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Lourdes Rodríguez-Mesa

Publications and source records attributed to Lourdes Rodríguez-Mesa.

At least 19 recordsLinked to original sources
arXiv2026

Extension theorems for logarithmic Schrödinger and discrete Laplacian operators

In this paper we consider logarithmic operators in two different contexts: the adapted to (continuous) Schrödinger operators and the classical discrete setting. The Schrödinger operator $\mathcal L_V$ on $\mathbb R^d$ is defined as $\mathcal L_V=-Δ+V$, where the potential $V$ is nonnegative and satisfies a reverse Hölder inequality and, as usual, $Δ$ denotes the Euclidean Laplacian, while the discrete Laplacian $Δ_d$ on $\mathbb Z$ is given by $(Δ_df)(n)=f(n+1)-2f(n)+f(n-1)$, $n\in \mathbb Z$. Both logarithmic operators $\log \mathcal L_V$ and $\log (-Δ_d)$ are nonlocal operators and we will define them through suitable extension problems. The extension problems for logarithmic operators are inspired by the one introduced by Caffarelli and Silvestre for the fractional Laplacian but, in this case, the logarithmic operators are obtained as the boundary values of the extension in a more involved way.

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arXiv2024

$L^p$-boundedness properties for some harmonic analysis operators defined by resolvents for a Laplacian with drift in Euclidean spaces

We consider the Laplacian with drift in $\mathbb R^n$ defined by $Δ_ν= \sum_{i=1}^n(\frac{\partial^2}{\partial x_i^2} + 2 ν_i\frac{\partial }{\partial{x_i}})$ where $ν=(ν_1,\ldots,ν_n)\in \mathbb R^n\setminus\{0\}$. The operator $Δ_ν$ is selfadjoint with respect to the measure $dμ_ν(x)=e^{2\langleν,x\rangle}dx$. This measure is not doubling but it is locally doubling in $\mathbb R^n$. We define, for every $M>0$ and $k \in \mathbb N$, the operators $$ W^k_{ν,M,*}(f) = \sup_{t>0}\left|A^k_{ν,M,t}(f)\right|,\hspace{5mm}g_{ν,M}^k(f) = \left(\int_0^\infty\left|A^k_{ν,M,t}(f)\right|^2\frac{dt}{t}\right)^{\frac{1}{2}},\,k\geq 1, $$ the $ρ$-variation operator $$ V_ρ\left( \{A^k_{ν,M,t}\}_{t>0}\right)(f)= \sup_{0 2, $$ and, if $\{t_j\}_{j\in \mathbb N}$ is a decreasing sequence in $(0,\infty)$, the oscillation operator $$ O(\{A_{ν,M,t}^k\}_{t>0},\{t_j\}_{j\in \mathbb N})(f)=\Big(\sum_{j\in \mathbb N}\;\;\sup_{t_{j+1}\leq \varepsilon <\varepsilon '\leq t_j}|A^k_{ν,M,\varepsilon}(f)-A^k_{ν,M,\varepsilon '}(f)|^2 \Big)^{1/2}. $$ where $A^k_{ν,M,t}=t^k\partial^k_t(I-tΔ_ν)^{-M}$, $t>0$. We denote by $T_{ν,M}^k$ any of the above operators. We analyze the boundedness of $T^k_{ν,M}$ on $L^p(\mathbb R^n,μ_ν)$ into itself, for every $1<p<\infty$, and from $L^1(\mathbb R^n,μ_ν)$ into $L^{1,\infty}(\mathbb R^n,μ_ν)$. In addition, we obtain boundedness properties for the operator $G_{ν,M}^{k,\ell}$, $1\leq \ell <2M$, defined by $$ G_{ν,M}^{k,\ell}(f)=\left(\int_0^\infty\left|t^{\ell /2+k}\partial _t^kD^{(\ell)}(I-tΔ_ν)^{-M}(f) \right|^2\frac{dt}{t}\right)^{\frac{1}{2}}, $$ for certain differentiation operator $D^{(\ell)}$.

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arXiv2023

Harmonic analysis operators in the rational Dunkl settings

In this paper we study harmonic analysis operators in Dunkl settings associated with finite reflection groups on Euclidean spaces. We consider maximal operators, Littlewood-Paley functions, $ρ$-variation and oscillation operators involving time derivatives of the heat semigroup generated by Dunkl operators. We establish the boundedness properties of these operators in $L^p(\mathbb R^d,ω_\mathfrak K)$, $1\leq p < 1$, Hardy spaces, BMO and BLO-type spaces in the Dunkl settings. The study of harmonic analysis operators associated to reflection groups need different strategies from the ones used in the Euclidean case since the integral kernels of the operators admit estimations involving two different metrics, namely, the Euclidean and the orbit metrics. For instance, the classical Calderón-Zygmund theory for singular integrals does not work in this setting.

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arXiv2022

Variation and oscillation operators on weighted Morrey-Campanato spaces in the Schrödinger setting

Let $\mathcal{L}$ be the Schrödinger operator with potential $V$, that is, $\mathcal L=-Δ+V$, where it is assumed that $V$ satisfies a reverse Hölder inequality. We consider weighted Morrey-Campanato spaces $BMO_{\mathcal L,w}^α(\mathbb R^d)$ and $BLO_{L,w}^α(\mathbb R^d)$ in the Schrödinger setting. We prove that the variation operator $V_σ(\{T_t\}_{t>0})$, $σ>2$, and the oscillation operator $O(\{T_t\}_{t>0}, \{t_j\}_{j\in \mathbb Z})$, where $t_j 0$, with $k\in \mathbb N$, are bounded operators from $BMO_{\mathcal L,w}^α(\mathbb R^d)$ into $BLO_{\mathcal L,w}^α(\mathbb R^d)$. We also establish the same property for the maximal operators defined by $\{t^k\partial_t^k e^{-t\mathcal L}\}_{t>0}$, $k\in \mathbb N$.

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arXiv2022

Littlewood-Paley functions associated with general Ornstein-Uhlenbeck semigroups

In this paper we establish $L^p(\mathbb{R}^d,γ_\infty)$-boundedness properties for square functions involving time and spatial derivatives of Ornstein-Uhlenbeck semigroups. Here $γ_\infty$ denotes the invariant measure. In order to prove the strong type results for $1<p<\infty$ we use $R$-boundedness. The weak type (1,1) property is established by studying separately global and local operators defined for the square Littlewood-Paley functions. By the way we prove $L^p(\mathbb{R}^d,γ_\infty)$-boundedness properties for maximal and variation operators for Ornstein-Uhlenbeck semigroups.

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arXiv2022

Maximal operator, Littlewood-Paley functions and variation operators associated with nonsymmetric Ornstein-Uhlenbeck operators

In this paper we establish $L^p$ boundedness properties for maximal operators, Littlewood-Paley functions and variation operators involving Poisson semigroups and resolvent operators associated with nonsymmetric Ornstein-Uhlenbeck operators. We consider the Ornstein-Uhlenbeck operators defined by the identity as the covariance matrix and having a drift given by the matrix $-λ(I+R)$, being $λ>0$ and $R$ a skew-adjoint matrix. The semigroup associated with these Ornstein-Uhlenbeck operators are the basic building blocks of all normal Ornstein-Uhlenbeck semigroups.

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arXiv2021

Quantitative weighted estimates for harmonic analysis operators in the Bessel setting by using sparse domination

In this paper we obtain quantitative weighted $L^p$-inequalities for some operators involving Bessel convolutions. We consider maximal operators, Littlewood-Paley functions and variational operators. We obtain $L^p(w)$-operator norms in terms of the $A_p$-characteristic of the weight $w$. In order to do this we show that the operators under consideration are dominated by a suitable family of sparse operators in the space of homogeneous type $((0,\infty),|\cdot |,x^{2λ}dx)$.

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arXiv2021

Variation operators associated with the semigroups generated by Schrödinger operators with inverse square potentials

By $\{T_t^a\}_{t>0}$ we denote the semigroup of operators generated by the Friedrichs extension of the Schrödinger operator with the inverse square potential $L_a=-Δ+\frac{a}{|x|^2}$ defined in the space of smooth functions with compact support in $\mathbb{R}^n\setminus\{0\}$. In this paper we establish weighted $L^p$-inequalities for the maximal, variation, oscillation and jump operators associated with $\{t^α\partial_t^αT_t^a\}_{t>0}$, where $α\geq 0$ and $\partial _t^α$ denotes the Weyl fractional derivative. The range of values $p$ that works is different when $a\geq 0$ and when $-\frac{(n-2)^2}{4}<a<0$.

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arXiv2020

Higher order Riesz transforms in the inverse Gaussian setting and UMD Banach spaces

In this paper we study higher order Riesz transforms associated with the inverse Gaussian measure given by $π^{n/2}e^{|x|^2}dx$ on $\mathbb{R}^n$. We establish $L^p(\mathbb{R}^n,e^{|x|^2}dx)$-boundedness properties and obtain representations as principal values singular integrals for the higher order Riesz transforms. New characterizations of the Banach spaces having the UMD property by means of the Riesz transforms and imaginary powers of the operator involved in the inverse Gaussian setting are given.

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arXiv2019

$L^p$--boundedness of Stein's square functions associated to Fourier--Bessel expansions

In this paper we prove $L^p$ estimates for Stein's square functions associated to Fourier-Bessel expansions. Furthermore we prove transference results for square functions from Fourier-Bessel series to Hankel transforms. Actually, these are transference results for vector-valued multipliers from discrete to continuous in the Bessel setting. As a consequence, we deduce the sharpness of the range of $p$ for the $L^p$-boundedness of Fourier-Bessel Stein's square functions from the corresponding property for Hankel-Stein square functions. Finally, we deduce $L^p$ estimates for Fourier-Bessel multipliers from that ones we have got for our Stein square functions.

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arXiv2018

BMO functions and Balayage of Carleson measures in the Bessel setting

By $BMO_o(R)$ we denote the space consisting of all those odd and bounded mean oscillation functions on R. In this paper we characterize the functions in $BMO_o(R)$ with bounded support as those ones that can be written as a sum of a bounded function on $(0,\infty )$ plus the balayage of a Carleson measure on $(0,\infty )\times (0,\infty )$ with respect to the Poisson semigroup associated with the Bessel operator $B_λ=-x^{-λ}Dx^{2λ}Dx^{-λ}$, $λ>0$. This result can be seen as an extension to Bessel setting of a classical result due to Carleson.

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arXiv2017

Local Hardy spaces with variable exponents associated to non-negative self-adjoint operators satisfying Gaussian estimates

In this paper we introduce variable exponent local Hardy spaces associated with a non-negative self-adjoint operator L. We define them by using an area square integral involving the heat semigroup associated to L. A molecular characterization is established and as an aplication of the molecular characterization we prove that our local Hardy space coincides with the (global) variable exponent Hardy space associated to L, provided that 0 does not belong to the spectrum of L. Also, we show that it coincides with the global variable exponent Hardy space associated to L+I.

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arXiv2017

Variable exponent Hardy spaces associated with discrete Laplacians on graphs

In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.

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arXiv2015

Discrete harmonic analysis associated with ultraspherical expansions

We study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted l^p-boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by certain difference operator. We also prove weighted l^p-boundedness properties of transplantation operators associated to the system of ultraspherical functions. In order to show our results we previously establish a vector-valued local Calderón-Zygmund theorem in our discrete setting.

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arXiv2014

Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions

We consider the Weinstein type equation $\mathcal{L}_λu=0$ on $(0,\infty )\times (0,\infty )$, where $\mathcal{L}_λ=\partial _t^2+\partial _x^2-\frac{λ(λ-1)}{x^2}$, with $λ>1$. In this paper we characterize the solutions of $\mathcal{L}_λu=0$ on $(0,\infty )\times(0,\infty )$ representable by Bessel-Poisson integrals of BMO-functions as those ones satisfying certain Carleson properties.

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arXiv2014

Conical square functions associated with Bessel, Laguerre and Schrödinger operators in UMD Banach spaces

In this paper we consider conical square functions in the Bessel, Laguerre and Schrödinger settings where the functions take values in UMD Banach spaces. Following a recent paper of Hytönen, van Neerven and Portal, in order to define our conical square functions, we use $γ$-radonifying operators. We obtain new equivalent norms in the Lebesgue-Bochner spaces $L^p((0,\infty ),\mathbb{B})$ and $L^p(\mathbb{R}^n,\mathbb{B})$, $1<p<\infty$, in terms of our square functions, provided that $\mathbb{B}$ is a UMD Banach space. Our results can be seen as Banach valued versions of known scalar results for square functions.

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arXiv2012

$γ$-radonifying operators and UMD-valued Littlewood-Paley-Stein functions in the Hermite setting on BMO and Hardy spaces

In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space $\B.$ If we denote by $H$ the Hilbert space $L^2((0,\infty),dt/t),γ(H,\B)$ represents the space of $γ$-radonifying operators from $H$ into $\B.$ We prove that the Hermite square function defines bounded operators from $BMO_\mathcal{L}(\R,\B)$ (respectively, $H^1_\mathcal{L}(\R, \B)$) into $BMO_\mathcal{L}(\R,γ(H,\B))$ (respectively, $H^1_\mathcal{L}(\R, γ(H,\B))$), where $BMO_\mathcal{L}$ and $H^1_\mathcal{L}$ denote $BMO$ and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in $BMO_\mathcal{L}(\R, \B)$ and $H^1_\mathcal{L}(\R,\B)$ by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.

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