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We prove weighted\({L^p}\)-Liouville theorems for a class of second-order hypoelliptic partial differential operators \({\mathcal{L}}\) on Lie groups \({\mathbb{G}}\) whose underlying manifold is \({n}\)-dimensional space. We show that a natural weight is the right-invariant measure \(\check{H}\) of \({\mathbb{G}}\). We also prove Liouville-type theorems for \({C^{2}}\) subsolutions in \({L^{p}(\mathbb{G},\check{H})}\). We provide examples of operators to which our results apply, jointly with an application to the uniqueness for the Cauchy problem for the evolution operator \({\mathcal{L}-\partial_{t}}\).

We prove a quantitative openness theorem for $C^1$ submersions under suitable assumptions on the differential. We then apply our result to a class of exponential maps appearing in Carnot-Carath\'eodory spaces and we improve a classical completeness result by Palais., Comment: 12 pages. Revised version. Minor changes. To appear on Annali di Matematica

The aim of this paper is to provide a comprehensive exposition of the early contributions to the so-called Campbell, Baker, Hausdorff, Dynkin Theorem during the years 1890–1950. Related works by Schur, Poincaré, Pascal, Campbell, Baker, Hausdorff, and Dynkin will be investigated and compared. For a full recovery of the original sources, many mathematical details will also be furnished. In particular, we rediscover and comment on a series of five notable papers by Pascal (Lomb Ist Rend, 1901–1902), which nowadays are almost forgotten.

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.

If $${\mathcal{L} = {\sum_{j=1}^m} {X_j^2} + X_0}$$ is a Hormander partial differential operator in $${\mathbb{R}^N}$$ , we give sufficient conditions on the vector fields X j ’s for the existence of a Lie group structure $${\mathbb{G} = (\mathbb{R}^N, *)}$$ (and we exhibit its construction), not necessarily nilpotent nor homogeneous, such that $${\mathcal{L}}$$ is left invariant on $${\mathbb{G}}$$ . The main tool is a formula of Baker-Campbell-Dynkin-Hausdorff type for the ODE’s naturally related to the system of vector fields {X 0, . . . , X m }. We provide a direct proof of this formula in the ODE’s context (which seems to be missing in literature), without invoking any result of Lie group theory, nor the abstract algebraic machinery usually involved in formulas of Baker-Campbell-Dynkin-Hausdorff type. Examples of operators to which our results apply are also furnished.

Let $${\mathbb{G}}$$ be a Carnot group of step r and m generators and homogeneous dimension Q. Let $${\mathbb{F}_{m,r}}$$ denote the free Lie group of step r and m generators. Let also $${\pi:\mathbb{F}_{m,r}\to\mathbb{G}}$$ be a lifting map. We show that any horizontally convex function u on $${\mathbb{G}}$$ lifts to a horizontally convex function $${u\circ \pi}$$ on $${\mathbb{F}_{m,r}}$$ (with respect to a suitable horizontal frame on $${\mathbb{F}_{m,r}}$$ ). One of the main aims of the paper is to exhibit an example of a sub-Laplacian $${\mathcal{L}=\sum_{j=1}^m X_j^2}$$ on a Carnot group of step two such that the relevant $${\mathcal{L}}$$ -gauge function d (i.e., d 2-Q is the fundamental solution for $${\mathcal{L}}$$ ) is not h-convex with respect to the horizontal frame {X 1, . . . , X m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).

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 \({\dot \Omega:= \Omega \setminus \{x_0\}}\).

In this paper, we provide a suitable theory for the energy where μ is a Radon measure and Γ is the fundamental solution of a sub-Laplacian on a stratified group As a significant application, we prove the quasi-continuity of superharmonic functions related to . The proofs are elementary and mostly rely on the use of appropriate mean-value formulas and mean-integral operators relevant to the Potential Theory for .

The aim of this paper is to prove a representation formula of Poisson–Jensen type on bounded domains Ω for subharmonic functions related to sub-Laplacians ΔG on stratified Lie groups G. Our result gives a complete answer to a question arisen in Math. Ann. 325 (2003), 97–122, where the additional hypothesis that Ω is regular for the Dirichlet problem related to ΔG was made: here we treat the case of arbitrary bounded domains Ω, omitting the hypothesis of regularity. In order to prove our representation formula, an ad-hoc theory of polar sets and capacity with respect to ΔG is also provided.

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 .

A generalization and some applications of the so-called Pizzetti's Formula (which expresses the integral mean of a smooth function over an Euclidean ball as a power series w.r.t. the radius of the ball, having the iterated of the ordinary Laplace operator as coefficients) is given for $$\Delta _{\mathbb{H}^N } $$ , the Kohn Laplace operator on the Heisenberg group. A formula expressing the n-th power of $$\Delta _{\mathbb{H}^N } $$ is also proved. In the case of the ordinary Laplace operator, by Pizzetti's formula, we prove in a simple way that the only nonnegative polyharmonic functions are polynomials.

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.