arXiv subjects

Santiago Gómez Cobos

Publications and source records attributed to Santiago Gómez Cobos.

5 recordsLinked to original sources

arXiv2026

Boundedness of spectral multipliers on locally compact groups and applications

We prove that the noncommutative Lorentz norm (associated to a semifinite von Neumann algebra) of a propagator of the form $φ(|\mathscr{L}|)$ can be estimated if the modulus of the Borel function $φ$ is bounded by a continuous positive monotonically decreasing function that vanishes at infinity $ψ$. As a consequence, we obtain the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates for the solutions of heat, wave, and Schrödinger type equations (new in this setting) on a locally compact separable unimodular group $G$ by using a non-local integro-differential operator in time and any positive left invariant operator (maybe unbounded and with discrete or continuous spectrum) on $G$. We also provide asymptotic estimates (large-time behavior) for the solutions, which in some cases can be claimed to be sharp. Illustrative examples are given for several groups.

math.AP↗

arXiv2025

Functional calculus for Safarov pseudo-differential operators

Given a smooth, closed Riemannian manifold $(M,g)$ equipped with a linear connection $\nabla$ (not necessarily metric), we develop the holomorphic functional calculus for operators belonging to the global pseudo-differential classes $Ψ_{ρ, δ}^m\left(Ω^κ, \nabla, τ\right)$ introduced by Safarov. As a consequence of our main result, we establish a Szegö type-theorem, derive asymptotic expansion of the heat kernel trace, and calculate some associated spectral $ζ$-functions.

math.AP↗

arXiv2024

Evolutionary integro-differential equations of scalar type on locally compact groups

We study existence, uniqueness, norm estimates and asymptotic time behaviour (in some cases can be claimed to be sharp) for the solution of a general evolutionary integral (differential) equation of scalar type on a locally compact separable unimodular group $G$ governed by any positive left invariant operator (unbounded and either with discrete or continuous spectrum) on $G$. We complement our studies by proving some time-space (Strichartz type) estimates for the classical heat equation, and its time-fractional counterpart, as well as for the time-fractional wave equation. The latter estimates allow us to give some results about the well-posedness of nonlinear partial integro-differential equations. We provide many examples of the results by considering particular equations, operators and groups through the whole paper.

math.AP↗

arXiv2024

$L^p-L^q$ estimates for non-local heat and wave type equations on locally compact groups

We prove the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates for the solutions of heat and wave type equations on a locally compact separable unimodular group $G$ by using an integro-differential operator in time and any positive left invariant operator (maybe unbounded) on $G$. We complement our studies by giving asymptotic time estimates for the solutions, which in some cases are sharp.

math.AP↗

arXiv2024

$L^p$-bounds in Safarov pseudo-differential calculus on manifolds with bounded geometry

Given a smooth complete Riemannian manifold with bounded geometry $(M,g)$ and a linear connection $\nabla$ on it (not necessarily a metric one), we prove the $L^p$-boundedness of operators belonging to the global pseudo-differential classes $Ψ_{ρ, δ}^m\left(Ω^κ, \nabla, τ\right)$ constructed by Safarov. Our result recovers classical Fefferman's theorem, and extends it to the following two situations: $ρ>1/3$ and $\nabla$ symmetric; and $\nabla$ flat with any values of $ρ$ and $δ$. Moreover, as a consequence of our main result, we obtain boundedness on Sobolev and Besov spaces and some $L^p-L^q$ boundedness. Different examples and applications are presented.

math.AP↗