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Wilfredo Urbina

Publications and source records attributed to Wilfredo Urbina.

18 recordsLinked to original sources
arXiv2025

On Tent Spaces for the Gaussian Measure

Following the scheme of tent spaces in classical harmonic analysis developed by R. Coifman, Y. Meyer, and E. Stein in \cite{cms}, we succeed in doing so for the Gaussian setting. In \cite{MNP}, part of this theory (an atomic decomposition) is developed for a specific tent space where functions are defined just in a proper subset of $\mathbb{R}^{n+1}_+,$ and without the use of an area function. In the present paper, using a variation of the area function considered in \cite{FSU}, we define the Gaussian area function and Gaussian tent spaces and prove both their atomic decompositions and the characterization of their dual spaces. Some applications are also considered.

math.AP↗
arXiv2021

The Boundedness of General Alternative Gaussian Singular Integrals on variable Lebesgue spaces with Gaussian measure

In a previous paper, we introduced a new class of Gaussian singular integrals, that we called the general alternative Gaussian singular integrals and study the boundedness of them on $L^p(γ_d)$, $ 1 < p < \infty.$ In this paper, we study the boundedness of those operators on Gaussian variable Lebesgue spaces under a certain additional condition of regularity on $p(\cdot)$ following a paper by E. Dalmasso and R. Scotto.

math.CA↗
arXiv2019

The Boundedness of the Ornstein-Uhlenbeck semigroup on variable Lebesgue spaces with respect to the Gaussian measure

The main result of this work is the proof of the boundedness of the Ornstein-Uhlenbeck semigroup $ \{T_t \}_{t\geq 0} $ in $ {\mathbb R}^d $ on Gaussian variable Lebesgue spaces under a condition of regularity on $p(\cdot)$ following previous papers by E. Dalmaso R. Scotto and S. Pérez. As a consequence of this result, we obtain the boundedness of Poisson-Hermite semigroup and the boundedness of the Gaussian Bessel potentials of order $β> 0$.

math.CA↗
arXiv2015

New real variable methods in H summability of Fourier series

In this paper we shall be concerned with $H_α$ summability, for $0 < α\leq 2$ of the Fourier series of arbitrary $L^1([-π,π]) $ functions. The methods to be employed here are a refinement of the real variable methods introduced by Marcinkiewicz in \cite{Marcin1}. In addition, we introduce maximal theorems with respect to the Lebesgue measure and $A_1$ weights.

math.CA↗
arXiv2014

On Cantor-like sets and Cantor-Lebesgue singular functions

In this paper we discuss several variations and generalizations of the Cantor set and study some of their properties. Also for each of those generalizations a Cantor-like function can be constructed from the set. We will discuss briefly the possible construction of those functions.

math.CA↗
arXiv2012

Riesz Potentials, Bessel Potentials and Fractional Derivatives on Triebel-Lizorkin spaces for the Gaussian Measure

In a previous paper the boundedness properties of Riesz Potentials, Bessel potentials and Fractional Derivatives were studied in detail on Gaussian Besov-Lipschitz spaces $B_{p,q}^α(γ_d)$. In this paper we will continue our study proving the boundedness of those operators on Gaussian Triebel-Lizorkin spaces $F_{p,q}^α(γ_d)$. Also these results can be extended to the case of Laguerre or Jacobi expansions and even further to the general framework of diffusions semigroups.

math.CA↗
arXiv2012

On Abel summability of Jacobi polynomials series, the Watson Kernel and applications

In this paper we return to the study of the Watson kernel for the Abel summabilty of Jacobi polynomial series. These estimates have been studied for over more than 30 years. The main innovations are in the techniques used to get the estimates that allow us to handle the case 0<αas well as -1< α<0, with essentially the same method; using an integral superposition of Poisson type kernel and Muckenhoupt A_p-weight theory. We consider a generalization of a theorem due to Zygmund in the context to Borel measures. The proofs are therefore different from the ones given in previous papers by several authors. We will also discuss in detail the Calderón-Zygmund decomposition for non-atomic Borel measures in the real line. Then, we prove that the Jacobi measure is doubling and therefore, following a work of A. P. Calderón, we study the corresponding A_p weight theory in the setting of Jacobi expansions, considering power weights of the form (1-x)^{\barα}, (1+x)^{\barβ}, -1 < {\barα}<0,\, -1 < {\barβ}<0 with negative exponents. Finally, as an application of the weight theory we obtain L^p estimates for the maximal operator of Abel summability of Jacobi function expansions for suitable values of p.

math.CA↗
arXiv2012

On the Theta semigroup

In this paper we consider a semigroup on trigonometric expansions that will be called the Theta semigroup since its kernel is a multiple of the third Jacobi theta function. We study properties of this semigroup and prove that it is a positive diffusion semigroup. We also obtain that its subordinated semigroup is the classical Poisson semigroup. The extensions to higher dimensions and to periodic ultra distributions are also considered.

math.CA↗
arXiv2008

Some results on Gaussian Besov-Lipschitz spaces and Gaussian Triebel-Lizorkin spaces

In this paper we define Besov-Lipschitz and Triebel-Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein-Uhlenbeck and the Poisson-Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces $L^p_α(γ_d)$ are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in the general framework of diffusions semigroups.

math.CA↗
arXiv2007

On Polar Legendre Polynomials

We introduce a new class of polynomials $\{P_{n}\}$, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with $n+1$ unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a differential operator and a discrete-continuous Sobolev type inner product.

math.CA↗
arXiv2007

Fractional Integration and Fractional Differentiation for d-dimensional Jacobi Expansions

In this paper we consider an alternative orthogonal decomposition of the space $L^2$ associated to the $d$-dimensional Jacobi measure and obtain an analogous result to P.A. Meyer's Multipliers Theorem for d-dimensional Jacobi expansions. Then we define and study the Fractional Integral, the Fractional Derivative and the Bessel potentials induced by the Jacobi operator. We also obtain a characterization of the potential spaces and a version of Calderon's reproduction formula for the d-dimensional Jacobi measure.

math.AP↗
arXiv2006

Non Tangential Convergence for the Ornstein-Uhlenbeck Semigroup

In this paper we are going to get the non tangential convergence, in an appropriated parabolic "gaussian cone", of the Ornstein-Uhlenbeck semigroup in providing two proofs of this fact. One is a direct proof by using the truncated non tangential maximal function associated. The second one is obtained by using a general statement. This second proof also allows us to get a similar result for the Poisson-Hermite semigroup.

math.CA↗
arXiv2006

Controllability of the Laguerre and the Jacobi Equations

In this paper we study the controllability of the controlled Laguerre equation and the controlled Jacobi equation. For each case, we found conditions which guarantee when such systems are approximately controllable on the interval $[0, t_1]$. Moreover, we show that these systems can never be exactly controllable

math.CA↗
arXiv2006

On the maximal function for the generalized Ornstein-Uhlenbeck semigroup

In this note we consider the maximal function for the generalized Ornstein-Uhlenbeck semigroup in $\RR$ associated with the generalized Hermite polynomials $\{H_n^μ\}$ and prove that it is weak type (1,1) with respect to $dλ_μ(x) = |x|^{2μ}e^{-|x|^2} dx,$ for $μ>-1/2$ as well as bounded on $L^p(dλ_μ) $ for $p>1$

math.CA↗