Lp{L^p}-Liouville Theorems for Invariant Partial Differential Operators in Rn{\mathbb{R}^n}

We prove some $L^p$-Liouville theorems for hypoelliptic second order Partial Differential Operators left translation invariant with respect to a Lie group composition law in $\mathbb{R}^n$. Results for both solutions and subsolutions are given

Subharmonic functions in sub-Riemannian settings

In this note we present mean value characterizations of subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution ¡. These characterizations are based on suitable average operators on the level sets of ¡. Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach and Reade. The results presented here generalize and carry forward former results of the authors in [6, 8].In this note we present mean value characterizations of subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution ¡. These characterizations are based on suitable average operators on the level sets of ¡. Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach and Reade. The results presented here generalize and carry forward former results of the authors in [6, 8]

Subelliptic harmonic morphisms

We study subelliptic harmonic morphisms i.e. smooth maps $\phi: \Omega \to \tilde\Omega$ among domains $\Omega \subset \mathbb{R}^n$ and $\tilde\Omega \subset \mathbb{R}^M$ endowed with Hörmander systems of vector fields $X$ and $Y$, that pull back local solutions to $H_Y v = 0$ into local solutions to $H_X u = 0$, where $H_X$ and $H_Y$ are Hörmander operators. We show that any subelliptic harmonic morphism is an open mapping. Using a subelliptic version of the Fuglede-Ishihara theorem (due to E. Barletta [5]) we show that given a strictly pseudoconvex CR manifold $M$ and a Riemannian manifold $N$ for any heat equation morphism $\Psi: M \times (0, \infty) \to N \times (0, \infty)$ of the form $\Psi(x,t) = ( \phi (x), h(t))$ the map $\phi : M \to N$ is a subelliptic harmonic morphism

On the Harmonic characterization of domains via mean value formulas

The Euclidean ball have the following harmonic characterization, via Gauss-mean value property: Let D be an open set with finite Lebesgue measure and let x0 be a point of D. If  for every nonnegative harmonic function u in D, then D is a Euclidean ball centered at x0. On the other hand, on every sufficiently smooth domain D and for every point x0 in D there exist Radon measures μ such thatfor every nonnegative harmonic function u in D. In this paper we give sufficient conditions so that this last mean value property characterizes the domain D

Dirichlet problem with L^p-boundary data for real sub-Laplacians

Let L be a real sub-Laplacian on a stratified Lie group G. In this note we present some results on the Dirichlet problem for L with L^p-boundary data, on domains which are contractible with respect to the natural dilations of G. One of the main difficulties we overcome is the presence of non-regular boundary points for the usual Dirichlet problem for L. A potential theoretical approach is followed

Liouville Theorems for a Class of Linear Second-Order Operators with Nonnegative Characteristic Form

We report on some Liouville-type theorems for a class of linear second-order partial differential equation with nonnegative characteristic form. The theorems we show improve our previous results

POTENTIAL ANALYSIS FOR A CLASS OF DIFFUSION EQUATIONS: A GAUSSIAN BOUNDS APPROACH

Let H be a linear second order partial differential operator with non-negative characteristic form in a strip S ⊂ R^N ×R. We assume that H as a fundamental solution, smooth out of its poles and bounded from above and from below by Gaussian kernels modeled on subriemannian doubling distances in R^N. Under these assumptions we show that H endows S with a structure of β-harmonic space. This allows us to study boundary value problems for L with a Perron-Wiener-Brelot-Bauer method, and to obtain pointwise regularity estimates at the boundary in terms of Wiener series modeled on the Gaussian kernels. Our analysis includes the proof of a scale invariant Harnack inequality for nonnegative solutions. We also show an application to the real hypersurphaces of C^{n+1} with given Levi-curvature.Let H be a linear second order partial differential operator with non-negative characteristic form in a strip S ⊂ RN ×R. We assume that H asa fundamental solution, smooth out of its poles and bounded from above and from below by Gaussian kernels modeled on subriemannian doubling distances in RN. Under these assumptions we show that H endows S with a structure of β-harmonic space. This allows us to study boundary value problems for L with a Perron-Wiener-Brelot-Bauer method, and to obtain pointwise regularity estimates at the boundary in terms of Wiener series modeled on the Gaussian kernels. Our analysis includes the proof of a scale invariant Harnack inequality for nonnegative solutions. We also show an application to the real hypersurphaces of Cn+1 with given Levi-curvature

Una base di insiemi risolutivi per l'equazione del calore: una costruzione elementare

By an easy “trick” taken from the caloric polynomial theory, we prove the existence of a basis of the Euclidean topology whose elements are resolutive sets of the heat equation. This result can be used to construct, with a very elementary approach, the Perron solution of the caloric Dirichlet problem on arbitrary bounded open subsets of the Euclidean space-time.Con un semplice espediente preso dalla teoria dei polinomi calorici, dimostriamo l'esistenza di una base della topologia euclidea i cui elementi sono insiemi risolutivi per l'equazione del calore. Questo risultato può essere utilizzato per costruire, con un approccio elementare, la soluzione di Perron del problema di Dirichlet calorico su arbitrari insiemi aperti limitati dello spazio-tempo euclideo