Related papers: Global hypoellipticity and global solvability for …

Hypoellipticity in Gevrey classes $G^s$ is characterized for a simple family of sums of squares of vector fields satisfying the bracket hypothesis, with analytic coefficients. It is shown that hypoellipticity holds if and only if $s$ is…

We develop the global moduli theory of symplectic varieties in the sense of Beauville. We prove a number of analogs of classical results from the smooth case, including a global Torelli theorem. In particular, this yields a new proof of…

We construct Morse homology groups associated with any regular function on a smooth complex algebraic variety, allowing singular and non-compact critical loci. These groups are generated by critical points of a certain large pertubation of…

In this work a theorical framework to apply the Poincar\'e compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the n-dimensional sphere, though…

This paper is part of a series of papers on differential geometry of $C^\infty$-ringed spaces. In this paper, we study vector fields and their flows on a class of singular spaces. Our class includes arbitrary subspaces of manifolds, as well…

We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the…

A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=0 is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important…

We establish a compactness result for solutions of a certain class of hypoelliptic equations. This result allows us to show the existence of global weak solutions to the non-homogeneous Landau-Fermi-Dirac equation with Coulomb potential.

Given a differential operator defined in terms of left-invariant vector fields on a Lie group, we prove that the local condition defining maximal hypoellipticity is equivalent to a global estimate if the operator is left invariant. As a…

We study the global hypoellipticity problem for certain linear operators in Komatsu classes of Roumieu and Beurling type on compact manifolds. We present an approach by combining a characterization of these spaces via eigenfuction…

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new…

We introduce new techniques to study the differential complexes associated to tube structures on $M \times \mathbb{T}^m$ of corank $m$, in which $M$ is a compact manifold and $\mathbb{T}^m$ is the $m$-torus. By systematically employing…

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…

The paper studies the complex 1-dimensional polynomial vector fields with real coefficients under topological orbital equivalence preserving the separatrices of the pole at infinity. The number of generic strata is determined, and a…

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…

This note presents an investigation on the global hypoellipticity problem for Cauchy operators on $\mathbb{T}^{n+1}$ belonging to the class \linebreak $L = \prod_{j=1}^{m}\left(D_t + c_j(t) P_j(D_x)\right)$, where $P_j(D_x)$ is a…

Let $X$ be a complex manifold and $M\subset X$ a compact, smooth, pseudoconvex CR manifold of dimension $2n-1$. (Here $n\ge 3$ or, in case $n=2$, it is made the extra assumption that $\dib_b$ has closed range on functions.) Assume that…

Let $M$ be a non-compact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$. We establish a Poincar\'{e}-Hopf theorem for a bounded vector field on $M$ satisfying a mild condition on zeros. As an…

For a $\mathbb{R}^{k}-$action generated by vector fields $X_{1},...,X_{k}$ we define an operator $-(X_{1}^{2}+...+X_{k}^{2})$, the orbitwise laplacian. In this paper, we study and classify $\mathbb{R}^{k}-$actions whose orbitwise laplacian…

We propose a generalization of Verbitsky's global Torelli theorem in the framework of compact K\"ahler irreducible holomorphically symplectic orbifolds by adapting Huybrechts' proof (arXiv:1106.5573). As intermediate step needed, we also…