arXiv subjects

J. P. Velasquez-Rodriguez

Publications and source records attributed to J. P. Velasquez-Rodriguez.

5 recordsLinked to original sources

arXiv2026

$L^r$-Multipliers on compact $p$-adic Lie groups

Let $p$ be a prime number, and let $\mathbb{G}$ be a compact $p$-adic Lie group. This work provides multiplier theorems for invariant operators on $\mathbb{G}$ acting on $L^r_α(\mathbb{G})$, $1 0$, in terms of the Ruzhansky-Turunen difference operators and Saloff-Coste's condition. As an application, a Littlewood-Paley decomposition is proven, together with the $L^r$-boundedness of bounded functions of the Vladimirov-Taibleson operator on compact Vilenkin groups.

math.RT↗

arXiv2024

The spectrum of the Vladimirov sub-Laplacian on the compact Heisenberg group

Let $p>2$ be a prime number. In this short note, we calculate explicitly the unitary dual and the matrix coefficients of the Heisenberg group over the $p$-adic integers. As an application, we consider directional Vladimirov-Taibleson derivatives, and some polynomials in these operators. In particular, we calculate explicitly the spectrum of the Vladimirov sub-Laplacian, and show how it provides a non-trivial example of a sub-elliptic operator on compact graded $p$-adic Lie groups.

math.RT↗

arXiv2024

The spectrum of the Vladimirov sub-Laplacian on the compact Engel group

Let $p>3$ be a prime number. In this note, we use p-adic Gaussian integrals to calculate explicitly the unitary dual and the matrix coefficients of the Engel group over the $p$-adic integers $\mathcal{B}_4(\mathbb{Z}_p)$. We use this information to calculate explicitly the spectrum of the Vladimirov sub-Laplacian, and show how it defines a globally hypoelliptic operator on $\mathcal{B}_4$.

math.RT↗

arXiv2024

Unitary dual and matrix coefficients of compact nilpotent p-adic Lie groups with dimension $d \leq 5$

Let p> 2 be a prime number, and let G be a compact nilpotent p-adic Lie group with nilpotency class N<p. In this note we calculate explicitly the unitary dual and the matrix coefficients of every compact nilpotent-adic Lie group with dimension less or equal than 5. As an application, we provide the corresponding spectral theorem for the Vladimirov sub-Laplacian, and show how this operator provides a non-trivial example of a globally hypoelliptic operator on compact nilpotent p-adic Lie groups.

math.RT↗

arXiv2023

Titchmarsh Theorems for Hölder-Lipschitz functions on profinite groups

In this note we extend to metrizable profinite groups the classical theorems of Titchmarsh on the Fourier transform of Hölder-Lipschitz functions. This generalizes the results of Younis on compact zero-dimensional abelian groups to the noncommutative case, and proves a relation between the Hölder-Lipschitz-continuity of functions and their Sobolev regularity given in terms of the Vladimirov-Taibleson operator. Since the class of profinite groups is fairly big, the formulation of our results requires to impose a special condition on the representation theory of the group. We prove that in particular such condition is satisfied by compact nilpotent metrizable profinite groups, which covers the case of compact nilpotent $\ell$-adic Lie groups. In addition, we study the modulus of continuity of $L^2$-functions on the group, the functional spaces related to it, and its relation to the $L^2$-based Hölder-Lipschitz spaces. Finally, we also derive a characterization for Dini-Lipschitz classes on metrizable profinite groups in terms of the behavior of their Fourier coefficients.

math.FA↗