Related papers: Hypoellipticity and Higher Order Levi Conditions

In this paper, we study Vekua-type operators associated with diagonal operators on compact Lie groups. Characterizations of global hypoellipticity and global solvability properties are presented on classes of Vekua-type operators with…

In this work, we study the two following minimization problems for $r \in \mathbb{N}^{*}$, \begin{equation*} \begin{array}{ccc} S_{0,r}(\varphi)=\displaystyle\inf_{u\in H_{0}^{r}(\Omega)\,|u+\varphi\|_{L^{2^{*r}}}=1}\|u\|_{r}^{2}&…

We propose simple conditions equivalent to the discreteness of the spectrum of the Laplace-Beltrami operator on a class of Riemannian manifolds close to warped products. For this class of manifolds we establish a relationship between…

In 1980 M{\'e}tivier characterized the analytic (and Gevrey) hypoellipticity of $L^2$-solvable partial linear differential operators by a-priori estimates. In this note we extend this characterization to ultradifferentiable hypoellipticity…

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…

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal{A}_{D,\varepsilon}$ with the…

We give a semiclassical analysis of a nonlinear eigenvalue problem arising from the study of the failure of analytic hypoellipticity and obtain a general family of hypoelliptic, but not analytic hypoelliptic operators.

This paper provides a complete characterization of global hypoellipticity and solvability with loss of derivatives for Fourier multiplier operators on the $n$-dimensional torus. We establish necessary and sufficient conditions for these…

We compare two C*-algebras that have been used to study the essential spectrum. This is done by considering a simple second order elliptic differential operator acting in L^2(R^N), which is affiliated with one or both of the algebras…

We construct parametrices for a class of pseudodifferential operators of infinite order acting on spaces of tempered ultradistributions of Beurling and Roumieu type. As a consequence we obtain a result of hypoellipticity in these spaces.

We prove local real analytic hypoellipticity for a sum of squares of complex vector fields studied by J.J. Kohn in a paper to appear in the Annals of Mathematics entitled "Hypoellipticity and loss of derivatives". The operator exhibits a…

We prove local hypoellipticity of the complex Laplacian $\Box$ and of the Kohn Laplacian $\Box_b$ in a pseudoconvex boundary when, for a system of cut-off $\eta$, the gradient $\partial_b\eta$ and the Levi form…

We prove that the algebraic condition $|p-2| |< {\mathscr Im}{\mathscr A}\xi,\xi>| \leq 2 \sqrt{p-1} < {\mathscr Re}{\mathscr A}\xi,\xi>$ (for any $\xi\in\mathbb{R}^{n}$) is necessary and sufficient for the $L^{p}$-dissipativity of the…

This study is an attempt at generalizing the class of partially hypoelliptic differential operators to a class of pseudodifferential operators, Symbol ideals are formed on the set of lineality and we discuss suitable topologies that allow…

We establish a strong unique continuation property for the subelliptic Baouendi operator under the presence of zero-order perturbations satisfying an almost Hardy-type growth condition. In particular, the admissible class includes both…

In the present paper we consider the Dirichlet problem for the second order differential operator $E=\nabla(A \nabla)$,where $A$ is a matrix with complex valued $L^\infty$ entries. We introduce the concept of dissipativity of $E$ with…

This paper presents a class of domains on which the Kohn Laplacian and the dib-bar-Neumann problem are hypoelliptic but not superlogarithmic and which, moreover, have a set of points of Levi-degeneracy with positive CR dimension.

We study a Laplacian operator related to the characteristic cohomology of a smooth manifold endowed with a distribution. We prove that this Laplacian does not behave very well: it is not hypoelliptic in general and does not respect the…

We obtain regularity conditions of a new type of problems of the calculus of variations with second-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main result asserts that autonomous integral…

For functions of a single complex variable, points of multiplicity greater than $k$ are characterized by the vanishing of the first $k$ derivatives. There are various quantitative generalizations of this statement, showing that for…