Hypoelliptic heat kernel inequalities on Lie groups

This paper discusses the existence of gradient estimates for the heat kernel of a second order hypoelliptic operator on a manifold. For elliptic operators, it is now standard that such estimates (satisfying certain conditions on coefficients) are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated “Ricci curvature” takes on the value −∞$-\infty$ at points of degeneracy of the semi-Riemannian metric. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting.     

Lower bounds for operators on graded Lie groups

In this note we present a symbolic pseudo-differential calculus on any graded (nilpotent) Lie group and, as an application, a version of the sharp Gårding inequality. As a corollary, we obtain lower bounds for positive Rockland operators with variable coefficients as well as their Schwartz-hypoellipticity.     

Multipliers for Besov spaces on graded Lie groups

In this note, we give embeddings and other properties of Besov spaces, as well as spectral and Fourier multiplier theorems, in the setting of graded Lie groups. We also present a Nikolskii-type inequality and the Littlewood–Paley theorem that play a role in this analysis and are also of interest on their own.     

Hardy and uncertainty inequalities on stratified Lie groups

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group G  . In particular, we show that the operators _Tα:f?|⋅|^−αL^−α/2f$T_{\alpha }:f?{|\cdot |}^{-\alpha }{L}^{-\alpha /2}f$, where |⋅|$|\cdot |$ is a homogeneous norm, 0<α<Q/p$0<\alpha <Q/p$, and L   is the sub-Laplacian, are bounded on the Lebesgue space L^p(G)$L^p(G)$. As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg–Pauli–Weyl inequality, relating the L^p$L^p$ norm of a function f   to the L^q$L^q$ norm of |⋅|^βf${|\cdot |}^{\beta }f$ and the L^r$L^r$ norm of L^δ/2f$L^{\delta /2}f$.     

Improved Sobolev inequalities and Muckenhoupt weights on stratified Lie groups

We study in this article the improved Sobolev inequalities with Muckenhoupt weights within the framework of stratified Lie groups. This family of inequalities estimate the Lq norm of a function by the geometric mean of two norms corresponding to Sobolev spaces and Besov spaces . When the value p which characterizes Sobolev space is strictly larger than 1, the required result is well known in Rn and is classically obtained by a Littlewood-Paley dyadic blocks manipulation. For these inequalities we will develop here another totally different technique. When p=1, these two techniques are not available anymore and following M. Ledoux (2003) [12], in Rn, we will treat here the critical case p=1 for general stratified Lie groups in a weighted functional space setting. Finally, we will go a step further with a new generalization of improved Sobolev inequalities using weak-type Sobolev spaces.     

Resolvent estimates and smoothing for homogeneous partial differential operators on graded Lie groups

By using commutator methods, we show uniform resolvent estimates and obtain globally smooth operators for self-adjoint injective homogeneous operators H on graded groups, including Rockland operators, sublaplacians, and many others. Left or right invariance is not required. Typically the globally smooth operator has the form T = V|H|^1∕2, where V only depends on the homogeneous structure of the group through Sobolev spaces, the homogeneous dimension and the minimal and maximal dilation weights. For stratified groups improvements are obtained, by using a Hardy-type inequality. Some of the results involve refined estimates in terms of real interpolation spaces and are valid in an abstract setting. Even for the commutative group ?^N some new classes of partial differential operators are treated.     

Refined Heinz operator inequalities

Motivated by the well-known Heinz norm inequalities, in this article we study the corresponding Heinz operator inequalities. We derive the whole series of refinements of these operator inequalities, first with the help of the well-known Hermite–Hadamard inequality, and then, utilizing the parametrized family of the so-called Heron means. In such a way, we obtain improvements of some recent results, known from the literature.     

Refined convex Sobolev inequalities

This paper is devoted to refinements of convex Sobolev inequalities in the case of power law relative entropies: a nonlinear entropy-entropy production relation improves the known inequalities of this type. The corresponding generalized Poincaré-type inequalities with weights are derived. Optimal constants are compared to the usual Poincaré constant.     

Caffarelli–Kohn–Nirenberg inequalities on Lie groups of polynomial growth

In the setting of a Lie group of polynomial volume growth, we derive inequalities of Caffarelli–Kohn–Nirenberg type, where the weights involved are powers of the Carnot–Caratheodory distance associated with a fixed system of vector fields which satisfy the Hörmander condition. The use of weak spaces is crucial in our proofs and we formulate these inequalities within the framework of Lorentz spaces (a scale of (quasi)‐Banach spaces which extend the more classical Lebesgue spaces) thereby obtaining a refinement of, for instance, Sobolev and Hardy–Sobolev inequalities.     

Sub-elliptic global high order Poincaré inequalities in stratified Lie groups and applications

Sharp Poincaré inequalities on balls or chain type bounded domains have been extensively studied both in classical Euclidean space and Carnot-Carathéodory spaces associated with sub-elliptic vector fields (e.g., vector fields satisfying Hörmander's condition). In this paper, we investigate the validity of sharp global Poincaré inequalities of both first order and higher order on the entire nilpotent stratified Lie groups or on unbounded extension domains in such groups. We will show that simultaneous sharp global Poincaré inequalities also hold and weighted versions of such results remain to be true. More precisely, let G be a nilpotent stratified Lie group and f be in the localized non-isotropic Sobolev space , where 1?p<Q/m and Q is the homogeneous dimension of the Lie group G. Suppose that the mth sub-elliptic derivatives of f is globally Lp integrable; i.e., is finite (but assume that lower order sub-elliptic derivatives are only locally Lp integrable). We denote the space of such functions as Bm,p(G). We prove a high order Poincaré inequality for f minus a polynomial of order m−1 over the entire space G or unbounded extension domains. As applications, we will prove a density theorem stating that smooth functions with compact support are dense in Bm,p(G) modulus a finite-dimensional subspace.     

Special symplectic Lie groups and hypersymplectic Lie groups

A special symplectic Lie group is a triple ${(G,\omega,\nabla)}$ such that G is a finite-dimensional real Lie group and ω is a left invariant symplectic form on G which is parallel with respect to a left invariant affine structure ${\nabla}$ . In this paper starting from a special symplectic Lie group we show how to “deform” the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure ${\nabla}$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.     

Lie groups

The survey deals with the investigations reviewed inReferativnyi Zhurnal Matematika between 1977–1981. In the survey there are reflected the investigations on the structure of Lie groups and Lie algebras, on their finite-dimensional linear representations and universal enveloping algebras, on the theory of invariants and Lie groups of transformations, and also on continuous and discrete subgroups of Lie groups.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 20, No. 153–192, 1982.     

Appell systems on Lie groups

Using the probabilistic interpretation of Appell polynomials as systems of moments, we show how to define them in the noncommutative case. The method is based on certain infinite-dimensional representations of local Lie groups. For processes, limit theorems play an essential role in the construction. Polynomial matrix representations of convolution semigroups are a principal feature.     

Ornstein-Uhlenbeck processes on Lie groups

We consider Ornstein-Uhlenbeck processes (OU-processes) associated to hypo-elliptic diffusion processes on finite-dimensional Lie groups: let L be a hypo-elliptic, left-invariant “sum of the squares”-operator on a Lie group G with associated Markov process X, then we construct OU-processes by adding negative horizontal gradient drifts of functions U. In the natural case U(x)=−logp(1,x), where p(1,x) is the density of the law of X starting at identity e at time t=1 with respect to the right-invariant Haar measure on G, we show the Poincaré inequality by applying the Driver-Melcher inequality for “sum of the squares” operators on Lie groups. The resulting Markov process is called the natural OU-process associated to the hypo-elliptic diffusion on G. We prove the global strong existence of these OU-type processes on G under an integrability assumption on U. The Poincaré inequality for a large class of potentials U is then shown by a perturbation technique. These results are applied to obtain a hypo-elliptic equivalent of standard results on cooling schedules for simulated annealing on compact homogeneous spaces M.     

Elliptic operators on Lie groups

We review the theory of strongly elliptic operators on Lie groups and describe some new simplifications. Let U be a continuous representation of a Lie group G on a Banach space and a ₁,...,a _d a basis of the Lie algebra g of G. Let A _i=dU(a _i) denote the infinitesimal generator of the continuous one-parameter group t U(exp(-ta _i)) and set % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqVa0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCaaale% qajeaObaGaeyySdegaaOGaeyypa0JaamyqamaaBaaajeaWbaGaaeyA% aaWcbeaajaaOdaWgaaqcbaAaamaaBaaajiaObaGaaiiBaaqabaaaje% aObeaakiaacElacaGG3cGaai4TaiaadgeadaWgaaqcbaCaaiaabMga% aSqabaGcdaWgaaWcbaWaaSbaaKGaahaacaGGUbaameqaaaWcbeaaaa% a!4897!\[A^\alpha = A_{\rm{i}} _{_l } \cdot\cdot\cdotA_{\rm{i}} _{_n } \], where =(i ₁,...,i _n) with _j and set ||=n. We analyze properties of mth order differential operators % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqFj0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2da9i% aabccadaaeqaqaaiaadogadaWgaaqcbaCaaiabgg7aHbWcbeaaaKqa% GgaacqGHXoqycaqG7aGaaeiiaiaabYhacqGHXoqycaqG8bGaeyizIm% QaaeyBaaWcbeqdcqGHris5aOGaamyqamaaCaaaleqajeaObaGaeyyS% degaaaaa!4A6C!\[H = {\rm{ }}\sum\nolimits_{\alpha {\rm{; |}}\alpha {\rm{|}} \le {\rm{m}}} {c_\alpha } A^\alpha \] with coefficients c . If H is strongly elliptic, i.e., % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqFj0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciOuaiaacwgacq% GH9aqpcaqGGaWaaabeaeaacaGGOaaajeaObaGaeyySdeMaae4oaiaa% bccacaqG8bGaeyySdeMaaeiFaiabg2da9iaab2gaaSqab0GaeyyeIu% oakiaabMgacqaH+oaEcaGGPaWaaWbaaSqabKqaGgaacqGHXoqyaaGc% cqGH+aGpcaaIWaaaaa!4C40!\[{\mathop{\rm Re}\nolimits} = {\rm{ }}\sum\nolimits_{\alpha {\rm{; |}}\alpha {\rm{|}} = {\rm{m}}} ( {\rm{i}}\xi )^\alpha > 0\] for all % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqVa0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdGNaeyicI4% SaeSyhHe6aaWbaaSqabeaacaWGKbaaaOGaaiixaiaacUhacaaIWaGa% aiyFaaaa!3EAA!\[\xi \in ^d \backslash \{ 0\} \], then we give a simple proof of the theorem that the closure of H generates a continuous (and holomorphic) semigroup on and the action of the semigroup is determined by a smooth, representation independent, kernel which, together with all its derivatives, satisfies mth order Gaussian bounds.     

Runge-Kutta methods on Lie groups

We construct generalized Runge-Kutta methods for integration of differential equations evolving on a Lie group. The methods are using intrinsic operations on the group, and we are hence guaranteed that the numerical solution will evolve on the correct manifold. Our methods must satisfy two different criteria to achieve a given order.

Berezin-Toeplitz quantization on Lie groups

Let K be a connected compact semisimple Lie group and KC its complexification. The generalized Segal-Bargmann space for KC is a space of square-integrable holomorphic functions on KC, with respect to a K-invariant heat kernel measure. This space is connected to the “Schrödinger” Hilbert space L²(K) by a unitary map, the generalized Segal-Bargmann transform. This paper considers certain natural operators on L²(K), namely multiplication operators and differential operators, conjugated by the generalized Segal-Bargmann transform. The main results show that the resulting operators on the generalized Segal-Bargmann space can be represented as Toeplitz operators. The symbols of these Toeplitz operators are expressed in terms of a certain subelliptic heat kernel on KC. I also examine some of the results from an infinite-dimensional point of view based on the work of L. Gross and P. Malliavin.