arXiv subjects

Bruno de Lessa Victor

Publications and source records attributed to Bruno de Lessa Victor.

4 recordsLinked to original sources

arXiv2026

Global properties of the differential complex associated to closed, nonsingular $1$-forms on compact manifolds

Given a closed, real, non-singular 1-form on a compact manifold $Ω$, global properties of the associated differential complex are studied. We completely characterize global solvability in the first and last levels of the complex. Furthermore, in the particular case where the $1$-form is rational, we prove global solvability for every degree and give a complete description of the cohomology spaces. Finally, a complete characterization for global hypoellipticity is obtained, building on the work of A. Meziani (Comm. PDE., 2002). In all cases, it is shown that the conditions depend exclusively on the arithmetic nature of the form's periods.

math.AP↗

arXiv2025

Relations between different types of Hypoellipticity: A systematic approach

We give a systematic treatment to the concept of hypoellipticity, putting it into an abstract form which allows us to deal with several different notions within the same framework. We then investigate when a notion of hypoellipticity implies another one and, in particular, when it can be extended for more general spaces. We also present a relation between certain types of hypoellipticity and local solvability (for the transpose) for a family of operators.

math.AP↗

arXiv2023

The Cauchy problem for a class of linear degenerate evolution equation on the torus

We study, in the periodic setting, the well-posedness of the Cauchy problem associated to the operator $P(t, D_{x}, D_{t}) = D_{t} - a_{2}(t) Δ_{x} + \sum_{j = 1}^{N} a_{1, j}(t) D_{x_{j}} + a_{0}(t)$, with $T> 0$, $t \in [0, T]$ and $a_{2}, a_{1,1}, \ldots, a_{1, N}, a_{0} \in C \left([0, T]; \mathbb{C} \right)$. Using Fourier analysis techniques, we obtain a complete characterization for the well-posedness of a class of degenerate initial-value problems in the Sobolev, Smooth, Gevrey and Real-analytic frameworks.

math.AP↗

arXiv2019

Global and Partial Fourier Series for Denjoy-Carleman Classes on the Torus

In this work, we develop Fourier Analysis for a family of classes of ultradifferentiable functions of Romieu type on the torus, usually known as Denjoy-Carleman classes. Then we are able to apply our results in order to generalize some results whose proofs rely almost exclusively on Fourier Series, such as the Greenfield-Wallach Theorem.

math.AP↗