weights associated with $\mathcal{R}$. For each $p \in (1, \infty)$ and each $w \in A_p^\mathcal{R}(\mathbb{T}^ω)$ we obtain that $M^{\mathcal{R}}$ is not bounded on $L^q(w)$ in the whole range $q \in [1,\infty)$. Since we are able to show that \[ \bigcup_{p \in (1, \infty)}A_p^\mathcal{R}(\mathbb{T}^ω) = \bigcup_{r \in (1, \infty)} \mathrm{RH}_r^\mathcal{R}(\mathbb{T}^ω), \] the unboundedness result applies also to all reverse Hölder weights.

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arXiv2020

Quantitative John-Nirenberg inequalities at different scales

We provide an abstract estimate of the form \[ \|f-f_{Q,μ}\|_{X \left(Q,\frac{\mathrm{d} μ}{Y(Q)}\right)}\leq c(μ,Y)ψ(X)\|f\|_{\mathrm{BMO}(\mathrm{d}μ)} \] for all cubes $Q$ in $\mathbb{R}^n$ and every function $f\in \mathrm{BMO}(\mathrm{d}μ)$, where $μ$ is a doubling measure in $\mathbb{R}^n$, $Y$ is some positive functional defined on cubes, $\|\cdot \|_{X \left(Q,\frac{\mathrm{d} w}{w(Q)}\right)}$ is a sufficiently good quasi-norm and $c(μ,Y)$ and $ψ(X)$ are positive constants depending on $μ$ and $Y$, and $X$, respectively. That abstract scheme allows us to recover the sharp estimate \[ \|f-f_{Q,μ}\|_{L^p \left(Q,\frac{\mathrm{d} μ(x)}{μ(Q)}\right)}\leq c(μ)p\|f\|_{\mathrm{BMO}(\mathrm{d}μ)}, \qquad p\geq1 \] for every cube $Q$ and every $f\in \mathrm{BMO}(\mathrm{d}μ)$, which is known to be equivalent to the John-Nirenberg inequality, and also enables us to obtain quantitative counterparts when $L^p$ is replaced by suitable strong and weak Orlicz spaces and $L^{p(\cdot)}$ spaces. Besides the aforementioned results we also generalize Theorem 1.2 in [OPRRR20] to the setting of doubling measures and obtain a new characterization of Muckenhoupt's $A_\infty$ weights.

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arXiv2020

Minimal conditions for BMO

We study minimal integrability conditions via Luxemburg-type expressions with respect to generalized oscillations that imply the membership of a given function $f$ to the space BMO. Our method is simple, sharp and flexible enough to be adapted to several different settings, like spaces of homogeneous type, non doubling measures on $\mathbb{R}^n$ and also BMO spaces defined over more general bases than the basis of cubes.

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arXiv2019

A note on generalized Fujii-Wilson conditions and BMO spaces

In this note we generalize the definition of Fujii-Wilson condition providing quantitative characterizations of some interesting classes of weights, such as $A_\infty$, $A_\infty^{weak}$ and $C_p$, in terms of BMO type spaces suited to them. We will provide as well some self improvement properties for some of those generalized BMO spaces and some quantitative estimates for Bloom's BMO type spaces.

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arXiv2019

Weighted estimates for maximal functions associated to skeletons

We provide quantitative weighted estimates for the $L^p(w)$ norm of a maximal operator associated to cube skeletons in $\mathbb{R}^n$. The method of proof differs from the usual in the area of weighted inequalities since there are no covering arguments suitable for the geometry of skeletons. We use instead a combinatorial strategy that allows to obtain, after a linearization and discretization, $L^p$ bounds for the maximal operator from an estimate related to intersections between skeletons and $k$-planes.

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arXiv2019

Degenerate Poincaré-Sobolev inequalities

We study weighted Poincaré and Poincaré-Sobolev type inequalities with an explicit analysis on the dependence on the $A_p$ constants of the involved weights. We obtain inequalities of the form $$ \left (\frac{1}{w(Q)}\int_Q|f-f_Q|^{q}w\right )^\frac{1}{q}\le C_w\ell(Q)\left (\frac{1}{w(Q)}\int_Q |\nabla f|^p w\right )^\frac{1}{p}, $$ with different quantitative estimates for both the exponent $q$ and the constant $C_w$. We will derive those estimates together with a large variety of related results as a consequence of a general selfimproving property shared by functions satisfying the inequality $$ \frac{1}{|Q|}\int_Q |f-f_Q| dμ\le a(Q), $$ for all cubes $Q\subset\mathbb{R}^n$ and where $a$ is some functional that obeys a specific discrete geometrical summability condition. We introduce a Sobolev-type exponent $p^*_w>p$ associated to the weight $w$ and obtain further improvements involving $L^{p^*_w}$ norms on the left hand side of the inequality above. For the endpoint case of $A_1$ weights we reach the classical critical Sobolev exponent $p^*=\frac{pn}{n-p}$ which is the largest possible and provide different type of quantitative estimates for $C_w$. We also show that this best possible estimate cannot hold with an exponent on the $A_1$ constant smaller than $1/p$. We also provide an argument based on extrapolation ideas showing that there is no $(p,p)$, $p\geq1$, Poincaré inequality valid for the whole class of $RH_\infty$ weights by showing their intimate connection with the failure of Poincaré inequalities, $(p,p)$ in the range $0<p<1$.

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arXiv2017

Improved Buckley's theorem on LCA groups

We present sharp quantitative weighted norm inequalities for the Hardy-Littlewood maximal function in the context of Locally Compact Abelian Groups, obtaining an improved version of the so-called Buckley's Theorem. On the way, we prove a precise reverse Hölder inequality for Muckenhoupt $A_\infty$ weights and provide a valid version of the "open property" for Muckenhoupt $A_p$ weights.

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arXiv2016

Asymptotically sharp reverse Hölder inequalities for flat Muckenhoupt weights

We present reverse Hölder inequalities for Muckenhoupt weights in $\mathbb{R}^n$ with an asymptotically sharp behavior for flat weights, namely $A_\infty$ weights with Fujii-Wilson constant $(w)_{A_\infty}\to 1^+$. That is, the local integrability exponent in the reverse Hölder inequality blows up as the weight becomes nearly constant. This is expressed in a precise and explicit computation of the constants involved in the reverse Hölder inequality. The proofs avoid BMO methods and rely instead on precise covering arguments. Furthermore, in the one-dimensional case we prove sharp reverse Hölder inequalities for one-sided and two sided weights in the sense that both the integrability exponent as well as the multiplicative constant appearing in the estimate are best possible. We also prove sharp endpoint weak-type reverse Hölder inequalities and consider further extensions to general non-doubling measures and multiparameter weights.

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arXiv2015

Reverse Hölder Property for strong weights and general measures

We present dimension-free reverse Hölder inequalities for strong $A^*_p$ weights, $1\le p < \infty$. We also provide a proof for the full range of local integrability of $A_1^*$ weights. The common ingredient is a multidimensional version of Riesz's "rising sun" lemma. Our results are valid for any nonnegative Radon measure with no atoms. For $p=\infty$, we also provide a reverse Hölder inequality for certain product measures. As a corollary we derive mixed $A_p^*-A_\infty^*$ weighted estimates.

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arXiv2013

Optimal exponents in weighted estimates without examples

We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator $T$ satisfies a bound like $$ \|T\|_{L^{p}(w)}\le c\, [w]^β_{A_p} \qquad w \in A_{p}, $$ then the optimal lower bound for $β$ is closely related to the asymptotic behaviour of the unweighted $L^p$ norm $\|T\|_{L^p(\mathbb{R}^n)}$ as $p$ goes to 1 and $+\infty$, which is related to Yano's classical extrapolation theorem. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a wide class of operators including maximal-type, Calderón--Zygmund and fractional operators. In particular, we obtain a lower bound for the best possible exponent for Bochner-Riesz multipliers. We also present a new result concerning a continuum family of maximal operators on the scale of logarithmic Orlicz functions. Further, our method allows to consider in a unified way maximal operators defined over very general Muckenhoupt bases.

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arXiv2013

Refined size estimates for Furstenberg sets via Hausdorff measures: a survey of some recent results

In this survey we collect and discuss some recent results on the so called "Furstenberg set problem", which in its classical form concerns the estimates of the Hausdorff dimension of planar sets containing, for any direction, a subset of an interval poitning in that direction of some prescribed dimension. This problem is closely related to the "Kakeya needle problem". In this work we approach this problem from a more general point of view, in terms of generalized Hausdorff measures associated to dimension functions. We generalize the known results in terms of "logarithmic gaps" and obtain analogues to the classical estimates. Moreover, these analogues allow us to extend our results to the zero dimensional endpoint. We also obtain results about the dimension of a variation of Furstenberg sets defined for a fractal set of directions. We prove analogous inequalities reflecting the interplay between the size of the set of directions and the size of each fiber. This problem is also studied in the general scenario of Hausdorff measures.

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