Manuel Alejandro Martínez Flores.

(2025) Submitted. With: Duván Cardona
In this paper we investigate \(H^p\)-\(L^p\)- and \(H^p\)-estimates for toroidal pseudo-differential operators with symbols in the \((\rho,\delta)\)-H"ormander classes defined on the torus. Our results are framed in the context of the toroidal symbolic calculus developed by Michael Ruzhansky and Ville Turunen. Here, we extend the methods of Álvarez and Hounie from \(R^n\) to the torus, which allow us to consider even the case \(\delta \geq \rho\). In order to obtain these properties in the context of toroidal pseudo-differential operators, we extend 'Alvarez and Milmans continuity results for the more general case of operators with operator-valued kernel in the context of the torus.

(2025) Submitted. With: Duván Cardona
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 H"ormander 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 'Alvarez and Hounie for the Euclidean case, considering even the case \(\rho\leq\delta\).

(2025) To appear in Trends in Mathematics. With: Duván Cardona
In this paper we investigate \(L^p\)-estimates for Hörmander classes of pseudo-differential operators on the torus \(\mathbb{T}^n\). Our results are framed within the global symbolic calculus developed by Ruzhansky and Turunen on \(\mathbb{T}^n \times \mathbb{Z}^n\), by using the discrete Fourier analysis on the torus. This approach extends the classical \((\rho, \delta)\)-Hörmander classes to the toroidal setting. The main contributions of this work generalize the method of Álvarez and Hounie for \(\mathbb{R}^n\) to the torus, while also extending Fefferman’s \(L^p\)-boundedness theorem to the toroidal context, even in cases where \(\delta \geq \rho\). When \(\delta \leq \rho\), our results align with and recover existing estimates found in the literature.

(2025) Submitted. With: Duván Cardona
In this paper we prove \(L^p\)-estimates for Hörmander classes of pseudo-differential operators on the torus \(\mathbb{T}^n\). Our results are framed within the global symbolic calculus developed by Ruzhansky and Turunen on \(\mathbb{T}^n \times \mathbb{Z}^n\), by using the discrete Fourier analysis on the torus. This approach extends the classical \((\rho, \delta)\)-Hörmander classes to the toroidal setting. The main contributions of this work generalize the method of Álvarez and Hounie for \(\mathbb{R}^n\) to the torus, while also extending Fefferman’s \(L^p\)-boundedness theorem to the toroidal context, even in cases where \(\delta \geq \rho\). When \(\delta \leq \rho\), our results align with and recover existing estimates found in the literature.

Advisor: Duván Cardona

We use In this repository we implement the ideas discussed in this document from MIT where they establish a correspondence between physical systems and graphs. Then we implement algorithms to generate the state equations.