arXiv subjects

Nicholas Braun Rodrigues

Publications and source records attributed to Nicholas Braun Rodrigues.

7 recordsLinked to original sources

arXiv2025

Regularity of infinitesimal automorphisms of involutive structures

In this paper, we prove that infinitesimal automorphisms of an involutive structure are smooth. For this, we build a regularity theory for sections of vector bundles over an involutive structure $(M,V)$ endowed with a connection compatible with $V$, which we call $V$-connection. We show that $V$-sections, i.e. sections which are parallel with respect to $V$ under the $V$-connection, satisfy an analogue of Hans Lewy's theorem as formulated for CR functions on an abstract CR manifold by Berhanu and Xiao, and introduce certain (generically satisfied) nondegeneracy conditions ensuring their smoothness.

math.CV↗

arXiv2025

On a class of globally analytic Hypoelliptic operators with non-negative characteristic form

The global analytic hypoellipticity is proved for a class of second order partial differential equations with non-negative characteristic form globally defined on the torus. The class considered in this work generalizes at some degree the class of sum of squares considered by Bove-Chinni and also by Cordaro-Himonas.

math.AP↗

arXiv2024

A class of globally analytic hypoelliptic operators on compact Lie groups

We obtain global analytic hypoellipticity for a class of differential operators that can be expressed as a zero-order perturbation of a sum of squares of vector fields with real-analytic coefficients on compact Lie groups. The key conditions are: the vector fields must satisfy Hörmander's finite type condition; there exists a closed subgroup whose action leaves the vector fields invariant; and the operator must be elliptic in directions transversal to the action of the subgroup. This paves the way for further studies on the regularity of sums of squares on principal fiber bundles.

math.AP↗

arXiv2023

Smooth and Gevrey Microlocal Hypoellipticity for a Class of Hypocomplex Tube Structures

We prove a smooth and Gevrey-$s$ microlocal hypoellipticity result for a system of complex vector fields associated with a real-analytic locally integrable structure of tube type, that is also microlocal hypocomplex. In order to so, we employ the use of a certain partial F.B.I. transform adapted to the locally integrable structure, first introduced by M. S. Baouendi, C. H. Chang and F. Treves, and we prove a microlocal characterization of the smooth and Gevrey-$s$ wave front set in terms of the decay of this partial F.B.I. transform.

math.AP↗

arXiv2022

Denjoy-Carleman Microlocal Regularity on Smooth Real Submanifolds of Complex Space

We prove the existence of approximate solutions in the (regular) Denjoy-Carleman sense for some systems of smooth complex vector fields. Such approximate solutions provide a well defined notion of Denjoy-Carleman wave front set of distributions on maximally real submanifolds in complex space which can be characterized in terms of the decay of the Fourier-Bros-Iagolnitzer transform. We also apply the approximate solutions to analyze the Denjoy-Carleman microlocal regularity of solutions of certain systems of first-order nonlinear partial differential equations.

math.AP↗

arXiv2020

An FBI characterization for Gevrey vectors on hypo-analytic structures and propagation of Gevrey singularities

In this work we prove an FBI characterization for Gevrey vectors on hypo-analytic structures, and we analyze the main differences of Gevrey regularity and hypo-analyticity concerning the FBI transform. We end with an application of this characterization on a propagation of Gevrey singularities result, for solutions of the non-homogeneous system associated with the hypo-analytic structure, for analytic structures of tube type.

math.CV↗

arXiv2018

Approximate solutions of vector fields and an application to Denjoy-Carleman regularity of solutions of a nonlinear PDE

In this paper we study microlocal regularity of a $\mathcal{C}^2$ solution $u$ of the equation \begin{equation*} u_t = f(x,t,u,u_x), \end{equation*} where $f(x,t,ζ_0, ζ)$ is ultradifferentiable in the variables $(x,t)\in \mathbb{R}^{N} \times \mathbb{R}$ and holomorphic in the variables $(ζ_0,ζ) \in \mathbb{C} \times \mathbb{C}^{N}$. We proved that if $\mathcal{C}^{\mathcal{M}}$ is a regular Denjoy-Carleman class (including the quasianalytic case) then: \begin{equation*} \mathrm{WF}_\mathcal{M} (u)\subset \mathrm{Char}(L^u), \end{equation*} where $\mathrm{WF}_\mathcal{M}(u)$ is the Denjoy-Carleman wave-front set of $u$ and $\mathrm{Char}(L^u)$ is the characteristic set of the linearized operator $L^u$: \begin{equation*} L^u = \dfrac{\partial}{\partial t} - \sum_{j=1}^{N}\frac{\partial f}{\partialζ_j}(x,t,u,u_x)\dfrac{\partial}{\partial x_j}. \end{equation*}

math.AP↗