# David Cruz-UribeUniversity of Alabama | UA · Department of Mathematics

David Cruz-Uribe

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147

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Introduction

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August 2015 - October 2015

July 1996 - July 2015

August 1993 - July 1996

## Publications

Publications (147)

A classical regularity result is that non-negative solutions to the Dirichlet problem Δu=f in a bounded domain Ω, where f∈Lq(Ω), q>n2, satisfy ‖u‖L∞(Ω)≤C‖f‖Lq(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 LA(Ω) that lies strictly between...

We extend the results of [5], where we proved an equivalence between weighted Poincar\'e 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 Poincar\'e inequalities in variable exponent spaces and solutions to a degenerate $p(x)$-Laplacia...

In this paper we consider two weight bump conditions for higher order commutators. Given $b$ and a Calder\'on-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 rep...

We consider weighted norm inequalities for multilinear multipliers whose symbols satisfy a product-type Hörmander condition. Our approach is to consider a more general family of multilinear singular integral operators associated to a family of smooth kernels that satisfy an Lr-Schwartz regularity condition. We give conditions for these operators to...

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.

A classical regularity result is that non-negative solutions to the Dirichlet problem $\Delta u =f$ in a bounded domain $\Omega$, where $f\in L^q(\Omega)$, $q>\frac{n}2$, satisfy $\|u\|_{L^\infty(\Omega)} \leq C\|f\|_{L^q(\Omega)}$. We extend this result in three ways: we replace the Laplacian with a degenerate elliptic operator; we show that we ca...

Given a space of homogeneous type $(X,\mu,d)$, 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-H\"older continuity con...

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 Ap(.)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 au...

We prove necessary conditions on pairs of measures $(\mu,\nu)$ 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 $\mu$ satisfies a weak doubling condition related to the non-degeneracy of the ker...

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 opera...

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,...

We prove quantitative matrix weighted endpoint estimates for the matrix weighted Hardy-Littlewood maximal operator, Calder\'on-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.

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.

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 o...

We characterize the weights for the Stieltjes transform and the Calder\'on 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-H\"older continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator d...

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 fir...

We prove a limited range, off-diagonal extrapolation theorem that generalizes a number of results in the theory of Rubio de Francia extrapolation, and use this to prove a limited range, multilinear extrapolation theorem. We give two applications of this result to the bilinear Hilbert transform. First, we give sufficient conditions on a pair of weig...

In this paper we extend the theory of two weight, Ap bump conditions to the setting of matrix weights. We prove two matrix weight inequalities for fractional maximal operators, fractional and singular integrals, sparse operators and averaging operators. As applications we prove quantitative, one weight estimates, in terms of the matrix Ap constant,...

In this note we prove a modular variable Orlicz inequality for the local maximal operator. This result generalizes several Orlicz and variable exponent modular inequalities that have appeared previously in the literature.

We prove versions of the Rubio de Francia extrapolation theorem in generalized Orlicz spaces. As a consequence, we obtain boundedness results for several classical operators as well as a Sobolev inequality in this setting. We also study complex interpolation in the same setting and use it to derive a compact embedding theorem. Our results include a...

We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of $p$-Laplacian equations. Given $\Omega\subset\mathbb{R}^n$, let $\rho$ be a quasi-metric on $\Omega$, and let $Q$ be an $n\times n$ semi-definite matrix function defined on $\Omega$. For an open set $\Theta\Subset\...

We prove a quantitative estimate on the -harmonic approximation of Sobolev functions. This improves a previous qualitative estimate of the second authors, Stroffolini and Verde.

Given a space of homogeneous type we give sufficient conditions on a variable
exponent {p(.)} so that the fractional maximal operator {M_{\eta}} maps
{L^{p(.)}(X)} to {L^{q(.)}(X)}, where {1/p(.) - 1/q(.) = {\eta}}. In the
endpoint case we also prove the corresponding weak type inequality. As an
application we prove norm inequalities for the fracti...

A now classical result in the theory of variable Lebesgue spaces due to Lerner [A. K. Lerner, On modular inequalities in variable $L^p$ spaces, Archiv der Math. 85 (2005), no. 6, 538-543] is that a modular inequality for the Hardy-Littlewood maximal function in $L^{p(\cdot)}(\mathbb{R}^n)$ holds if and only if the exponent is constant. We generaliz...

We establish the boundedness of the multilinear Calder\'on-Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton (Collect. Math. 2001) and recent work by the third author, Grafakos, Nakamura, and Sawano. As part of our...

We prove an equivalence between weighted Poincare inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p- Laplacian. The Poincare inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-La...

We give new and elementary proofs of one weight norm inequalities for fractional integral operators and commutators. Our proofs are based on the machinery of dyadic grids and sparse operators used in the proof of the A2 conjecture.

We give conditions on the exponent function $p(\cdot)$ that imply the existence of embeddings between grand, small and variable Lebesgue spaces. We construct examples to show that our results are close to optimal. Our work extends recent results by the second author, Rakotoson and Sbordone.

A modestly revised version of lecture notes that were distributed to accompany my four lectures at the 2017 Spring School of Analysis at Paseky, sponsored by Charles University, Prague. They are an introductory survey of Rubio de Francia extrapolation, Jones factorization, and applications.

In this note we prove a multilinear version of the reverse H\"older inequality in the theory of Muckenhoupt $A_p$ weights. We give two applications of this inequality to the study of multilinear weighted norm inequalities. First, we prove a structure theorem for multilinear $A_{\vec{p}}$ weights; second, we give a new sufficient condition for multi...

In this article we extend recent results by the first author on the necessity of $BMO$ for the boundedness of commutators on the classical Lebesgue spaces. We generalize these results to a large class of Banach function spaces. We show that with modest assumptions on the underlying spaces and on the operator $T$, if the commutator $[b,T]$ is bounde...

In these lecture notes we describe some recent work on two weight norm inequalities for fractional integral operators, also known as Riesz potentials, and for commutators of fractional integrals. These notes are based on three lectures delivered at the 6th International Course of Mathematical Analysis in Andalucia, held in Antequera, Spain, Septemb...

We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calder\'on-Zygmund operators. Namely, given $1<p<q<\infty$ and a pair of weights $(u,v)$, if the Hardy-Littlewood maximal function satisfies the following two weight inequalities: $$ M : L^p(v) \rightarrow L^q(u) \quad \text{and} \quad M: L^{q'}(u^{1-q'}) \rightarrow L^{p'}(v^{1-p'}), $$...

We consider a well known calculus question, and show that the solution of this problem is equivalent to finding integer solutions to a Diophantine equation. We generalize the calculus question, which in turn leads to a more general Diophantine equation. We give solutions to all of these and describe some of the historical background.

We develop a theory of extrapolation for weights that satisfy a generalized reverse H\"older inequality in the scale of Orlicz spaces. This extends previous results by Auscher and Martell [2] on limited range extrapolation. As an application, we show that a number of weighted norm inequalities for linear and bilinear operators can be proved using o...

We compute the right and left democracy functions of admissible wavelet bases in variable Lebesgue spaces defined on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\...

We prove fractional Leibniz rules and related commutator estimates in the settings of weighted and variable Lebesgue spaces. Our main tools are uniform weighted estimates for sequences of square-function-type operators and a bilinear extrapolation theorem. We also give applications of the extrapolation theorem to the boundedness on variable Lebesgu...

We study degenerate Sobolev spaces where the degeneracy is controlled by a matrix (Formula presented.) weight. This class of weights was introduced by Nazarov, Treil and Volberg, and degenerate Sobolev spaces with matrix weights have been considered by several authors for their applications to PDEs. We prove that the classical Meyers–Serrin theorem...

In this article we study the Kato problem for degenerate divergence form
operators, This was begun by Cruz-Uribe and Rios who proved that given an
operator $L_w=-w^{-1}{\rm div}(A\nabla)$, where $w\in A_2$ and $A$ is a
$w$-degenerate elliptic measure (i.e, $A=w\,B$ with $B$ an $n\times n$ bounded,
complex-valued, uniformly elliptic matrix), then $L...

We consider second order operators ℒw = −w−1 div Aw ∇ with ellipticity controlled by a Muckemphout A2 weight w. We prove that the Kato square root estimate holds in the weighted space ℒ2 (w).

We give a necessary and sufficient condition on the exponent function p(⋅) for approximate identities to converge in measure in the variable Lebesgue spaces.

We study degenerate Sobolev spaces where the degeneracy is controlled by a
matrix $A_p$ weight. This class of weights was introduced by Nazarov, Treil and
Volberg, and degenerate Sobolev spaces with matrix weights have been considered
by several authors for their applications to PDEs. We prove that the classical
Meyers-Serrin theorem, H = W, holds...

We establish two-weight norm inequalities for singular integral operators defined on spaces of homogeneous type. We do so first when the weights satisfy a double bump condition and then when the weights satisfy separated logarithmic bump conditions. Our results generalize recent work on the Euclidean case, but our proofs are simpler even in this se...

In these lecture notes we describe some recent work on two weight norm
inequalities for fractional integral operators, also known as Riesz potentials,
and for commutators of fractional integrals. These notes are based on three
lectures delivered at the 6th International Course of Mathematical Analysis in
Andalucia, held in Antequera, Spain, Septemb...

We rectify an error in the proof of the Gaussian estimates for the heat kernel associated to certain weighted elliptic equations.

We prove a priori estimates for strong solutions to the Dirichlet problem for
a divergence form elliptic operator. We give $L^p$ estimates for the second
derivative for $p<2$. Our work generalizes results due to Miranda [24].

We extend the theory of Rubio de Francia extrapolation, including
off-diagonal, limited range, and $A_{\infty}$ extrapolation, to the weighted
variable Lebesgue spaces. As a consequence we are able to show that a number of
different operators from harmonic analysis are bounded on these spaces. The
proofs of our extrapolation results are developed i...

We begin with an intuitive introduction to the variable Lebesgue spaces, briefly sketch their history, and give some of the contemporary motivations for studying these spaces.

Before we pursue the analysis of equations and systems with time-dependent coefficients, it is instructive to understand what happens in the case of equations with constant coefficients. One of the very helpful observations available in this case is that after a Fourier transform in the spatial variable x we obtain an ordinary differential equation...

If lower order terms are too large to be controlled, it becomes important to investigate the behaviour of solutions for bounded frequencies. We will restrict ourselves to situations where an asymptotic construction for ξ → 0 becomes important and provide some essential estimates for this.

In this chapter we will provide a diagonalisation based approach to obtain the high-frequency asymptotic properties of the representation of solutions for more general uniformly strictly hyperbolic systems. The exposition is based on ideas from the authors’ paper [52].

We now look at equations with time-dependent coefficients. In this chapter we will review two scale invariant model cases, which can be treated by means of special functions. They both highlight a structural change in the behaviour of solutions when lower order terms become effective. This change is a true variable coefficient phenomenon, it can no...

Most of the results presented here were based on diagonalisation procedures in order to deduce asymptotic information on the representations of solutions. This is natural and has a long history in the study of hyperbolic equations and coupled systems. For diagonalisation schemes in broader sense and their application we also refer to [21]. Some mor...

In this chapter we develop the function space properties of variable Lebesgue spaces. We begin with the basic properties and notation for exponent functions. We then define the modular and the norm, and prove that L
p(.) is a Banach space. We prove a version of Hölder’s inequality, define the associate norm, and then characterize the dual space whe...

Both in Chapters 4 and 5 we made symbol like assumptions on coefficients, e.g., we considered hyperbolic systems $${\rm{D}}_tU = {\sum \limits_{k=1}^{n}}A_k(t){{\rm D}_{x_k}}U$$ (6.0.1) with coefficient matrices \(A_k(t)\in\mathcal{T}\left\{{0}\right\}\), meaning that derivatives of the coefficients are controlled by $$\|{\rm D}_t^lA_k(t)\|\leq \,...

In this chapter we develop a general theory for proving norm inequalities for the other classical operators in harmonic analysis. Our main result is a powerful generalization of the Rubio de Francia extrapolation theorem. This approach, first developed in [22] and then treated as part of a more general framework in [27], lets us use the theory of w...

We establish two-weight norm inequalities for singular integral operators
defined on spaces of homogeneous type. We do so first when the weights satisfy
a double bump condition and then when the weights satisfy separated logarithmic
bump conditions. Our results generalize recent work on the Euclidean case, but
our proofs are simpler even in this se...

In the 1970s Muckenhoupt and Wheeden made several conjectures relating two weight norm inequalities for the Hardy-Littlewood maximal operator to such inequalities for singular integrals. Using techniques developed for the recent proof of the A
2 conjecture we prove a related pair of conjectures linking the Riesz potential and the fractional maximal...

This book provides an accessible introduction to the theory of variable Lebesgue spaces. These spaces generalize the classical Lebesgue spaces by replacing the constant exponent p with a variable exponent p(x). They were introduced in the early 1930s but have become the focus of renewed interest since the early 1990s because of their connection wit...

In this chapter we consider some of the classical operators of harmonic analysis: convolution operators, singular integral operators, and Riesz potentials. Rather than treat each operator separately, we develop a general theory that builds upon the Rubio de Francia theory of extrapolation from the theory of weighted norm inequalities. The advantage...

In this chapter we turn to the study of harmonic analysis on the variable Lebesgue spaces. Our goal is to establish sufficient conditions for the Hardy–Littlewood maximal operator to be bounded on L
p(.); in the next chapter we will show how this can be used to prove norm inequalities on L
p(.) for the other classical operators of harmonic analysis...

In this chapter we continue our study of the Hardy-Littlewood maximal operator. In Chap. 3 we showed that the log Hölder continuity conditions LH 0 and LH ∞ are sufficient for the maximal operator to be bounded. In this chapter we will show that they are not necessary, even though they are the best possible pointwise decay conditions. To find weake...

In this chapter we give a precise definition of the variable Lebesgue spaces and establish their structural properties as Banach function spaces. Throughout this chapter we will generally assume that \(\Omega \) is a Lebesgue measurable subset of \({\mathbb{R}}^{n}\) with positive measure. Occasionally we will have to assume more, but we make it ex...

In this chapter we present the elementary theory of variable Sobolev spaces. Unlike Chap. 2, where we systematically developed a complete theory of variable Lebesgue spaces, our goal here is less ambitious. Our aim is to illustrate how the theorems and techniques given in previous chapters can be applied to the variable Sobolev spaces. Consequently...

We develop the theory of variable exponent Hardy spaces. Analogous to the
classical theory, we give equivalent definitions in terms of maximal operators.
We also show that distributions in these spaces have an atomic decomposition
including a "finite" decomposition; this decomposition is more like the
decomposition for weighted Hardy spaces due to...

We prove weighted strong and weak-type norm inequalities for the Hardy–Littlewood maximal operator on the variable Lebesgue space Lp(⋅)Lp(⋅). Our results generalize both the classical weighted norm inequalities on LpLp and the more recent results on the boundedness of the maximal operator on variable Lebesgue spaces.

We consider weighted norm inequalities for the Riesz potentials $I_\alpha$,
also referred to as fractional integral operators. First we prove mixed
$A_p$-$A_\infty$ type estimates in the spirit of [13, 15, 17]. Then we prove
strong and weak type inequalities in the case $p<q$ using the so-called log
bump conditions. These results complement the str...

We prove an interpolation theorem for integral operators with positive kernel on the variable Lebesgue spaces. As an application we show that the set of exponents for spaces on which the Hardy-Littlewood maximal operator is bounded is convex.

We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for
Calder\'on-Zygmund operators. Namely, given $1<p<q<\infty$ and a pair of
weights $(u,v)$, if the Hardy-Littlewood maximal function satisfies the
following two weight inequalities: $$ M : L^p(v) \rightarrow L^q(u) \quad
\text{and} \quad M: L^{q'}(u^{1-q'}) \rightarrow L^{p'}(v^{1-p'}), $$...

In this paper we establish a continuity result for local minimizers of some quasilinear functionals that satisfy degenerate elliptic bounds. The non-negative function which measures the degree of degeneracy is assumed to be exponentially integrable. The minimizers are shown to have a modulus of continuity controlled by log log(1/|x|)−1. Our proof a...

We prove that if a pair of weights $(u,v)$ satisfies a sharp $A_p$-bump
condition in the scale of log bumps and certain loglog bumps, then Haar shifts
map $L^p(v)$ into $L^p(u)$ with a constant quadratic in the complexity of the
shift. This in turn implies the two weight boundedness for all
Calder\'on-Zygmund operators. This gives a partial answer...

We prove regularity results for solutions of the equation \[div(< AXu,X
u>^{(p-2)/2} AX u) = 0,\] $1<p<\infty$, where $X=(X_1,...,X_m)$ is a family of
vector fields satisfying H\"ormander's ellipticity condition, $A$ is an
$m\times m$ symmetric matrix that satisfies degenerate ellipticity conditions.
If the degeneracy is of the form \[\lambda w(x)^...

We study the boundedness of the maximal operator on the weighted variable exponent Lebesgue spaces L
ωp(·) (Ω). For a given log-Hölder continuous exponent p with 1 < inf p ⩽ supp < ∞ we present a necessary and sufficient condition on the weight ω for the boundedness of M. This condition is a generalization of the classical Muckenhoupt condition.

We study the Hardy type, two-weight inequality for the multidimensional Hardy operator in the variable exponent Lebesgue space
L
p(.)(ℝ
n
). We prove equivalent conditions for L
p(.)→L
q(.) boundness of the Hardy operator in the case of so called “mixed” exponents: q(0)≥p(0), q(∞)p(∞) or q(0)p(0), q(∞)≥p(∞). We show that a necessary and suffici...

Our primary goal in this chapter is to define the appropriate analogs of A
1 weights and prove a “reverse factorization” property for pairs of weights that satisfy the A
p bump conditions defined in Chapter 5. As we discussed in Chapter 2, reverse factorization is an essential tool in the proof
of the Rubio de Francia extrapolation theorem; in the...

The extrapolation theorem of Rubio de Francia is one of the deepest results in the study of weighted norm inequalities in
harmonic analysis: it is simple to state but has profound and diverse applications. The goal of this book is to give a systematic
development of the theory of extrapolation, one which unifies known results and expands them in ne...

In this chapter we give our new proof of the Rubio de Francia extrapolation theorem, Theorem 1.4, and discuss how our proof
allows a number of powerful generalizations. For the convenience of the reader we restate it here.

In this chapter we prove our fundamental generalization of the Rubio de Francia extrapolation theorem. Our result combines
the two most important generalizations we discussed in Chapter 2: generalized maximal operators and the elimination of the
operator. We then consider the results we get by rescaling, particularly A
∞ extrapolation. Next, we pro...

In this chapter we continue to apply the extrapolation theorems in Chapters 7 and 8 to the theory of two-weight norm inequalities.
We consider two operators: the dyadic square function and the vector-valued maximal operator. There are two different approaches
to these operators. On the one hand, they can be treated as vector-valued singular integra...

In this chapter we consider further variations of the two-weight extrapolation theorem proved in Chapter 7.

In this chapter we extend our theory of extrapolation to get norm inequalities on Banach function spaces starting from inequalities
in weighted L
p.

In this chapter we give our generalization of the Rubio de Francia extrapolation theorem to the two-weight setting.

In this chapter we gather together many of the basic facts we will use throughout Part II: Orlicz spaces, A
p-type conditions, maximal operators, and fractional maximal operators. Fractional maximal operators are treated separately even though a unified presentation is possible; there is some duplication but we believe that the exposition is made c...

The local behavior of solutions to a degenerate elliptic equation where andw(x)2|ξ|⩽〈A(x)ξ,ξ〉⩽v(x)2|ξ| for weights w(x)⩾0 and v(x), has been studied by Chanillo and Wheeden. In Chanillo and Wheeden (1986) [7], they generalize the results of Fabes, Kenig, and Serapioni (1961) [8] relative to the case v(x)=Λw(x).We consider the case where and v(x)=K(...

Preface.- Preliminaries.- Part I. One-Weight Extrapolation.- Chapter 1. Introduction to Norm Inequalities and Extrapolation.- Chapter 2. The Essential Theorem.- Chapter 3. Extrapolation for Muckenhoupt Bases.- Chapter 4. Extrapolation on Function Spaces.- Part II. Two-Weight Factorization and Extrapolation.- Chapter 5. Preliminary Results.- Chapter...

We prove several sharp weighted norm inequalities for commutators of
classical operators in harmonic analysis. We find sufficient $A_p$-bump
conditions on pairs of weights $(u,v)$ such that $[b,T]$, $b\in BMO$ and $T$ a
singular integral operator (such as the Hilbert or Riesz transforms), maps
$L^p(v)$ into $L^p(u)$. Because of the added degree of...

We consider the relationship in the variable Lebesgue space $L^{p(\cdot)}(\Omega)$ between convergence in norm, convergence in modular, and convergence in measure, for both bounded and unbounded exponent functions.

We give a new proof of the sharp weighted $L^2$ inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent re...

We give a new proof of the sharp one weight $L^p$ inequality for any operator
$T$ that can be approximated by Haar shift operators such as the Hilbert
transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids
the Bellman function technique and two weight norm inequalities. We use instead
a recent result due to A. Lerner to est...

We prove the Kato conjecture for degenerate elliptic operators on R-n. More precisely, we consider the divergence form operator L-w = -w(-1) divA del, where w is a Muckenhoupt A(2) weight and A is a complex-valued n x n matrix such that w(-1)A is bounded and uniformly elliptic. We show that if the heat kernel of the associated semigroup e(-tLw) sat...