Schwartz regularity of differential operators on the cylinder

This article presents an investigation of global properties of a class of differential operators on \T^1\times\R. Our approach involves the utilization of a mixed Fourier transform, incorporating both partial Fourier series on the torus and partial Fourier transform in Euclidean space. By examining the behavior of the mixed Fourier coefficients, we obtain necessary and sufficient conditions for the Schwartz global hypoellipticity of this class of differential operators, as well as conditions for the Schwartz global solvability of said operators.Comment: arXiv admin note: text overlap with arXiv:2306.1557

Fourier Analysis on Tm×Rn\mathbb{T}^m\times\mathbb{R}^n and Applications to Global Hypoellipticity

This article presents a convenient approach to Fourier analysis for the investigation of functions and distributions defined in $\mathbb{T}^m \times \mathbb{R}^n$. Our approach involves the utilization of a mixed Fourier transform, incorporating both partial Fourier series on the torus for the initial variables and partial Fourier transform in Euclidean space for the remaining variables. By examining the behaviour of the mixed Fourier coefficients, we achieve a comprehensive characterization of the spaces of smooth functions and distributions in this context. Additionally, we apply our results to derive necessary and sufficient conditions for the global hypoellipticity of a class first order differential operators defined on $\mathbb{T}^m \times \mathbb{R}^n$, including all constant coefficient first order differential operators

Global hypoellipticity of GG-invariant operators on homogeneous vector bundles

We establish necessary and sufficient conditions for the global hypoellipticity of $G$-invariant operators on homogeneous vector bundles. These criteria are established in terms of the corresponding matrix-valued symbols as developed by Ruzhansky and Turunen and extended in [7] to homogeneous vector-bundles.Comment: 15 page

Grupos de lIE

Orientador: Professor Hudson do Nascimento LimaTrabalho de Conclusão de Curso (graduação) - Universidade Federal do Paraná, Setor de Ciências Exatas, Curso de Graduação em MatemáticaInclui referênciasResumo : O trabalho reúne definições e resultados básicos da teoria de Grupos de Lie, entre eles o Teorema do Subgrupo Fechado e os 3 Teoremas de Lie. Mostra-se que é possível definir um funtor entre a categoria dos Grupos de Lie e a das Álgebras de Lie de dimensão finita, que, quando restrito a subcategoria de grupos simplesmente conexos, é fiel. Também se classificam os Grupos de Lie abelianos e conexos, e são estudadas algumas propriedades de Grupos e Álgebras compactas. Por fim, reúnem-se exemplos dos principais grupos de LieAbstract: The paper compiles basic results and definitions from the thoery of Lie Groups, among them the Closed Subgroup Theorem and the Lie’s 3 Theorems. It is shown that its possible to define the a functor between the category of Lie Groups e the category of Lie Algebras of finite dimension and, when this functor is restricted to simply connected Groups, this is a faithful functor. The abelian connected subgroups are classified and some properities of compact Lie Groups and Algebras are also exibihite

Global Properties for first order differential operators on Tr+1×(S3)s\mathbb{T}^{r+1}\times(\mathbb{S}^{3})^s

In this paper, we study the global properties of a class of evolution-like differential operator with a 0-order perturbation defined on the product of $r+1$ tori and $s$ spheres $\mathbb{T}^{r+1}\times(\mathbb{S}^{3})^s$, with $r$ and $s$ non-negative integers. By varying the values of $r$ and $s$, we show that it is possible to recover results already known in the literature and present new results. The main tool used in this study is Fourier analysis, taken partially with respect to each copy of the torus and sphere. We obtain necessary and sufficient conditions related to Diophantine inequalities, change of sign and connectivity of level sets associated the operator's coefficients