Related papers: Fundamental solution for higher order homogeneous …

We prove the existence of a global fundamental solution $\Gamma(x;y)$ (with pole $x$) for any H\"ormander operator $\mathcal{L}=\sum_{i=1}^m X_i^2$ on $\mathbb{R}^n$ which is $\delta$-homogeneous of degree $2$. By means of a global Lifting…

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…

By using commutator methods, we show uniform resolvent estimates and obtain globally smooth operators for self-adjoint injective homogeneous operators $H$ on graded groups, including Rockland operators, sublaplacians and many others. Left…

Let $\mathcal{L}=\sum_{j=1}^{m}X_{j}^{2}$ be a H\"{o}rmander sum of squares of vector fields in $\mathbb{R}^{n}$, where any $X_{j}$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in $\mathbb{R}^{n}$. Then…

We prove an index theorem for inhomogeneous differential operators satisfying the Rockland condition (hence hypoelliptic). This theorem extends an index theorem for contact manifolds by Van-Erp.

Let $X_{1},...,X_{m}$ be a family of real smooth vector fields defined in $\mathbb{R}^{n}$, $1$-homogeneous with respect to a nonisotropic family of dilations and satisfying H\"{o}rmander's rank condition at $0$ (and therefore at every…

We investigate the global hypoellipticity and global solvability of systems of left-invariant differential operators on compact Lie groups. Focusing on diagonal systems, we establish necessary and sufficient conditions for these global…

In this paper we give several global characterisations of the Hormander class of pseudo-differential operators on compact Lie groups. The result is applied to give criteria for the ellipticity and the global hypoellipticity of…

The aim of this paper is to prove the existence and several selected properties of a global fundamental Heat kernel $\Gamma$ for the parabolic operators $\mathcal{H}=\sum_{j=1}^m X_j^2-\partial_t$, where $X_1,\ldots,X_m$ are smooth vector…

We compute temperate fundamental solutions of homogeneous differential operators with real-principal type symbols. Via analytic continuation of meromorphic distributions, fundamental solutions for these non-elliptic operators can be…

We present, for the isospectral family of oscillator Hamiltonians, a systematic procedure for constructing raising and lowering operators satisfying any prescribed `distorted' Heisenberg algebra (including the $q$-generalization). This is…

We compute fundamental solutions of homogeneous elliptic differential operators, with constant coefficients, on $\mathbb{R}^n$ by mean of analytic continuation of distributions. The result obtained is valid in any dimension, for any degree…

We extend the classical regularity theorem of elliptic operators to maximally hypoelliptic differential operators. More precisely, given vector fields $X_1,\ldots,X_m$ on a smooth manifold which satisfy H\"ormander's bracket generating…

This paper deals with well-known higher-order generalizations of Hankel operators. We show that higher-order Hankel operators can be written explicitly as linear differential operators, and give the exact form of these differential…

We construct left, right and bilateral fundamental solutions for generalized steady Stokes' operators $S$ with smooth coefficients coefficients, associated with the de Rham complex of differentials on differential forms over a domain $X$ in…

The main result established in this paper is the existence and uniqueness of strong solutions to the obstacle problem for a class of subelliptic operators in non-divergence form. The operators considered are structured on a set of smooth…

We study the global hypoellipticity and solvability of strongly invariant operators and systems of strongly invariant operators on closed manifolds. Our approach is based on the Fourier analysis induced by an elliptic pseudo-differential…

We consider a class of nonvariational degenerate elliptic operators of the kind \[ Lu=\sum_{i,j=1}^{m}a_{ij}\left( x\right) X_{i}X_{j}u \] where $\left\{ a_{ij}\left( x\right) \right\} _{i,j=1}^{m}$ is a symmetric uniformly positive matrix…

We prove global analytic hypoellipticity on a product of tori for partial differential operators which are constructed as rigid (variable coefficient) quadratic polynomials in real vector fields satisfying the H\"ormander condition and…

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…