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Maximum Principles on unbounded domains play a crucial r\^ole in several problems related to linear second-order PDEs of elliptic and parabolic type. In this paper we consider a class of sub-elliptic operators $\mathcal{L}$ in $\mathbb{R}^N$ and we establish some criteria for an unbounded open set to be a Maximum Principle set for $\mathcal{L}$. We extend some classical results related to the Laplacian (by Deny, Hayman and Kennedy) and to the sub-Laplacians on stratified Lie groups (by Bonfiglioli and the second-named author).

We prove a quantitative stability result for the Gauss mean value formula. We also show by an example that the estimate proved here is sharp.

By an easy trick taken from caloric polynomial theory we construct a family $\mathscr{B}$ of $almost\ regular$ domains for the caloric Dirichlet problem. $\mathscr{B}$ is a basis of the Euclidean topology. This allows to build, with a basically elementary procedure, the Perron solution to the caloric Dirichlet problem on every bounded domain.

In this paper we give a general proof of Mean Value formulas for solutions to second order linear PDEs, only based on the local properties of their fundamental solution Gamma. Our proof requires a kind of pointwise vanishing integral condition for the intrinsic gradient of Gamma. Combining our Mean Value formulas with a "descent method" due to Kuptsov, we obtain formulas with improved kernels. As an application, we implement our general results to heat operators on stratified Lie groups and to Kolmogorov operators.

Let D be an open subset of $${\mathbb {R}}^n$$ with finite measure, and let $$x_0 \in D$$ . We introduce the p-Gauss gap of D w.r.t. $$x_0$$ to measure how far are the averages over D of the harmonic functions $$ u \in L^p(D)$$ from $$u(x_0)$$ . We estimate from below this gap in terms of the ball gap of D w.r.t. $$x_0$$ , i.e., the normalized Lebesgue measure of $$D \setminus B$$ , being B the biggest ball centered at $$x_0$$ contained in D. From these stability estimates of the mean value formula for harmonic functions in $$L^p$$ -spaces, we straightforwardly obtain rigidity properties of the Euclidean balls. We also prove a continuity result of the p-Gauss gap in the Sobolev space $$W^{1,p'}$$ , where $$p'$$ is the conjugate exponent of p.

We prove, with a purely analytic technique, a one-side Liouville theorem for a class of Ornstein--Uhlenbeck operators ${\mathcal L_0}$ in $\mathbb{R}^N$, as a consequence of a Liouville theorem at “$t=- \infty$” for the corresponding Kolmogorov operators ${\mathcal L_0} - \partial_t$ in $\mathbb{R}^{N+1}$. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to $({\mathcal L_0} - \partial_t) u = 0$ which seems to have an independent interest in its own right. We stress that our Liouville theorem for ${\mathcal L_0}$ cannot be obtained by a probabilistic approach based on recurrence if $N>2$. We provide a self-contained proof of a Liouville theorem involving recurrent Ornstein--Uhlenbeck stochastic processes in the Appendix.

We establish a necessary and sufficient condition for a boundary point to be regular for the Dirichlet problem related to a class of Kolmogorov-type equations. Our criterion is inspired by two classical criteria for the heat equation: the Evans-Gariepy's Wiener test, and a criterion by Landis expressed in terms of a series of caloric potentials.

Inverse problem results, related to densities with the mean value property for the harmonic functions, were recently proved by the authors. In the present paper we improve and extend them to the sub-Laplacians on stratified Lie groups.

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 R n . Results for both solutions and subsolutions are given.

In this article we present an elementary introduction to the Levi curvature equations, together with a survey on some existence, symmetry and regularity results for their solutions.

We prove some one-side Liouville-type theorems in halfspaces for a class of evolution hypoelliptic equations. The operators we deal with are left translation invariant, and homogeneous of degree two, on homogeneous Lie groups on $\mathbb{R}^{N+1}$.

A one-side Liouville Theorem is proved for second order hypoelliptic operators homogenous with respect to a group of dilations

In this paper we furnish mean value characterizations for subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution 0. These characterizations are based on suitable average operators on the level sets of 0. Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach, Reade and Saks.We analyze as well the notion of subharmonic function in the sense of distributions, and we show how to approximate subharmonic functions by smooth ones. The classes of operators involved are wide enough to contain, as very special cases, the sub-Laplacians on Carnot groups. The results presented here generalize and carry forward former results of the authors.

The Newtonian potential of an Euclidean ball $B$ of $\mathbb{R}^n$ centered at $x_0$ is proportional, outside $B$, to the Newtonian potential of a mass concentrated at $x_0$. Vice-versa, as proved by Aharonov, Schiffer and Zalcman, if $D$ is a bounded open set in $\mathbb{R}^n$, containing $x_0$, whose Newtonian potential is proportional, outside $D$, to the one of a mass concentrated at $x_0$, then $D$ is an Euclidean ball with center $x_0$. In this paper we generalize this last result to more general measures and domains.

We prove some existence, nonexistence and regularity results for the boundary value problem Δ λ u + f ( u ) = 0 in Ω , u ∣ ∂ Ω = 0 , where Ω is a bounded subset of R N , N ≥ 2 , and Δ λ is a Δ λ -Laplacian, i.e. a “degenerate” elliptic operator of the kind Δ λ : = ∑ i = 1 N ∂ x i ( λ i 2 ( x ) ∂ x i ) , λ = ( λ 1 , … , λ N ) . Together with some assumptions made in Franchi and Lanconelli (1984) [1] , the family λ is supposed to verify a condition making Δ λ homogeneous of degree two with respect to a group of dilations in R N .

In this note we extend, to the sub-Laplacian setting, a theorem of Aharonov, Schiffer and Zalcman regarding an inverse property for harmonic functions. As a byproduct, a harmonic characterization of the gauge balls is proved, thus extending a theorem of Kuran concerning the Euclidean balls.

Aim of this paper is to provide new examples of Hormander operators \({\mathcal{L}}\) to which a Lie group structure can be attached making \({\mathcal{L}}\) left invariant. Our class of examples contains several subclasses of operators appearing in literature and arising both in theoretical and in applied fields: evolution Kolmogorov operators, degenerate Ornstein–Uhlenbeck operators, Mumford and Fokker–Planck operators, Ornstein–Uhlenbeck operators with time-dependent periodic coefficients. Our examples basically come from exponential of matrices, as well as from linear constant-coefficient ODE’s, in \({\mathbb{R}}\) or in \({\mathbb{C}}\) . Furthermore, we describe how these groups can be combined together to obtain new structures and new operators, also having an interest in the applied field.

If \( \mathcal{L} = \sum\limits_{j = 1}^m {X_j^2} + {X_0} \) is a Hormander partial differential operator in \( {\mathbb{R}^N} \), we give sufficient conditions on the \( {X_{{j^{\text{S}}}}} \) for the existence of a Lie group structure \( \mathbb{G} = \left( {{\mathbb{R}^N},*} \right) \), not necessarily nilpotent, such that \( \mathcal{L} \) is left invariant on \( \mathbb{G} \). We also investigate the existence of a global fundamental solution Γ for \( \mathcal{L} \), providing results that ensure a suitable left-invariance property of Γ. Examples are given for operators \( \mathcal{L} \) to which our results apply: some are new; some have appeared in recent literature, usually quoted as Kolmogorov–Fokker–Planck-type operators. Nontrivial examples of homogeneous groups are also given.

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.

We establish Holder estimates of second derivatives for a class of sub-elliptic partial differential operators in \({\mathbb{R}^{N}}\) of the kind $$\mathcal L=\sum_{i,j=1}^{m}a_{ij}(x)X_{i}X_{j}+X_{0},$$ where the Xj’s are smooth vector fields in \({\mathbb{R}^{N}}\), and aij is a uniformly elliptic matrix. It is assumed that the Xj’s satisfy homogeneity conditions with respect to a group of dilations δr which yield the existence of a composition law \({\circ}\) in \({\mathbb{R}^{N}}\) making the triplet \({\mathbb G=(\mathbb{R}^{N},\circ,\delta_{r})}\) an homogeneous Lie group on which the Xj’s are left translation invariant. The Holder norms are defined in terms of this composition law. The main tools used are the Taylor formula for smooth functions on \({\mathbb{G}}\), some properties of the corresponding Taylor polynomials, and an orthogonality theorem that extends to homogeneous Lie groups a classical theorem of Calderon and Zygmund in the Euclidean setting.

We prove a Liouville-type theorem for a class of degenerate elliptic operators of the form L u ≔ ∑ i , j = 1 N ∂ x i ( a i j ∂ x j u ) + ∑ i = 1 N b i ∂ x i u . L is supposed to be X -elliptic, with respect to a family X = ( X 1 , … , X m ) of locally Lipschitz continuous vector fields, in the sense introduced in [E. Lanconelli, A.E. Kogoj, X -elliptic operators and X -control distances, Contributions in Honor of the Memory of Ennio De Giorgi, Ricerche di Matematica 49 (Suppl.) (2000) 223–243].

reserved 3 no In this paper, we are concerned with hypoelliptic diffusion operators H. Our main aim is to show, with an axiomatic approach, that a Wiener-type test of H-regularity of boundary points can be derived starting from the following basic assumptions: Gaussian bounds of the fundamental solution of H with respect to a distance satisfying doubling condition and segment property. As a main step toward this result, we establish some estimates at the boundary of the continuity modulus for the generalized Perron–Wiener solution to the relevant Dirichlet problem. The estimates involve Wiener-type series, with the capacities modeled on the Gaussian bounds. We finally prove boundary Hölder estimates of the solution under a suitable exterior cone condition. mixed Lanconelli, Ermanno; Tralli, Giulio; Uguzzoni, Francesco Lanconelli, Ermanno; Tralli, Giulio; Uguzzoni, Francesco

In R 3 , Lipschitz continuous viscosity solutions to the K -prescribed Levi curvature equation are smooth and strictly pseudoconvex if K is smooth and strictly positive; see [5] . We show here that in R 2 n + 1 , n > 1 , a similar result does not hold; that is, we prove the existence in C n + 1 , n > 1 , of nonsmooth pseudoconvex hypersurfaces with smooth Levi curvature.

The aim of this paper is to give some representation formulas of Riesz and Poisson-Jensen type for super-solutions to a class of hypoelliptic ultraparabolic operators \(\mathcal{L}\) on a homogeneous Lie group \(\mathbb{L}\). Our results complete the ones obtained in Cinti (Math Scand 100:1–21, 2007). We also provide a suitable theory for \(\mathcal{L}\)-Green functions and for \(\mathcal{L}\)-Green potentials of Radon measures. The proofs mostly rely on the use of appropriate techniques relevant to the Potential Theory for \(\mathcal{L}\).

Let \({\mathcal{L} = \sum_{i=1}^m X_i^2}\) be a real sub-Laplacian on a Carnot group \({\mathbb{G}}\) and denote by \({\nabla_\mathcal{L} = (X_1,\ldots,X_m)}\) the intrinsic gradient related to \({\mathcal{L}}\). Our aim in this present paper is to analyze some features of the \({\mathcal{L}}\)-gauge functions on \({\mathbb{G}}\), i.e., the homogeneous functions d such that \({\mathcal{L}(d^\gamma) = 0}\) in \({\mathbb{G} \setminus \{0\}}\) , for some \({\gamma \in \mathbb{R} \setminus \{0\}}\). We consider the relation of \({\mathcal{L}}\)-gauge functions with: the \({\mathcal{L}}\)-Eikonal equation \({|\nabla_\mathcal{L} u| = 1}\) in \({\mathbb{G}}\); the Mean Value Formulas for the \({\mathcal{L}}\)-harmonic functions; the fundamental solution for \({\mathcal{L}}\); the Bocher-type theorems for nonnegative \({\mathcal{L}}\)-harmonic functions in “punctured” open sets \({\dot \Omega:= \Omega \setminus \{x_0\}}\).

We prove the existence of a fundamental solution for a class of Hormander heat-type operators. For this fundamental solution and its derivatives we obtain sharp Gaussian bounds that allow to prove an invariant Harnack inequality. To cite this article: M. Bramanti et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).

We study the first order linear system $u_{\tilde{z}} + \bar{v}_{w} = 0, u_{\bar{w}} - \bar{v}_{z} = 0$ in a domain $\Omega \subset \mathbb{C}^{2}$ (first considered by G. Cimmino, [3]). We prove a Morera type theorem, emphasizing the analogy to the Cauchy-Riemann system, and a representation formula yielding a result on removable singularities of solutions to (2). We derive (by a Hilbert space technique outlined in [5]) compatibility relations among the free terms and boundary data in the boundary value problem $u_{\tilde{z}}+ \bar{v}_{w} = f, u_{\bar{w}} - \bar{v}_{z} = g$ in $\Omega$, and $u = \varphi,v = \psi$ on $\partial \Omega$. If $F = (u, v) : \Omega \to \mathbb{C}^{2}$ is a solution to (2) such that $\sup_{\varepsilon \gt O}$\int_{\partial \Omega_{\varepsilon}}|F(z, w)|^{p} d\sigma_{\varepsilon}(z, w} \lt \infty$ for some $p \geq 2$, then we show that $F$ admits non tangential limits at almost every $(\xi, \omega) \in \partial\Omega$.

We construct the fundamental solutionsΓ\Gammaandγ\gammafor the non-divergence form operators∑i,jai,j(x,t)XiXj−∂t\,{\textstyle \sum _{i,\,j}\,} a_{i,\,j}(x,t)\,X_iX_j\,-\,\partial _t\,and∑i,jai,j(x)XiXj{\,\textstyle \sum _{i,\,j}}\,a_{i,\,j}(x)\,X_iX_j, where theXiX_i’s are Hörmander vector fields generating a stratified groupG\mathbb {G}and(ai,j)i,j(a_{i,j})_{i,j}is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates ofΓ\Gammaand its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates ofΓ\Gammaallow us to constructγ\gammausing bothtt-saturation and approximation arguments.

We construct the fundamental solutions for the sub-elliptic operators in non-divergence form∑i,jai,j(x,t)XiXj−∂t{\textstyle \sum _{i,j}} a_{i,j}(x,t)\,X_iX_j-\partial _tand∑i,jai,j(x)XiXj{\textstyle \sum _{i,j}}a_{i,j}(x)\,X_iX_j, where theXiX_i’s form a stratified system of Hörmander vector fields andai,ja_{i,j}are Hölder continuous functions belonging to a suitable class of ellipticity.

Let be a family of vector fields in , where are locally Lipschitz continuous functions in . As usual, we identify the vector field Xj with the first order differential operator In this article we c...

Many results of classical Potential Theory are extended to sub-Laplacians ▵𝔾 on Carnot groups 𝔾. Some characterizations of ▵𝔾-subharmonicity, representation formulas of Poisson-Jensen's kind and Nevanlinna-type theorems are proved. We also characterize the Riesz-measure related to bounded-above ▵𝔾-subharmonic functions in ℝ N .

The maximum principle on a wide class of unbounded domains is proved for solutions to the partial differential inequality Δ G u + cu ≥ 0, where c ≤ 0 and Δ G is a real sub-Laplacian. A potential theory approach is followed.

An inequality generalizing the classical Liouville and Harnack Theorems for real sub-Laplacians ℒ is proved. A representation formula for functions $u$ for which ℒu is a polynomial is also showed. As a consequence, some conditions are given ensuring that u is a polynomial whenever ℒu is a polynomial. Finally, an application of this last result is given: if ψ is a C 2 map commuting with ℒ, then any of its component is a polynomial function.

We prove the Poincaré inequality for vector fields on the balls of the control distance by integrating along subunit paths. Our method requires that the balls are representable by means of suitable “controllable almost exponential maps”.

(2000). Non-existence results for semilinear kohn-laplace equations in unbounded domains. Communications in Partial Differential Equations: Vol. 25, No. 9-10, pp. 1703-1739.

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind \[\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}\left( x\right) \partial _{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}\] where $\left( a_{ij}\right) $ is symmetric uniformly positive definite on $\mathbb{R}^{p_{0}}$ ($p_{0}\leq N$), with uniformly continuous and bounded entries, and $\left( b_{ij}\right) $ is a constant matrix such that the frozen operator $\mathcal{A}_{x_{0}}$ corresponding to $a_{ij}\left( x_{0}\right) $ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1

We extend Dudley’s characterization of convex functions to the framework of h-convex distributions on stratified groups. Precisely, we prove that every distribution with nonnegative horizontal Hessian is defined by an h-convex function.

In this paper, we consider a wide class of second order hypoelliptic partial differential operators with nonnegative characteristic form. We prove a monotone smoothing theorem and a representation formula for superparabolic functions.