Manuel Alejandro Martínez Flores.

(2025) With: Duván Cardona. To appear in Journal of Mathematical Analysis and Applications.
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 \((\rho,\delta)\)-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 \(\delta\geq\rho\). When \(\delta \leq \rho\), our results recover the available estimates in the literature.

(2025) With: Duván Cardona. To appear in Trends in Mathematics, Birkhäuser.

(2025) With: Duván Cardona. Submitted for review.
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 \((\rho, \delta)\)-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 \(\rho\leq\delta\).

(2025) With: Duván Cardona. To appear in Trends in Mathematics, Birkhäuser.

(2025) With: Duván Cardona. Submitted for review.
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 and we consider its toroidal extension here for the completeness of the boundedness of toroidal pseudo-differential operators with respect to the current literature.