Papers by Wilfredo Urbina
Société mathématique de France eBooks, 2012
Contemporary Mathematics, 1995
arXiv (Cornell University), Nov 15, 2022
The ternary Cantor set C, constructed by George Cantor in 1883, is probably the best known exampl... more The ternary Cantor set C, constructed by George Cantor in 1883, is probably the best known example of a perfect nowhere-dense set in the real line, but as we will see later, it is not the only one. The present article will delve into the richness and the peculiarities of C through exploration of several variants and generalizations, and will provide an example of a non-centered asymmetric Cantor-like set.
arXiv (Cornell University), Sep 27, 2012
In [7] the boundedness properties of Riesz Potentials, Bessel potentials and Fractional Derivativ... more In [7] the boundedness properties of Riesz Potentials, Bessel potentials and Fractional Derivatives were studied in detail on Gaussian Besov-Lipschitz spaces B α p,q (γ d). In this paper we will continue our study proving the boundedness of those operators on Gaussian Triebel-Lizorkin spaces F α p,q (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.
arXiv (Cornell University), Feb 26, 2012
In this paper using the well known asymptotic relations between Jacobi polynomials and Hermite an... more In this paper using the well known asymptotic relations between Jacobi polynomials and Hermite and Laguerre polynomials. we develop a transference method to obtain the L p-continuity of the Gaussian-Riesz transform and the L p-continuity of the Laguerre-Riesz transform from the L p-continuity of the Jacobi-Riesz transform, in dimension one.
arXiv (Cornell University), May 23, 2022
In this paper we study the regularity properties of the Gaussian Bessel potentials and Gaussian B... more In this paper we study the regularity properties of the Gaussian Bessel potentials and Gaussian Bessel fractional derivatives on variable Gaussian Besov-Lipschitz spaces B α p(•),q(•) (γ d), that were defined in a previous paper [11], under certain conditions on p(•) and q(•).
Springer eBooks, 2012
In [5] Gaussian Lipschitz spaces Lip α (γ d) were considered and then the boundedness properties ... more In [5] Gaussian Lipschitz spaces Lip α (γ d) were considered and then the boundedness properties of Riesz Potentials, Bessel potentials and Fractional Derivatives were studied in detail. In this paper we will study the boundedness of those operators on Gaussian Besov-Lipschitz spaces B α p,q (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.
arXiv (Cornell University), Jun 20, 2020
In a previous paper [13], we introduced a new class of Gaussian singular integrals, that we calle... more In a previous paper [13], 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 < ∞. In this paper, we study the boundedness of those operators on Gaussian variable Lebesgue spaces under a certain additional condition of regularity on p(•) following [6].
arXiv (Cornell University), Dec 16, 2019
In this paper we introduce a new class of Gaussian singular integrals, the general alternative Ga... more In this paper we introduce a new class of Gaussian singular integrals, the general alternative Gaussian singular integrals and study the boundedness of them in L p (γ d), 1 < p < ∞ and its weak (1, 1) boundedness with respect to the Gaussian measure following [6] and [1], respectively.
arXiv (Cornell University), Sep 28, 2007
We introduce a new class of polynomials {P n }, that we call polar Legendre polynomials, they app... more 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.
arXiv (Cornell University), Jul 18, 2012
In this paper we return to the study of the Watson kernel for the Abel summabilty of Jacobi polyn... more 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 < α as well as −1 < α < 0, with essentially the same method; using an integral superposition of Poisson type kernel and Muckenhoupt Ap-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 [7], [8], [9] and [12]. We will also discuss in detail the Calderón-Zygmund decomposition for non-atomic Borel measures in R. Then, we prove that the Jacobi measure is doubling and therefore, following [10], we study the corresponding Ap weight theory in the setting of Jacobi expansions, considering power weights of the form (1 − x) α , (1 + x) β , −1 < α < 0, −1 < β < 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.
arXiv (Cornell University), Aug 25, 2022
In this paper we prove the boundedness of the Gaussian Riesz potentials I β , for β ≥ 1 on L p(•)... more In this paper we prove the boundedness of the Gaussian Riesz potentials I β , for β ≥ 1 on L p(•) (γ d), the Gaussian variable Lebesgue spaces under a certain additional condition of regularity on p(•) following [5]. Additionally, this result trivially gives us an alternative proof of the boundedness of Gaussian Riesz potentials I β on Gaussian Lebesgue spaces L p (γ d).
arXiv (Cornell University), Aug 24, 2021
The ternary Cantor set C, constructed by George Cantor in 1883, is the best known example of a pe... more The ternary Cantor set C, constructed by George Cantor in 1883, is the best known example of a perfect nowhere-dense set in the real line. The present article we study the basic properties C and also study in detail the ternary expansion characterization C. We then consider the Cantor-Lebesgue function defined on C, prove its basic properties and study its continuous extension to [0, 1]. We also consider the geometric construction of F as the uniform limit of polygonal functions. Finally, we consider the Lebesgue's function defined from C onto [0, 1] 2 and onto [0, 1] 3 , as well as their continuous extension to [0, 1], i.e., obtained the Lebesgue's space filling curves. Finally we discuss Hausdorff's theorem, which is a natural generalization of the definition of Lebesgue's functions, that states that any compact metric space is a continuous image of the Cantor set C. This notes are the outgrowth of a (zoom)-seminar during the 2020 spring and summer semesters based on H. Sagan's book [14], thus there is not any pretension of originality and/or innovation in this notes and its main objective is a systematic exposition of these topics, in other words its objective is being a review.
Association for Women in Mathematics series, 2017
Focusing on the groundbreaking work of women in mathematics past, present, and future, Springer's... more Focusing on the groundbreaking work of women in mathematics past, present, and future, Springer's Association for Women in Mathematics Series presents the latest research and proceedings of conferences worldwide organized by the Association for Women in Mathematics (AWM). All works are peer-reviewed to meet the highest standards of scientific literature, while presenting topics at the cutting edge of pure and applied mathematics. Since its inception in 1971, The Association for Women in Mathematics has been a non-profit organization designed to help encourage women and girls to study and pursue active careers in mathematics and the mathematical sciences and to promote equal opportunity and equal treatment of women and girls in the mathematical sciences. Currently, the organization represents more than 3000 members and 200 institutions constituting a broad spectrum of the mathematical community, in the United States and around the world.
Quaestiones Mathematicae, 2021
Abstract In this paper we are going to prove that the Hardy-Litllewood maximal operators on varia... more Abstract In this paper we are going to prove that the Hardy-Litllewood maximal operators on variable Lebesgue spaces L p(·)(µ) with respect to a probability Borel measure µ, are weak type and strong type for two conditions of regularity on the exponent function p(·); following [2] and [3]. Also following [2] we extend some important properties of this operator to a probability Borel measure µ, the key to extend these results is using the Besicovitch covering lemma instead of the Calderón-Zygmund decomposition.
Acta Cientifica Venezolana, 1997
Springer Proceedings in Mathematics & Statistics, 2014
Preface (Constantine Georgakis).- Remembrances and Silhouettes.- The Calderon brothers, a happy m... more Preface (Constantine Georgakis).- Remembrances and Silhouettes.- The Calderon brothers, a happy mathematical relation.- Calixto Calderon as I knew him.- An Appraisal of Calixto Calderon's Work in Mathematical Biology.- Remarks on various generalized derivatives.- Some non standard applications of the Laplace method.- Fejer Polynomials and Chaos.- A note on Widder's Inequality.- Solyanik Estimates in Harmonic Analysis.- Some open problems related with generalized Fourier series.- Modeling the Mechanics of Aneurysm Development and Rupture Computational Simulation of Aneurysm Evolution, Growth and Rupture.- Singular Integral Operators on C1 Manifolds and C1 Curvilinear Polygons.- Towards a unified theory of Sobolev inequalities.- Transference of fractional Laplacian regularity.- Local sharp maximal functions.- Weighted norm estimates for singular integrals with L log L kernels Regularity of weak solutions of some degenerate quasilinear equations.
In this work we construct and study families of generalized orthogonal polynomials with hermitian... more In this work we construct and study families of generalized orthogonal polynomials with hermitian matrix argument associated to a family of orthogonal polynomials on R. Different normalizations for these polynomials are considered and we obtain some classical formulas for orthogonal polynomials from the corresponding formulas for the one–dimensional polynomials. We also construct semigroups of operators associated to the generalized orthogonal polynomials and we give an expression of the infinitesimal generator of this semigroup and, in the classical cases, we prove that this semigroup is also Markov. For d–dimensional Jacobi expansions we study the notions of fractional integral (Riesz potentials), Bessel potentials and fractional derivatives. We present a novel decomposition of the L2 space associated with the d–dimensional Jacobi measure and obtain an analogous of Meyer's multiplier theorem in this setting. Sobolev Jacobi spaces are also studied.
Revista Colombiana de Matemáticas, 2021
The main result of this paper is the proof of the boundedness of the Maximal Function T* of the O... more The main result of this paper is the proof of the boundedness of the Maximal Function T* of the Ornstein-Uhlenbeck semigroup {Tt}t≥ 0 in Rd, on Gaussian variable Lebesgue spaces Lp(.) (γd); under a condition of regularity on p(.) following [5] and [8]. As an immediate consequence of that result, the Lp(.) (γd)-boundedness of the Ornstein-Uhlenbeck semigroup {Tt}t≥ 0 in Rd is obtained. Another consequence of that result is the Lp(.) (γd)-boundedness of the Poisson-Hermite semigroup and the Lp(.) (γd)- boundedness of the Gaussian Bessel potentials of order β > 0.
Cornell University - arXiv, Oct 8, 2015
In this paper we shall be concerned with H α summability, for 0 < α ≤ 2 of the Fourier series of ... more In this paper we shall be concerned with H α summability, for 0 < α ≤ 2 of the Fourier series of arbitrary L 1 ([−π, π]) functions. The method to be employed is a refinement of the real variable method introduced by Marcinkiewicz in [8].
Uploads
Papers by Wilfredo Urbina