For the hypoelliptic differential operators $P={\partial }_x^2+{\left(x^k{\partial }_y-x^l{\partial }_t\right)}^2$ introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of $k$ and $l$ left open in the analysis, the operators $P$ also fail to be analytic hypoelliptic (except for $(k,l)=(0,1)$), in accordance with Treves’ conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of eigenvalues of a non-self-adjoint operator.

The sub-Laplacian on the Heisenberg group is first decomposed into twisted Laplacians parametrized by Planck's constant. Using Fourier-Wigner transforms so parametrized, we prove that the twisted Laplacians are globally hypoelliptic in the setting of tempered distributions. This result on global hypoellipticity is then used to obtain Liouville's theorems for harmonic functions for the sub-Laplacian on the Heisenberg group.

Precedenti risultati riguardanti il principio di massimo generalizzato e la valutazione del primo autovalore per operatori uniformemente ellittici di tipo variazionale vengono estesi agli operatori subellittici di tipo Heisenberg non simmetrici e a coefficienti discontinui.

The problems of Gevrey hypoellipticity for a class of degenerated quasi-elliptic operators are studied by several authors (see [1]–[5]). In this paper we obtain the Gevrey hypoellipticity for a degenerated quasi-elliptic operator in ${ℝ}^2$, without any restriction on the characteristic polyhedron.

We consider a class of possibly degenerate second order elliptic operators on ℝⁿ. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Hölder spaces both for elliptic equations and for parabolic Cauchy problems involving . The Hölder spaces in question are defined with respect to a possibly non-Euclidean metric related to the operator . Schauder estimates are deduced by sharp...

Let $X=(X_1,X_2,\cdots ,X_q)$ be a system of vector fields satisfying the Hörmander condition. We prove $L_X^{2,\lambda }$ local regularity for the gradient $Xu$ of a solution of the following strongly elliptic system $$-X_{\alpha }^{*}\left(a_{ij}^{\alpha \beta }\left(x\right)X_{\beta }{u}^j\right)=g_i-X_{\alpha }^{*}{f}_i^{\alpha }\left(x\right)\forall i=1,2,\cdots ,N,$$ where $a_{ij}^{\alpha \beta }\left(x\right)$ are bounded functions and belong to Vanishing Mean Oscillation space.

We prove some Hardy-type inequalities related to quasilinear second-order degenerate elliptic differential operators $L_{p}u:=-{\nabla }_L^{*}\left({\left|{\nabla }_{L}u\right|}^{p-2}{\nabla }_{L}u\right)$. If $\phi$ is a positive weight such that $-L_p\phi \ge 0$, then the Hardy-type inequalityholds. We find an explicit value of the constant involved, which, in most cases, results optimal. As particular case we derive Hardy inequalities for subelliptic operators on Carnot Groups.

On a real hypersurface $M$ in ${ℂ}^{n+1}$ of class $C^{2,\alpha }$ we consider a local CR structure by choosing $n$ complex vector fields $W_j$ in the complex tangent space. Their real and imaginary parts span a $2n$-dimensional subspace of the real tangent space, which has dimension $2n+1.$ If the Levi matrix of $M$ is different from zero at every point, then we can generate the missing direction. Under this assumption we prove interior a priori estimates of Schauder type for solutions of a class of second order partial differential equations...

The aim of this paper is to show how, in order to prove regularity theorems, Hölder estimates, i.e. estimates involving products of powers of different semi-norms, can be used as well as standard estimates, and may in some instances be casier to prove.