arXiv subjects

David Cruz-Uribe

Publications and source records attributed to David Cruz-Uribe.

At least 37 records · Page 2Linked to original sources

arXiv2024

Matrix weights, singular integrals, Jones factorization and Rubio de Francia extrapolation

In this article we give an overview of the problem of finding sharp constants in matrix weighted norm inequalities for singular integrals, the so-called matrix A2 conjecture. We begin by reviewing the history of the problem in the scalar case, including a sketch of the proof of the scalar A2 conjecture. We then discuss the original, qualitative results for singular integrals with matrix weights and the best known quantitative estimates. We give an overview of new results by the author and Bownik, who developed a theory of harmonic analysis on convex set-valued functions. This led to the proof the Jones factorization theorem and the Rubio de Francia extrapolation theorem for matrix weights, two longstanding problems. Rubio de Francia extrapolation was expected to be a major tool in the proof of the matrix A2 conjecture; however, this conjecture was very recently proved false. We discuss this problem.

math.CA↗

arXiv2023

Weighted weak-type inequalities for maximal operators and singular integrals

We prove quantitative, one-weight, weak-type estimates for maximal operators, singular integrals, fractional maximal operators and fractional integral operators. We consider a kind of weak-type inequality that was first studied by Muckenhoupt and Wheeden and later by Cruz-Uribe, Martell and Perez. We obtain quantitative estimates for these operators in both the scalar and matrix weighted setting using sparse domination techniques. Our results extend those obtained by Cruz-Uribe, Isralowitz, Moen, Pott, and Rivera-Ríos for singular integrals and maximal operators when $p=1$.

math.CA↗

arXiv2023

Convolution Operators in Matrix Weighted, Variable Lebesgue Spaces

We extend the theory of matrix weights to the variable Lebesgue spaces. The theory of matrix $\mathcal{A}_p$ weights has attracted considerable attention beginning with the work of Nazarov, Treil, and Volberg in the 1990s. We extend this theory by generalizing the matrix $\mathcal{A}_p$ condition to the variable exponent setting. We prove boundedness of the convolution operator $\mathbf{f}\mapsto ϕ\ast \mathbf{F}$ for $ϕ\in C_c^\infty(Ω)$, and show that the approximate identity defined using $ϕ$ converges in matrix weighted, variable Lebesgue spaces $L^{p(\cdot)}(W,Ω)$ for $W$ in matrix $\mathcal{A}_{p(\cdot)}$. Our approach to this problem is through averaging operators; these results are of interest in their own right. As an application of our work, we prove a version of the classical $H=W$ theorem for matrix weighted, variable exponent Sobolev spaces.

math.CA↗

arXiv2023

Sobolev meets Riesz: a characterization of weighted Sobolev spaces via weighted Riesz bounded variation space

We introduce weighted Riesz bounded variation spaces defined on an open subset of the $n$-dimensional Euclidean space and use them to characterize weighted Sobolev spaces when the weight belongs to the Muckenhoupt class. As an application, using Rubio de Francia's extrapolation theory, a similar characterization of the variable exponent Sobolev spaces via variable exponent Riesz bounded variation spaces is obtained.

math.CA↗

arXiv2023

Hardy-Leray inequalities in variable Lebesgue spaces

In this paper, we prove the Hardy-Leray inequality and related inequalities in variable Lebesgue spaces. Our proof is based on a version of the Stein-Weiss inequality in variable Lebesgue spaces derived from two weight inequalities due to Melchiori and Pradolini. We also discuss an application of our results to establish an existence result for the degenerate $\pp$-Laplace operator.

math.CA↗

arXiv2022

The ε-Maximal Operator and Haar Multipliers on Variable Lebesgue Spaces

C. Stockdale, P. Villarroya, and B. Wick introduced the $ε$-maximal operator to prove the Haar multiplier is bounded on the weighted spaces $L^p(w)$ for a class of weights larger than $A_p$. We prove the $ε$-maximal operator and Haar multiplier are bounded on variable Lebesgue spaces $\Lpp(\R^n)$ for a larger collection of exponent functions than the log-Holder continuous functions used to prove the boundedness of the maximal operator on $\Lpp(\R^n)$. We also prove that the Haar multiplier is compact when restricted to a dyadic cube $Q_0$.

math.CA↗

arXiv2021

Poincaré Inequalities and Neumann Problems for the Variable Exponent Setting

We extend the results of [5], where we proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $p$-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $p(x)$-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.

math.AP↗

arXiv2021

New oscillation classes and two weight bump conditions for commutators

In this paper we consider two weight bump conditions for higher order commutators. Given $b$ and a Calderón-Zygmund operator $T$, define the commutator $T^1_bf=[T,b]f= bTf-T(bf)$, and for $m\geq 2$ define the iterated commutator $T^m_b f = [b,T_b^{m-1}]f$. Traditionally, commutators are defined for functions $b\in BMO$, but we show that if we replace $BMO$ by an oscillation class first introduced by Pérez [31], we can give a range of sufficient conditions on a pair of weights $(u,v)$ for $T^m_b : L^p(v)\rightarrow L^p(u)$ to be bounded. Our results generalize work of the first two authors in [10], and more recent work by Lerner, et al. [28]. We also prove necessary conditions for the iterated commutators to be bounded, generalizing results of Isralowitz, et al.[20].

math.CA↗

arXiv2020

A note on the limit of Orlicz norms

We generalize the well-known inequality that the limit of the $L^p$ norm of a function as $p\rightarrow\infty$ is the $L^\infty$ norm to the scale of Orlicz spaces.

math.CA↗

arXiv2020

Bounded weak solutions to elliptic PDE with data in Orlicz spaces

A classical regularity result is that non-negative solutions to the Dirichlet problem $Δu =f$ in a bounded domain $Ω$, where $f\in L^q(Ω)$, $q>\frac{n}2$, satisfy $\|u\|_{L^\infty(Ω)} \leq C\|f\|_{L^q(Ω)}$. We extend this result in three ways: we replace the Laplacian with a degenerate elliptic operator; we show that we can take the data $f$ in an Orlicz space $L^A(Ω)$ that lies strictly between $L^{\frac{n}{2}}(Ω)$ and $L^q(Ω)$, $q>\frac{n}2$; and we show that that we can replace the $L^A$ norm in the right-hand side by a smaller expression involving the logarithm of the "entropy bump" $\|f\|_{L^A(Ω)}/\|f\|_{L^{\frac{n}{2}}(Ω)}$, generalizing a result due to Xu.

math.AP↗

arXiv2020

Weighted norm inequalities for the maximal operator on $\Lpp$ over spaces of homogeneous type

Given a space of homogeneous type $(X,μ,d)$, we prove strong-type weighted norm inequalities for the Hardy-Littlewood maximal operator over the variable exponent Lebesgue spaces $L^\pp$. We prove that the variable Muckenhoupt condition $\App$ is necessary and sufficient for the strong type inequality if $\pp$ satisfies log-Hölder continuity conditions and $1 < p_- \leq p_+ < \infty$. Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved by Cruz-Uribe, Fiorenza and Neugebuaer (2012).

math.CA↗

arXiv2020

Neccessary conditions for two weight inequalities for singular integral operators

We prove necessary conditions on pairs of measures $(μ,ν)$ for a singular integral operator $T$ to satisfy weak $(p,p)$ inequalities, $1\leq p<\infty$, provided the kernel of $T$ satisfies a weak non-degeneracy condition first introduced by Stein, and the measure $μ$ satisfies a weak doubling condition related to the non-degeneracy of the kernel. We also show similar results for pairs of measures $(μ,σ)$ for the operator $T_σf = T(f\,dσ)$, which has come to play an important role in the study of weighted norm inequalities. Our major tool is a careful analysis of the strong type inequalities for averaging operators; these results are of interest in their own right. Finally, as an application of our techniques, we show that in general a singular operator does not satisfy the endpoint strong type inequality $T : L^1(ν) \rightarrow L^1(μ)$. Our results unify and extend a number of known results.

math.CA↗

arXiv2019

On the dual of variable Lebesgue spaces with unbounded exponent

We study the dual space of the variable Lebesgue space $\Lp$ with unbounded exponent function $\pp$ and provide an answer to a question posed in~[fiorenza-cruzuribe2013]. Our approach is to decompose the dual into a topological direct sum of Banach spaces. The first component corresponds to the dual in the bounded exponent case, and the second is, intuitively, the dual of functions that live where the exponent is unbounded (in a heuristic sense). The second space is extremely complicated, and we illustrate this with a series of examples. In the special case of the variable sequence space $\ell^\pp$, we show that this piece can be further decomposed into two spaces, one of which can be characterized in terms of a generalization of finitely additive measures. As part of our work, we also considered the question of dense subsets in $\Lp$ for unbounded exponents. We constructed two examples, one for general variable Lebesgue spaces $\Lp$ and one in the sequence space $\lp$. This gives an answer to another question from~[fiorenza-cruzuribe2013].

math.CA↗

arXiv2019

Weighted norm inequalities for the bilinear maximal operator on variable Lebesgue spaces

We extend the theory of weighted norm inequalities on variable Lebesgue spaces to the case of bilinear operators. We introduce a bilinear version of the variable $\A_\pp$ condition, and show that it is necessary and sufficient for the bilinear maximal operator to satisfy a weighted norm inequality. Our work generalizes the linear results of the first author, Fiorenza and Neugebauer \cite{dcu-f-nPreprint2010} in the variable Lebesgue spaces and the bilinear results of Lerner {\em et al.} \cite{MR2483720} in the classical Lebesgue spaces. As an application we prove weighted norm inequalities for bilinear singular integral operators in the variable Lebesgue spaces.

math.CA↗

arXiv2019

Weak endpoint bounds for matrix weights

We prove quantitative matrix weighted endpoint estimates for the matrix weighted Hardy-Littlewood maximal operator, Calderón-Zygmund operators, and commutators of CZOs with scalar BMO functions, when the matrix weight is in the class $A_1$ introduced by M.~Frazier and S.~Roudenko.

math.CA↗

arXiv2019

Multilinear fractional Calderón-Zygmund operators on weighted Hardy spaces

We prove norm estimates for multilinear fractional integrals acting on weighted and variable Hardy spaces. In the weighted case we develop ideas we used for multilinear singular integrals [7]. For the variable exponent case, a key element of our proof is a new multilinear, off-diagonal version of the Rubio de Francia extrapolation theorem.

math.CA↗

arXiv2019

A new approach to norm inequalities on weighted and variable Hardy spaces

We give new proofs of Hardy space estimates for fractional and singular integral operators on weighted and variable exponent Hardy spaces. Our proofs consist of several interlocking ideas: finite atomic decompositions in terms of $L^\infty$ atoms, vector-valued inequalities for maximal and other operators, and Rubio de Francia extrapolation. Many of these estimates are not new, but we give new and substantially simpler proofs, which in turn significantly simplifies the proofs of the Hardy spaces inequalities.

math.CA↗

arXiv2019

The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights

We characterize the weights for the Stieltjes transform and the Calderón operator to be bounded on the weighted variable Lebesgue spaces $L_w^{p(\cdot)}(0,\infty)$, assuming that the exponent function $p(\cdot)$ is log-Hölder continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervals $\{ (0,b) : b>0\}$ on $(0,\infty)$. Our results extend those in \cite{DMRO1} for the constant exponent $L^p$ spaces with weights. We also give two applications: the first is a weighted version of Hilbert's inequality on variable Lebesgue spaces, and the second generalizes the results in \cite{SW} for integral operators to the variable exponent setting.

math.CA↗