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In Section 2, taking the dimension to be one, we establish a covariance inequality that is valid for any measure on R and that indeed generalizes the L1-Poincar´e inequality (1.4). Then we will consider in Section 3 extensions of our covariance inequalities that are related to Lp-Poincar´e inequalities, for p ≥ 1. In particular, we will ...We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carré du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which ...In this note we state weighted Poincaré inequalities associated with a family of vector fields satisfying Hörmander rank condition. Then, applications are given to relative isoperimetric inequalities and to local regularity (Harnack's inequality) for a class of degenerate elliptic equations with measurable coefficients.We present an improved version of the second-order Gaussian Poincaré inequality, first introduced in Chatterjee (Probab Theory Relat Fields 143(1):1-40, 2009) and Nourdin et al. (J Funct Anal 257(2):593-609, 2009). These novel estimates are used in order to bound distributional distances between functionals of Gaussian fields and normal random variables. Several applications are developed ...We examine the validity of the Poincaré inequality for degenerate, second-order, elliptic operators H in divergence form on \({L_2(\mathbf{R}^{n}\times \mathbf{R}^{m})}\).We assume the coefficients are real symmetric and \({a_1H_\delta\geq H\geq a_2H_\delta}\) for some \({a_1,a_2>0}\) where H δ is a generalized Grušin operator,The results show that Poincare inequalities over quasimetric balls with given exponents and weights are self-improving in the sense that they imply global inequalities of a similar kind, but with ...We prove a Poincaré inequality for Orlicz–Sobolev functions with zero boundary values in bounded open subsets of a metric measure space. This result generalizes the (p, p)-Poincaré inequality for Newtonian functions with zero boundary values in metric measure spaces, as well as a Poincaré inequality for Orlicz–Sobolev …We present an improved version of the second-order Gaussian Poincaré inequality, first introduced in Chatterjee (Probab Theory Relat Fields 143(1):1-40, 2009) and Nourdin et al. (J Funct Anal 257(2):593-609, 2009). These novel estimates are used in order to bound distributional distances between functionals of Gaussian fields and normal random variables. Several applications are developed ...Extensions of the classical Poincaré inequality to non-Euclidean settings have widely been studied in the last decades.A thorough overview of the literature would go out of the scope of the present paper, so we refer the reader to the milestone [] and the references therein.For what concerns Lie groups, a Poincaré inequality on unimodular groups can be obtained by combining [16, §8.3] and ...3 The weighted one dimensional inequality The goal of this section is to prove that the inequality (2.2) holds and to flnd the best possible constant C1. The key point in our argument is the following lemma which gives an inequality for concave functions. Lemma 3.1 Let ‰ be a non negative concave function on [0;1] such that R1 0 ‰(x)dx = 1 ...On the Poincare inequality´ 891 (h1) There exists R >0 such that Ω⊂B(0,R). (h2) There exists a fixed finite cone Csuch that each point x ∈ ∂Ωis the vertex of a cone C x congruent to Cand contained in Ω. (h3) There exists δ 0 >0 such that for any δ∈ (0,δ 0), Ωδis a connected set.Take the square of the inverse of (4a 2 r 2 + 1 e + 2)m (r − 1) as 1 2 β (s) for the desired conclusion. a50 In [24] Eberle showed that a local Poincaré inequality holds for loops spaces over a compact manifold. However the computation was difficult and complicated and there wasn't an estimate on the blowing up rate.Perspective. Poincar e inequalities are central in the study of the geomet-rical analysis of manifolds. It is well known that carrying a Poincar e inequal-ity has strong geometric consequences. For instance, a complete, doubling, non-compact, Riemannian manifold admitting a (1;1;1)-uniform Poincar e inequality satis es an isoperimetric inequality.In this paper we will establish different weighted Poincaré inequalities with variable exponents on Carnot-Carathéodory spaces or Carnot groups. We will use different techniques to obtain these inequalities. For vector fields satisfying Hörmander's condition in variable non-isotropic Sobolev spaces, we consider a weight in the variable Muckenhoupt class $% A_{p(\\cdot ),p^{\\ast }(\\cdot ...Generalized Poincar´e Inequalities Lemma 4.1 (Generalized Poincar´e inequality: Homogeneous case). Let K⊂R3 be a cube of side length L, and define the average of a function f ∈ L1(K) by f K = 1 L3 K f(x)dx. There exists a constant C such that for all measurable sets Ω ⊂Kand all f ∈ H1(K) the inequality K |f(x)−f K|2dx ≤ C L2 Ω ...Nobody has time to read an 80 page paper [LE20]. Therefore I doubt most readers realized the manifold Langevin algorithm paper actually contains a novel technique for establishing functional inequalities. And I really doubt anyone had time to interpret the intuitive consequences of such results on perturbed gradient descent, and definitely not …Here, the Inequality is defined as. Definition. Let p ∈ [1; ∞). A metric measure space (X, d, μ) supports a p -Poincaré inequality, if every ball in X has positive and finite measure ant if there exist constants C > 0 and λ ≥ 1 such that 1 μ(B)∫B | u(x) − uB | dμ(x) ≤ Cdiam(B)( 1 μ(λB)∫λBρ(x)pdμ(x))1 p for every open ...A NOTE ON POINCARE- AND FRIEDRICHS-TYPE INEQUALITIES 5 3. Poincar e-type inequalities in Hm() Now we consider Poincar e-type inequalities in Hm() with m2N 0. Throughout this section let ˆRdbe a bounded domain with Lipschitz boundary. On Hm() we use the inner product (u;v) m= X jsj m Z DsuDsvdx and the induced norm kkLangevin diffusions are rapidly convergent under appropriate functional inequality assumptions. Hence, it is natural to expect that with additional smoothness conditions to handle the discretization errors, their discretizations like the Langevin Monte Carlo (LMC) converge in a similar fashion. This research program was initiated by …On equivalent conditions for the validity of Poincaré inequality on weighted Sobolev space with applications to the solvability of degenerated PDEs involving p-Laplacian. Journal of Mathematical Analysis and Applications, Vol. 432, Issue. 1, p. 463.In this paper, we prove that, in dimension one, the Poincare inequality is equivalent to a new transport-chi-square inequality linking the square of the quadratic Wasserstein distance with the … Expand. 8. PDF. Save. Analysis and Geometry of Markov Diffusion Operators. D. Bakry, I. Gentil, M. Ledoux.The additional assumption on the Poincaré inequality in the second The first part of the Sobolev embedding theor The Poincare inequality appears similar to the "uncertainty principle" except that it is independent of dimension. Both inequalities can be obtained by con-sidering the spectral resolution of a second-order selfadjoint differential operator acting on …The author first reviews the classical Korn inequality and its proof. Following recent works of S. Kesavan, P. Ciarlet, Jr., and the author, it is shown how the Korn inequality can be recovered by an entirely different proof. This new proof hinges on appropriate weak versions of the classical Poincaré and Saint-Venant lemma. In fine, both proofs essentially depend on a crucial lemma of J. L ... The assumption on the measure is the fact that it sa POINCARE INEQUALITIES ON CONVEX SETS´ BY OPTIMAL TRANSPORT METHODS LORENZO BRASCO AND FILIPPO SANTAMBROGIO Abstract. We show that a class of Poincar´e-Wirtinger inequalities on bounded convex sets can be obtained by means of the dynamical formulation of Optimal Transport. ThisStack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Counter example for analogous Poincare ine

1 Answer. Poincaré inequality is true if Ω Ω is bounded in a direction or of finite measure in a direction. ∥φn∥2 L2 =∫+∞ 0 φ( t n)2 dt = n∫+∞ 0 φ(s)2ds ≥ n ‖ φ n ‖ L 2 2 = ∫ 0 + ∞ φ ( t n) 2 d t = n ∫ 0 + ∞ φ ( s) 2 d s ≥ n. ∥φ′n∥2 L2 = 1 n2 ∫+∞ 0 φ′( t n)2 dt = 1 n ∫+∞ 0 φ′(s)2ds ...The Buser inequality is a reverse Cheeger inequality in case of non-negative Ricci curvature stating that λ 1 ≤ C h 2 where λ 1 is the smallest positive eigenvalue of the Laplacian, and h is the Cheeger constant, and C is a constant, see Theorem 3.2.2.Therefore, fractional Poincare inequality hold for all s ∈ (0, 1). Example 2 D as in Theorem 1.2. For s ∈ (1 2, 1) there is an easy geometric characterization for any domain Ω to satisfy LS (s) condition. A domain Ω satisfies LS(s) condition if and only if sup x 0 ∈ R n, ω ∈ σ B C (L Ω (x 0, ω)) < ∞, where the sets L Ω (x 0, ω ...Probability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger …

Equivalent definitions of Poincare inequality. Hot Network Questions Calculate NDos-size of given integer Balancing Indexing and Database Performance: How Many Indexes Are Too Many? Dropping condition from conditional probability How did early computers deal with calculations involving pounds, shillings, and pence? ...$\begingroup$ It seems to me that the Poincare inequality on bounded domains is strictly weaker than (GN)S. Could you confirm whether the exponents in the (1) Poincare-Wirtinger inequality for oscillations around the mean on bounded domains (2) Poincare inequality for functions on domains bounded in only one direction, are optimal (for smooth domains even?)?Inequality (1.1) can be seen as a Poincaré inequality with trace term. The main result of the paper states that balls are the sets which minimize the constant in (1.1) among domains with a given volume. Theorem 1.1 The main result. Let p ∈ [1, + ∞ [.…

Reader Q&A - also see RECOMMENDED ARTICLES & FAQs. Poincare type inequality is one of the main t. Possible cause: Here we show existence of many subsets of Euclidean spaces that, despite having.