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Let $L=sum_{j=1}^m X_j^2$ be a Hörmander sum of squares of vector fields in $R^n$, where any $X_j$ is homogeneous of degree 1 with respect to a family of non-isotropic dilations in $R^n$. Then, $L$ is known to admit a global fundamental solution Γ(x;y) that can be represented as the integral of a fundamental solution of a sublaplacian operator on a lifting space $R^n×R^p$, equipped with a Carnot group structure. The aim of this paper is to prove global pointwise (upper and lower) estimates of Γ, in terms of the Carnot–Carathéodory distance induced by $X={X_1,…,X_m}$ on $R^n$, as well as global pointwise (upper) estimates for the X-derivatives of any order of Γ, together with suitable integral representations of these derivatives. The least dimensional case n=2 presents several peculiarities which are also investigated. Applications to the potential theory for L and to singular-integral estimates for the kernel $X_iX_jΓ$ are also provided. Finally, most of the results about Γ are extended to the case of Hörmander operators with drift $L+X_0$, where $X_0$ is 2-homogeneous and $X_1,…,X_m$ are 1-homogeneous.

We investigate some topics related to the celebrated Baker-Campbell-Hausdorff Theorem: a non-convergence result and prolongation issues. Given a Banach algebra $\mathcal{A}$ with identity $I$, and given $X,Y\in \mathcal{A}$, we study the relationship of different issues: the convergence of the BCH series $\sum_n Z_n(X,Y)$, the existence of a logarithm of $e^Xe^Y$, and the convergence of the Mercator-type series $\sum_n {(-1)^{n+1}}(e^Xe^Y-I)^n/n$ which provides a selected logarithm of $e^Xe^Y$. We fix general results and, by suitable matrix counterexamples, we show that various pathologies can occur, among which we provide a non-convergence result for the BCH series. This problem is related to some recent results, of interest in physics, on closed formulas for the BCH series: while the sum of the BCH series presents several non-convergence issues, these closed formulas can provide a prolongation for the BCH series when it is not convergent. On the other hand, we show by suitable counterexamples that an analytic prolongation of the BCH series can be singular even if the BCH series itself is convergent., Comment: 18pp Additional results and comments added

Let L = ∑ j = 1 m X j 2 be a Hormander sum of squares of vector fields in space R n , where any X j is homogeneous of degree 1 with respect to a family of non-isotropic dilations in space. In this paper we prove global estimates and regularity properties for L in the X-Sobolev spaces W X k , p ( R n ) , where X = { X 1 , … , X m } . In our approach, we combine local results for general Hormander sums of squares, the homogeneity property of the X j 's, plus a global lifting technique for homogeneous vector fields.

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 the existence of a global fundamental solution Γ(x;y) (with pole x) for any Hormander operator L=∑i=1mXi2 on Rn which is δλ-homogeneous of degree 2. Here homogeneity is meant with respect to a family of non-isotropic diagonal maps δλ of the form δλ(x)=(λσ1x1,…,λσnxn), with 1=σ1⩽⋯⩽σn. Due to a global lifting method for homogeneous operators proved by Folland [Comm. Partial Differential Equations 2 (1977) 161–207], there exists a Carnot group G and a polynomial surjective map π:G→Rn such that L is π-related to a sub-Laplacian LG on G. We show that it is always possible to perform a (global) change of variable on G such that the lifting map π becomes the projection of G≡Rn×Rp onto Rn. We prove that an integration argument over the (non-compact) fibers of π provides a fundamental solution for L. Indeed, if ΓG(x,x′;y,y′) (x,y∈Rn; x′,y′∈Rp) is the fundamental solution of LG, we show that ΓG(x,0;y,y′) is always integrable with respect to y′∈Rp, and its y′-integral is a fundamental solution for L.

We outline several results of Potential Theory for a class of linear partial differential operators \(\mathcal {L}\) of the second order in divergence form. Under essentially the sole assumption of hypoellipticity, we present a non-invariant homogeneous Harnack inequality for \(\mathcal {L}\); under different geometrical assumptions on \(\mathcal {L}\) (mainly, under global doubling/Poincare assumptions), it is described how to obtain an invariant, non-homogeneous Harnack inequality. When \(\mathcal {L}\) is equipped with a global fundamental solution \(\varGamma \), further Potential Theory results are available (such as the Strong Maximum Principle). We present some assumptions on \(\mathcal {L}\) ensuring that such a \(\varGamma \) exists.

We consider a class of hypoelliptic second-order operators L in divergence form, arising from CR geometry and Lie group theory, and we prove the Strong and Weak Maximum Principles and the Harnack Inequality. The operators are not assumed in the Hörmander hypoellipticity class, nor to satisfy subelliptic estimates or Muckenhoupt-type degeneracy conditions; indeed our results hold true in the infinitely-degenerate case and for operators which are not necessarily sums of squares. We use a Control Theory result on hypoellipticity to recover a meaningful geometric information on connectivity and maxima propagation, in the absence of any maximal rank condition. For operators L with C^ω coefficients, this control-theoretic result also implies a Unique Continuation property for the L-harmonic functions. The Harnack theorem is obtained via a weak Harnack inequality by means of a Potential Theory argument and the solvability of the Dirichlet problem for L.

Motivated by the physical applications ofq-calculus and ofq-deformations, the aim of this paper is twofold. Firstly, we prove theq-deformed analogue of the celebrated theorem by Baker, Campbell, and Hausdorff for the product of two exponentials. We deal with theq-exponential functionexpq⁡(x)=∑n=0∞(xn/[n]q!), where[n]q=1+q+⋯+qn-1denotes, as usual, thenthq-integer. We prove that ifxandyare any noncommuting indeterminates, thenexpq⁡(x)expq⁡(y)=expq⁡(x+y+∑n=2∞Qn(x,y)), whereQn(x,y)is a sum of iteratedq-commutators ofxandy(on the right and on the left, possibly), where theq-commutator[y,x]q≔yx-qxyhas always the innermost position. When[y,x]q=0, this expansion is consistent with the known result by Schützenberger-Cigler:expq⁡(x)expq⁡(y)=expq⁡(x+y). Our result improves and clarifies some existing results in the literature. Secondly, we provide an algorithmic procedure for obtaining identities between iteratedq-commutators (of any length) ofxandy. These results can be used to obtain simplified presentation for the summands of theq-deformed Baker-Campbell-Hausdorff Formula.

From the introduction to the monograph: Students of scientific disciplines usually meet vector fields during their undergraduate courses in connection with physics, when studying conservative forces and potentials. Then the vector fields reappear in the context of the differential geometry, in the manifold theory, and again in the theory of Lie groups and in the advanced theory of ordinary differential equations. The aim of this book is to provide the reader with a gentle path through the multifaceted theory of vector fields, starting from the definition and the basic properties of vector fields and flows, and ending with some of their countless applications. The building blocks of rich background material comprise the following topics: ∙ commutators and Lie derivatives; the semi-group property and the equation of variation; global vector fields, C^0,C^k,C^ω-dependence; ∙ relatedness, invariance and commutability; Hadamard-type formulas; ∙ composition of flows: Taylor approximation and exact formulas; ∙ the algebraic Campbell-Baker-Hausdorff-Dynkin Theorem; the relevant series and convergence; the Campbell-Baker-Hausdorff-Dynkin operation and its local associativity; Poincaré-type ordinary differential equations; ∙ iterated commutators and the Hörmander bracket-generating condition; connectivity; sub-unit curves and the control distance. Once the background material is established in the first part of the monograph, the application mainly deals, according to the choice of the authors, with the following three settings: ∙ ordinary differential equations theory; ∙ weak and strong maximum principles, propagation principles; ∙ Lie groups, with an emphasis on the construction of Lie groups. This book also provides an introduction to the basic theory of geometrical analysis with a new foundational presentation based on ordinary differential equations techniques, in a unitary and self-contained way.

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 .

We announce an analogue of the celebrated theorem by Campbell, Baker, Hausdorff, and Dynkin for the $q$-exponential $\exp_q(x)=\sum_{n=0}^{\infty} \frac{x^n}{[n]_q!}$, with the usual notation for $q$-factorials: $[n]_q!:=[n-1]_q!\cdot(q^n-1)/(q-1)$ and $[0]_q!:=1$. Our result states that if $x$ and $y$ are non-commuting indeterminates and $[y,x]_q$ is the $q$-commutator $yx-q\,xy$, then there exist linear combinations $Q_{i,j}(x,y)$ of iterated $q$-commutators with exactly $i$ $x$'s, $j$ $y$'s and $[y,x]_q$ in their central position, such that $\exp_q(x)\exp_q(y)=\exp_q\!\big(x+y+\sum_{i,j\geq 1}Q_{i,j}(x,y)\big)$. Our expansion is consistent with the well-known result by Schutzenberger ensuring that one has $\exp_q(x)\exp_q(y)=\exp_q(x+y)$ if and only if $[y,x]_q=0$, and it improves former partial results on $q$-deformed exponentiation. Furthermore, we give an algorithm which produces conjecturally a minimal generating set for the relations between $[y,x]_q$-centered $q$-commutators of any bidegree $(i,j)$, and it allows us to compute all possible $Q_{i,j}$.

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

Some cryptogenic strokes are caused by undetected paroxysmal atrial fibrillation (AF) and could benefit from oral anticoagulation. In this study, we searched for echocardiographic parameters associated with first diagnosed AF, to form a scoring system for the identification of patients with AF. We examined 571 patients with ischemic stroke (72.7 ± 13.5 years, 50.6% women), subdivided into 4 groups: documented cause without AF, first diagnosed AF, known paroxysmal AF, and permanent AF. All patients underwent transthoracic echocardiography, brain computed tomography scan, carotid/vertebral ultrasound, and continuous electrocardiographic monitoring. Eight factors independently characterized first diagnosed AF and formed the "MrWALLETS" score: mitral regurgitation, mild-to-moderate (+1), white matter lesions (-1), age ≥75 years (+1), left atrium ≥4 cm (+1), cerebral lesion diameter ≥4 cm (+1), left ventricular end-diastolic volume

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 aim of this paper is to study some classes of second-order divergence-form partial differential operators L of sub-Riemannian type. Our main assumption is the C^infinity-hypoellipticity of L, together with the existence of a well-behaved fundamental solution Gamma(x, y) for L. We consider the mean-integral operator Mr naturally associated to the mean-value theorem for the L-harmonic functions and we investigate the following topics: the positivity set of the kernel associated to M_r; the role of M_r in solving the homogeneous Dirichlet problem related to L in the Perron–Wiener–Brelot sense; the existence of an inverse mean-value theorem characterizing the sub-Riemannian ‘balls’ Omega_r(x), superlevel sets of Gamma(x, ·). This last result extends a previous theorem by Kuran [Bull. London Math. Soc. 1972]. As side-results, we provide a short proof of the Strong Maximum Principle for L using M_r, a Poisson–Jensen formula for the L-subharmonic functions and several results concerning the geometry of the sets Omega_r(x).

Objectives: The ultrasound investigation of carotid and vertebral arteries is routinely performed in stroke patients to determine the etiopathogenetic classification and possible need of revascularization. However, the medium and long-term prognostic implications of carotid and vertebral ultrasound in ischemic stroke patients are not yet known. Methods: This study included 309 ischemic stroke patients (mean age 76.3; 160 men). They all had undergone carotid and vertebral ultrasound (carotid stenoses were measured according to the European Carotid Surgery Trial [ECST] method). After a median interval of 9.4months, a telephone follow-up was performed to determine their outcome. Dependency or death (modified Rankin scale-mRS >2) and all cause mortality were the study end-points. Results: At follow-up, 158 patients had a mRS >2. In multivariate analysis, of 13 variables univariately predictive of dependency or death, only National Institutes of Health Stroke Scale (NIHSS) score (P2. Sixty-nine patients had died. In a Cox proportional hazards regression, of 10 variables univariately predictive of mortality, only NIHSS score (P&nbsp

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

We prove lower bounds for the fundamental solutions of the non-divergence form operators \[ ∑ i , j a i , j ( x , t ) X i X j − ∂ t and ∑ i , j a i , j ( x ) X i X j , {\textstyle \sum _{i,j}} a_{i,j}(x,t)\,X_iX_j-\partial _t \quad \text {and}\quad {\textstyle \sum _{i,j}}a_{i,j}(x)\,X_iX_j, \] where the X i X_i ’s are Hörmander vector fields generating a stratified group G \mathbb {G} and ( a i , j ) i , j (a_{i,j})_{i,j} is a positive-definite matrix with Hölder continuous entries. We then prove an invariant Harnack inequality for such operators. As a byproduct we also study some relevant properties of the Green functions on bounded domains.

In this survey we consider a general Hormander type operator, represented as a sum of squares of vector fields plus a drift and we outline the central role of the fundamental solution in developing Potential and Regularity Theory for solutions of related PDEs. After recalling the Gaussian behavior at infinity of the kernel, we show some mean value formula on the level set of the fundamental solution, which allow to obtain a comprehensive parallel of the classical Potential Theory. Then we show that a precise knowledge of the fundamental solution leads to global regularity results: estimates at the boundary or on the whole space. Finally in the problem of regularity of non linear differential equations we need an ad hoc modification of the parametrix method, based on the properties of the fundamental solution of an approximating problem.

We prove suitable versions of the weak maximum principle and of the maximum propagation for solutions u of a differential inequality H u ⩾ 0 . Here H = ∑ i , j a i , j ( z ) Z i Z j + Z 0 is a differential operator structured on the vector fields Z j 's, whereas u belongs to an appropriate intrinsic class of regularity modelled on the Z j 's.

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 .

Background: A reduced prevalence of cardiovascular diseases has been reported in liver cirrhosis. However, studies focusing on supraortic district of cirrhotic patients are lacking. Methods: By ultrasound duplex scanning, the presence and severity of atherosclerotic plaques and flow pattern abnormalities were assessed in carotid and vertebral arteries of 118 cirrhotics aged 60.7±12.8 (1 standard deviation) years, and in 236 controls matched with cirrhotic patients according to age, sex and cigarette smoking. Results: Previous ischemic strokes were significantly less numerous in cirrhotic patients than in controls (0.8% vs. 10.5%; P=0.0009); also the prevalence of myocardial infarction was significantly reduced (1.7% vs. 6.4%; P=0.0532). Moreover, cirrhotic patients differed from controls for a lower prevalence of hypertension and hypercholesterolemia, and for a greater proportion of diabetics. Although the presence and severity of atherosclerotic plaques was similar in the two groups, liver cirrhosis was associated with a lower prevalence of abnormal flow patterns (13.6% vs. 29.2%; P=0.0011). The inverse association of hemodynamic changes with liver cirrhosis persisted after all main risk factors were simultaneously taken into account by multiple logistic regression. However, in the presence of hypertension, hypercholesterolemia and cigarette smoking, the ‘protective’ effect of cirrhosis on the occurrence of abnormal flow patterns was no longer detectable. Conclusions: Advanced liver disease is associated with a reduced prevalence of ischemic stroke, which seems to be related to a decreased prevalence of abnormal flow patterns in the supraortic vessels, especially among non-smokers.

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.

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.

We discuss a non-existence result for a semilinear sub-elliptic Dirichlet problem with critical growth on half-spaces of stratified groups of step two. Our result improves a recent theorem (N. Garofalo, D. Vassilev, Math. Ann. 318 (3) (2000) 453–516). To cite this article: A. Bonfiglioli, F. Uguzzoni, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

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.

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.

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.

A classical theorem of Montel states that a family of holomorphic functions on a domain Ω ⊆ C , uniformly bounded on the compact subsets of Ω , is a normal family. The aim of this paper is to obtain a generalization of this result in the subelliptic setting of families of solutions u to L u = 0 , where L belongs to a wide class of real divergence-form PDOs, comprising sub-Laplacians on Carnot groups, subelliptic Laplacians on arbitrary Lie groups, as well as the Laplace–Beltrami operator on Riemannian manifolds. To this end, we extend another remarkable result, due to Koebe: we characterize the solutions to L u = 0 as fixed points of suitable mean-value operators with non-trivial kernels. A suitable substitute for the Cauchy integral formula is also provided. Finally, the local-boundedness assumption is relaxed, by replacing it with L loc 1 -boundedness.

We prove a convergence result for the Campbell–Baker–Hausdorff–Dynkin series in infinite-dimensional Banach–Lie algebras L. In the existing literature, this topic has been investigated when L is the Lie algebra Lie(G) of a finite-dimensional Lie group G (see [Blanes and Casas, 2004]) or of an infinite-dimensional Banach–Lie group G (see [Mérigot, 1974]). Indeed, one can obtain a suitable ODE, which follows from the well-behaved formulas for the differential of the Exponential Map of the Lie group G. The novelty of our approach is to derive this ODE in any infinite-dimensional Banach–Lie algebra, not necessarily associated to a Lie group, as a consequence of an analogous abstract ODE first obtained in the most natural algebraic setting: that of the formal power series in two commuting indeterminates over the free unital associative algebra generated by two non-commuting indeterminates.

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 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.

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).

THE aim of this chapter is to collect all the missing proofs of the results in 3 Chap. 2. The chapter is divided into several sections, corresponding to 4 those of Chap. 2. Finally, Sect. 7.8 collects some proofs from Chaps. 4 and 6 5 too, considered as less crucial in favor of economy of presentation in the 6 chapters they originally belonged to.

THE aim of this chapter is twofold. On the one hand, we aim to study the 4 convergence of the Dynkin series $$ \begin{array}{*{20}c} {uv: = } & {\sum\limits_{j = 1}^\infty {\left( {\sum\limits_{n = 1}^j {\frac{{\left( { - 1} \right)^{n + 1} }}{n}\,\sum\limits_{\begin{array}{*{20}c} {(h_1,k_1 ), \cdots (h_n,k_n ) \ne (0,0)} \\ {h_1 + k_1 + \cdots + h_n + k_n = j} \\ \end{array}} \times \frac{{(ad\,u)^{h_1 } (ad\,\upsilon )^{k_1 } \cdots (ad\,u)^{h_n } (ad\,\upsilon )^{k_n - 1} (\upsilon )}}{{h_1 ! \cdots h_n !k_1 ! \cdots k_n !(\sum\nolimits_{i = 1}^n {(h_i + k_i )} )}}} } \right),} } \\ \end{array} $$ in various contexts. For instance, this series can be investigated in any nilpotent Lie algebra (over a field of characteristic zero) where it is actually a finite sum, or in any finite dimensional real or complex Lie algebra and, more generally, its convergence can be studied in any normed Banach-Lie algebra (over R or C). For example, the case of the normed Banach algebras (becoming normed Banach-Lie algebras if equipped with the associated commutator) will be extensively considered here.