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Víctor Almeida

Publications and source records attributed to Víctor Almeida.

15 recordsLinked to original sources
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

Variation and oscillation for harmonic operators in the inverse Gaussian setting

We prove variation and oscillation $L^p$-inequalities associated with fractional derivatives of certain semigroups of operators and with the family of truncations of Riesz transforms in the inverse Gaussian setting. We also study these variational $L^p$-inequalities in a Banach-valued context by considering Banach spaces with the UMD-property and whose martingale cotype is fewer than the variational exponent. We establish $L^p$-boundedness properties for weighted difference involving the semigroups under consideration.

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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

Discrete Hardy spaces and heat semigroup associated with the discrete Laplacian

In this paper we study the behavior of some harmonic analysis operators associated with the discrete Laplacian $Δ_d$ in discrete Hardy spaces $\mathcal H^p(\mathbb Z)$. We prove that the maximal operator and the Littlewood-Paley $g$ function defined by the semigroup generated by $Δ_d$ are bounded from $\mathcal H^p(\mathbb Z)$ into $\ell^p(\mathbb Z)$, $0<p\leq 1$. Also, we establish that every $Δ_d$-spectral multiplier of Laplace transform type is a bounded operator from $\mathcal H^p(\mathbb Z)$ into itself, for every $0<p\leq 1$.

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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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arXiv2013

Riesz transforms and multipliers for the Bessel-Grushin operator

We establish that the spectral multiplier $\frak{M}(G_α)$ associated to the differential operator $$ G_α=- Δ_x +\sum_{j=1}^m{{α_j^2-1/4}\over{x_j^2}}-|x|^2 Δ_y \; \text{on} (0,\infty)^m \times \R^n,$$ which we denominate Bessel-Grushin operator, is of weak type $(1,1)$ provided that $\frak{M}$ is in a suitable local Sobolev space. In order to do this we prove a suitable weighted Plancherel estimate. Also, we study $L^p$-boundedness properties of Riesz transforms associated to $G_α$, in the case $n=1$.

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