arXiv subjects
Publications and source records attributed to Wilfredo O. Urbina.
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 the exploration of several variants and generalizations and will provide an example of a non-centered asymmetric Cantor-like set.
In this paper we prove the boundedness of the Gaussian Riesz potentials $I_β$, for $β\geq 1$ on $L^{p(\cdot)}(γ_d)$, the Gaussian variable Lebesgue spaces under a certain additional condition of regularity on $p(\cdot)$ following \cite{DalSco}. Additionally, this result trivially gives us an alternative proof of the boundedness of Gaussian Riesz potentials $I_β$ on Gaussian Lebesgue spaces $L^p(γ_d)$.
In this paper we study the regularity properties of the Gaussian Bessel potentials and Gaussian Bessel fractional derivatives on variable Gaussian Besov-Lipschitz spaces $B_{p(\cdot),q(\cdot)}^α(γ_{d}),$ that were defined in a previous paper \cite{Pinrodurb}, under certain conditions on $p(\cdot)$ and $q(\cdot)$.
The ternary Cantor set $\mathcal{C}$, 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 $\mathcal{C}$ and also study in detail the ternary expansion characterization $\mathcal{C}$. We then consider the Cantor-Lebesgue function defined on $\mathcal{C},$ 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 $\mathcal{C}$ 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 $\mathcal{C}$. This notes are the outgrowth of a (zoom)-seminar during the 2020 spring and summer semesters based on H. Sagan's book \cite{sagan}, 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.
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 we will explore the richness, the peculiarities and the generalities that has $C$ and explore some variants and generalizations of it. For a more systematic treatment the Cantor like sets we refer to our previous paper.
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.