arXiv subjects

Manuel Alejandro Martínez

Publications and source records attributed to Manuel Alejandro Martínez.

3 recordsLinked to original sources

arXiv2025

Estimates for pseudo-differential operators on the torus revisited. III

This paper finishes the goal of the authors started in two previous manuscripts dedicated to revisiting the continuity properties of toroidal pseudo-differential operators with symbols in the Hörmander classes. Here we prove pointwise estimates in terms of the Fefferman-Stein sharp maximal function and of the Hardy-Littlewood maximal function. Combining these estimates with the properties of Muckenhoupt's weight class $A_p$ we obtain boundedness theorems for pseudo-differential operators between weighted Lebesgue spaces on the torus $L^p(w)$. These results are given in the context of the global symbolic analysis defined on $\mathbb{T}^n\times \mathbb{Z}^n$ as developed by Ruzhansky and Turunen by using discrete Fourier analysis, and extend those of Park and Tomita available in the Euclidean case. Moreover, we include continuity results on Sobolev spaces $W^s_p$ and on Besov spaces $B^s_{p,q}$ on the torus. Our techniques are taken from Park and Tomita \cite{park-tomita} and we consider its toroidal extension here for the completeness of the boundedness of toroidal pseudo-differential operators with respect to the current literature.

math.AP↗

arXiv2025

Boundedness of pseudo-differential operators on the torus revisited. II

In this paper we continue our program of revisiting the new aspects about the boundedness properties of pseudo-differential operators on the torus. Here we prove $H^p$-$L^p$ and $H^p$-estimates for Hörmander classes of pseudo-differential operators on the torus $\mathbb{T}^n$ for $p\leq 1$. The results are presented in the context of the global symbolic analysis developed by Ruzhansky and Turunen on $\mathbb{T}^n \times \mathbb{Z}^n$ by using the discrete Fourier analysis, which extends the $(ρ, δ)$-Hörmander classes on $\mathbb{T}^n$ defined by local coordinate systems. These results extend those proved by Álvarez and Hounie for the Euclidean case, considering even the case $ρ\leqδ$.

math.AP↗

arXiv2025

Estimates for pseudo-differential operators on the torus revisited. I

In this paper we prove $L^p$-estimates for Hörmander classes of pseudo-differential operators on the torus $\mathbb{T}^n$. The results are presented in the context of the global symbolic calculus of Ruzhansky and Turunen on $\mathbb{T}^n\times\mathbb{Z}^n$ by using the discrete Fourier analysis on the torus which extends the usual ($ρ,δ$)-Hörmander classes on $\mathbb{T}^n$. The main results extend Álvarez and Hounie's method for $\mathbb{R}^n$ to the torus, and Fefferman's $L^p$-boundedness theorem in the toroidal setting allowing the condition $δ\geqρ$. When $δ\leq ρ$, our results recover the available estimates in the literature.

math.AP↗