Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Let ? = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d _W (·, ·) associated to the generating set {a, b, a ^?1, b ^?1}. Letting B _n = {x ∈ ?: d _W (x, e _?) ? n} denote the corresponding closed ball of radius n ∈ ?, and writing c = [a, b] = aba ^?1 b ^?1, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ? → X satisfies $$\sum\limits_{k = 1}^{{n^2}} {\sum\limits_{x \in {B_n}} {\frac{{\left\| {f(x{c^k}) - f(x)} \right\|_X^q}}{{{k^{1 + q/2}}}}} } \leqslant K\sum\limits_{x \in {B_{21n}}} {(\left\| {f(xa) - f(x)} \right\|_X^q + \left\| {f(xb) - f(x)} \right\|_X^q)} $$ . It follows that for every n ∈ ? the bi-Lipschitz distortion of every f: B _n → X is at least a constant multiple of (log n)^1/q , an asymptotically optimal estimate as n → ∞.     

Hardy’s Inequalities for the Heisenberg Group

Potential Analysis - This article presents two types of Hardy’s inequalities for the Heisenberg group. The proofs are mainly based on estimates of the Fourier transform for atomic functions...     

A Poincaré Cone Condition in the Poincaré Group

Ben Arous and Gradinaru (Potential Anal 8(3):217–258, 1998) described the singularity of the Green function of a general sub-elliptic diffusion. In this article we first adapt their proof to the more general context of a hypoelliptic diffusion. In a second time, we deduce a Wiener criterion and a Poincaré cone condition for a relativistic diffusion with values in the Poincaré group (i.e the group of affine direct isometries of the Minkowski space-time).     

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

A carpet is a metric space homeomorphic to the Sierpiński carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poincaré inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.     

Global bifurcation of fixed points and the Poincaré translation operator on manifolds

Let M be a differentiable manifold and [0, )×MM be a C¹ map satisfying the condition (0, p)=p for all pM. Among other results, we prove that when the degree (also called Hopf index or Euler characteristic) of the tangent vector field wMTM, given by w(p)=(/)(0, p), is well defined and nonzero, then the set (of nontrivial pairs) admits a connected subset whose closure is not compact and meets the slice {0}×M of [0, )×M. This extends known results regarding the existence of harmonic solutions of periodic ordinary differential equations on manifolds.     

Bilinear Sobolev–Poincaré Inequalities and Leibniz-Type Rules

The dual purpose of this article is to establish bilinear Poincaré-type estimates associated with an approximation of the identity and to explore the connections between bilinear pseudo-differential operators and bilinear potential-type operators. The common underlying theme in both topics is their applications to Leibniz-type rules in Sobolev and Campanato–Morrey spaces under Sobolev scaling.     

Duality and the Poincaré–Hopf Inequalities

In this article we obtain a duality result for an n-manifold N with boundary ∂N = N + ⊔N ⁻ a disjoint union, where N ⁺ and N ⁻ are arbitrarily chosen parts in ∂N and need not be compact. This duality result is used to generalize the Poincaré–Hopf inequalities in a non-compact setting.     

Littlewood Paley g-function on the Heisenberg Group

We consider the g-function related to a class of radial functions which gives a characterization of the L^p-norm of a function on the Heisenberg group.     

Super Poincaré Inequalities,Orlicz Norms and Essential Spectrum

We prove some results about the super Poincaré inequality (SPI) and its relation to the spectrum of an operator: we show that it can be alternatively written with Orlicz norms instead of L ¹ norms, and we use this to give an alternative proof that a bound on the bottom of the essential spectrum implies a SPI. Finally, we apply these ideas to give a spectral proof of the log Sobolev inequality for the Gaussian measure.     

Poincaré and Logarithmic Sobolev Inequalities for Nearly Radial Measures

Acta Mathematica Sinica, English Series - Poincaré inequality has been studied by Bobkov for radial measures, but few are known about the logarithmic Sobolev inequality in the radial case. We...     

Higher-Order Multilinear Poincaré and Sobolev Inequalities in Carnot Groups

The notions of higher-order weighted multilinear Poincaré and Sobolev inequalities in Carnot groups are introduced. As an application, weighted Leibniz-type rules in Campanato-Morrey spaces are established.     

Hörmander Multipliers on the Heisenberg Group

 Let be the Heisenberg group of dimension . Let be the partial sub-Laplacians on and T the central element of the Lie algebra of . For any we prove that the operator is bounded on the Hardy spaces , if the function m satisfies a Hrmander-type condition on which depends on . We also obtain analogous results for the operators and , where the function m satisfies analogous H?rmander-type conditions on and on , respectively. Here is the Kohn-Laplacian on . (Received 28 July 1999; in final form 6 March 2000)     

The Compactness of Commutators on the Heisenberg Group

We consider the commutators of the HSrmander multiplier with CMO-functions on the Heisenberg group. The result of compactness on L^P spaces is proved.     

Modulus and the Poincaré inequality on metric measure spaces

The purpose of this paper is to develop the understanding of modulus and the Poincaré inequality, as defined on metric measure spaces. Various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces. Consequently, modulus is shown to be upper semi-continuous with respect to the limit of a sequence of curve families contained in a converging sequence of metric measure spaces. Moreover, several competing definitions for the Poincaré inequality are shown to coincide, if the underlying measure is doubling. One such characterization considers only continuous functions and their continuous upper gradients, and extends work of Heinonen and Koskela. Applications include showing that the p-Poincaré inequality (with a doubling measure), for p1, persists through to the limit of a sequence of converging pointed metric measure spaces — this extends results of Cheeger. A further application is the construction of new doubling measures in Euclidean space which admit a 1-Poincaré inequality. Mathematics Subject Classification (2000):31C15, 46E35.     

Self-improvement of Poincaré Type Inequalities Associated with Approximations of the Identity and Semigroups

The purpose of this paper is to present a general method that allows us to study self-improving properties of generalized Poincaré inequalities. When measuring the oscillation in a given cube, we replace the average by an approximation of the identity or a semigroup scaled to that cube and whose kernel decays fast enough. We apply the method to obtain self-improvement in the scale of Lebesgue spaces of Poincaré type inequalities. In particular, we propose some expanded Poincaré estimates that take into account the lack of localization of the approximation of the identity or the semigroup. As a consequence of this method we are able to obtain global pseudo-Poincaré inequalities.     

Poincar and Weak Poincar Inequalities for the Mixed Poisson Measure

By using the Mecke identity, we study a class of birth-death type Dirichlet forms associated with the mixed Poisson measure. Both Poincar and weak Poincar inequalities are established, while another Poincar type inequality is disproved under some reasonable assumptions.     

Sharp Weighted Young's Inequalities and Moser–Trudinger Inequalities on Heisenberg Type Groups and Grushin Spaces

We obtain sharp weighted Moser–Trudinger inequalities for first-layer symmetric functions on groups of Heisenberg type, and for -symmetric functions on the Grushin plane. To this end, we establish weighted Young's inequalities in the form , for first-layer radial weights on a general Carnot group and functions with first-layer symmetric. The proofs use some sharp estimates for hypergeometric functions.Research supported by NSF grant DMS-0228807.     

Integral representations and the generalized Poincaré inequality on Carnot groups

We give some integral representations of the form f(x) = P(f)+K(?f) on two-step Carnot groups, where P(f) is a polynomial and K is an integral operator with a specific singularity. We then obtain the weak Poincaré inequality and coercive estimates as well as the generalized Poincaré inequality on the general Carnot groups.     

Weyl Pseudo-Differential Operator and Wigner Transform on the Poincaré Disk

The purpose of this paper is to investigate some relations between the kernel of a Weyl pseudo-differential operator and the Wigner transform on Poincaré disk defined in our previous paper [11]. The composition formula for the class of the operators defined in [11] has not been proved yet. However, some properties and relations, which are analogous to the Euclidean case, between the Weyl pseudo-differential operator and the Wigner transform have been investigated in [11]. In the present paper, an asymptotic formula for the Wigner transform of the kernel of a Weyl pseudo-differential operator as 0 is given. We also introduce a space of functions on the cotangent bundle T ^* D whose definition is based on the notion of the Schwartz space on the Poincaré disk. For an S ¹-invariant symbol in that space, we obtain a formula to reproduce the symbol from the kernel of the Weyl pseudo-differential operator.