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The aim of this paper is to prove an invariant, non-homogeneous Harnack inequality for a class of subelliptic operators L in divergence form, with low-regular coefficients. The main assumption, whose geometric meaning is well known in the literature on Harnack inequalities, is the requirement that L be naturally associated with a Carnot–Caratheodory doubling metric space, where a Poincare inequality also holds. Both doubling and Poincare conditions are assumed to hold globally for every CC-ball: accordingly, the Harnack inequality will hold true on every CC-ball. Applications to inner and boundary Holder estimates are provided, together with pertinent results on the Green function for L . An explicit example of a class of operators for which our results are fulfilled is also given. Via the Green function for L , the global nature of the Harnack inequality can be applied to the study of the existence of a fundamental solution Γ for L , globally defined out of the diagonal of R N × R N .

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.

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

We prove that every homogeneous Carnot group can be lifted to a free homogeneous Carnot group. Though following the ideas of Rothschild and Stein, we give simple and self-contained arguments, providing a constructive proof, as shown in the examples.

In this paper, we provide a non-existence result for a semilinear sub-elliptic Dirichlet problem with critical growth on the half-spaces of any group of Heisenberg-type. Our result improves a recent theorem in (Math. Ann. 315 (3) (2000) 453).

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.

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 characterization of convex functions in RN states that an upper semicontinuous function u is convex if and only if u(Ax) is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix A. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups G when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps x → Ax of the Euclidean case must be replaced by suitable group isomorphisms x → T_A(x), whose differential preserves the first layer of the stratification of Lie(G).

In this paper, we provide a Taylor formula with integral remainder in the setting of homogeneous groups in the sense of Folland and Stein (Hardy spaces on homogeneous groups. Mathematical notes, vol 28. Princeton University Press, Princeton, 1982). This formula allows us to give a simplified proof of the so-called ‘Taylor inequality’. As a by-product, we furnish an explicit expression for the relevant Taylor polynomials. Applications are provided. Among others, it is given a sufficient condition for the real-analiticity of a function whose higher order derivatives (in the sense of the Lie algebra) satisfy a suitable growth condition. Moreover, we prove the ‘L-harmonicity’ of the Taylor polynomials related to a ‘L-harmonic’ function, when L is a general homogenous left-invariant differential operator on a homogeneous group. (This result is one of the ingredients for obtaining Schauder estimates related to L).

Motivated by some problems coming from non-linear subelliptic PDE's, we investigate the equivalence of sub-Laplacians on Carnot groups. We show that all the sub- Laplacians are equivalent if the group is free, while this is not true in the general case. We directly construct dieomorphisms turning general sub-Laplacians into the canonical one, deriving uniform estimates when the operators vary in suitable classes.