Poincare inequality

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 ≥

You haven't exactly followed the hint, but your proof seems correct. As pointed out by Chee Han, you could follow the hint by squaring the given identity (using the Cauchy-Schwarz inequality like you did), integrating from $0$ to $1$ and exchanging the order of integration.Jul 8, 2010 · MATHEMATICS OF COMPUTATION Volume 80, Number 273, January 2011, Pages 119–140 S 0025-5718(2010)02296-3 Article electronically published on July 8, 2010 Poincaré inequalities for Markov chains: a meeting with Cheeger, Lyapunov and Metropolis Christophe Andrieu, Anthony Lee, Sam Power, Andi Q. Wang School of Mathematics, University of Bristol August 11, 2022 Abstract We develop a theory of weak Poincaré inequalities to characterize con-vergence rates of ergodic Markov chains.This caused me to investigate the 1913 edition of Websters Dictionary - which is now in the public domain. However, after a day's work wrangling it into a ...Studying the heat semigroup, we prove Li–Yau-type estimates for bounded and positive solutions of the heat equation on graphs. These are proved under the assumption of the curvature-dimension inequality CDE′⁢(n,0){\\mathrm{CDE}^{\\prime}(n,0)}, which can be considered as a notion of curvature for graphs. We further show that non …Abstract. In an \(n\)-dimensional bounded domain \(\Omega_n\), \(n\ge 2\), we prove the Steklov-Poincaré inequality with the best constant in the case where \(\Omega_n\) is an \(n\)-dimensional ball.We also consider the case of an unbounded domain with finite measure, in which the Steklov-Poincaré inequality is proved on the basis of a Sobolev inequality.Apr 13, 2018 · For what it's worth, I'm looking at the book and Evans writes "This estimate is sometimes called Poincare's inequality." (Page 282 in the second edition.) See also the Wiki article or Wolfram Mathworld, which have somewhat divergent opinions on what should or shouldn't be called a Poincare inequality.

DOI: 10.1214/ECP.V13-1352 Corpus ID: 18581137; A simple proof of the Poincaré inequality for a large class of probability measures @article{Bakry2008ASP, title={A simple proof of the Poincar{\'e} inequality for a large class of probability measures}, author={Dominique Bakry and Franck Barthe and Patrick Cattiaux and Arnaud Guillin}, journal={Electronic Communications in Probability}, year ...From Poincar\'e Inequalities to Nonlinear Matrix Concentration. June 2020. This paper deduces exponential matrix concentration from a Poincar\'e inequality via a short, conceptual argument. Among ...Henri Poincaré was a mathematician, theoretical physicist and a philosopher of science famous for discoveries in several fields and referred to as the last polymath, one who could make significant contributions in multiple areas of mathematics and the physical sciences. This survey will focus on Poincaré's philosophy.Matteo Levi, Federico Santagati, Anita Tabacco, Maria Vallarino. We prove local Lp -Poincaré inequalities, p ∈ [1, ∞], on quasiconvex sets in infinite graphs endowed with a family of locally doubling measures, and global Lp -Poincaré inequalities on connected sets for flow measures on trees. We also discuss the optimality of our results.Poincar e inequalities and geometric bounds themodern era : Lichnerowicz’s bound (1958) (M;g)compact Riemannian manifold normalized Riemannian volume elementfor all Ω ∈ C, all Lipschitz continuous functions f on Ω, and all weights w which are any positive power of a non-negative concave function on Ω is the same as the best constant for the corresponding one-dimensional situation, where C reduces to the class of bounded intervals. Using facts from 'Sharp conditions for weighted 1-dimensional Poincaré inequalities', by S.-K. Chua and R. L ...

The Poincaré inequality need not hold in this case. The region where the function is near zero might be too small to force the integral of the gradient to be large enough to control the integral of the function. For an explicit counterexample, let. Ω = {(x, y) ∈ R2: 0 < x < 1, 0 < y < x2} Ω = { ( x, y) ∈ R 2: 0 < x < 1, 0 < y < x 2 }Theorem 1. ForanysimpleconnectedgraphG,if isasetofcanonicalpathsthatsatisfies8 ,then 4d2b jEj,hencethePoincaréboundissuperiortotheCheegerboundforthischoiceofpaths.The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincaré inequality. The key point is the implementation of a refinement of the classical Pólya-Szegö inequality for the symmetric decreasing rearrangement which yields an optimal weighted Wirtinger inequality.(i) It suffices to prove the inequality (1) for all f ∈ C∞. 0 (Ω). In this context we need the generalized H ̈ older inequality, namely, if fj ∈ Lpj(Ω), = 1, · · · , m, such that p−1 + . · · · …reverse poincare inequality for polynomials with vanishing boundary. Hot Network Questions Early 1980s short story (in Asimov's, probably) - Young woman consults with "Eliza" program, and gives it anxiety Understanding TLS Protections Against DNS Spoofing and Fake Websites Eliminating one variable from two simple polynomial equations ...

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Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze. We study the validity of the Lp inequality for the Riesz transform when p > 2 and of its reverse inequality when 1 < p < 2 on complete Riemannian manifolds under the doubling property and some Poincare inequalities. MSC numbers 2000: 58J35, 42B20. View PDF on arXiv.Poincar´e inequality, this paper studies the weaker Orlicz–Poincar´e inequality. More precisely, for any Young function Φ whose growth is slower than quadric, the Orlicz–Poincar´e inequality f 2 Φ CE(f,f),µ(f):= f dµ =0 is studied by using the well-developed weak Poincar´e inequalities, where E is a conservative DirichletThe first part of the Sobolev embedding theorem states that if k > ℓ, p < n and 1 ≤ p < q < ∞ are two real numbers such that. and the embedding is continuous. In the special case of k = 1 and ℓ = 0, Sobolev embedding gives. This special case of the Sobolev embedding is a direct consequence of the Gagliardo-Nirenberg-Sobolev inequality.A NOTE ON WEIGHTED IMPROVED POINCARÉ-TYPE INEQUALITIES 2 where C > 0 is a constant independent of the cubes we consider and w is in the class A∞ of all Muckenhoupt weights. The authors remark that, although the A∞ condition is assumed, the A∞ constant, which is defined by (1.3) [w]A∞:= sup Q∈QJun 27, 2023 · In mathematics, the Poincaré inequality [1] is a result in the theory of Sobolev spaces, named after the France mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, direct methods ... The rest of the paper is arranged as follows. In Section 2, Poincaré-type inequalities are proved for functions in W1,p(Ω) which vanish on the boundary ∂Ω or in ω. In Sec-tion 3, Friedrichs-type inequalities are proved inW1,p(Ω) with respect to two integral functionals. 2. Poincaré-type inequalities

tain the exact constants in the inequalities. Finally, we apply our result to study solutions of ordinary differential equations with given average value. 1. Introduction. If 1 < q, p < <x>, and -oo < a < b < oo, there is a constant TPA depending on a, b such that the one dimensional Poincaré type inequality / (b i rb i \l/ci / rb \1'pIn functional analysis, Sobolev inequalities and Morrey's inequalities are a collection of useful estimates which quantify the tradeoff between integrability and smoothness. The ability to compare such properties is particularly useful when studying regularity of PDEs, or when attempting to show boundedness in a particular space in order to ...WEIGHTED POINCARE INEQUALITY AND THE POISSON EQUATION 5´ as (1.5) for each annulus. However, instead of the weighted Poincar´e inequality, we now use Poincar´e inequality by appealing to a result of Li and Schoen [15] on the estimate of the bottom spectrum of a geodesic ball in terms of the Ricci curvature lower bound and its radius.For what it's worth, I'm looking at the book and Evans writes "This estimate is sometimes called Poincare's inequality." (Page 282 in the second edition.) See also the Wiki article or Wolfram Mathworld, which have somewhat divergent opinions on what should or shouldn't be called a Poincare inequality.Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products.sequence of this inequality, one obtains immediately the "existence" part of the Fredholm alternative for the positive Dirichlet Laplacian −Δ at the first eigenvalue λ1. In this article we replace the power 2 by p (2 ≤ p<∞) and thus extend inequality (1.1) to the "degenerate" case 2 <p<∞. A simplified version ofIn mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré.In the link above, the generalization of the Poincare inequality to general measure spaces is considered as well. I searched for papers myself but was not able to find anything specialized to Gaussian measures. Could anyone please help me? pr.probability; inequalities; gaussian; Share.Analogous to , higher order Poincaré inequality involving higher order derivatives also holds in \(\mathbb {H}^{N}\). In this context, a worthy reference on this inequality is [22, Lemma 2.4] where it has been shown that for k and l be non-negative integers with \(0\le l<k\) there holdsplete manifolds with weighted Poincar´e inequality which is of independent interest. In [17], Li and Wang studied complete manifolds with satisfying property (P ρ) and obtained many theorems on rigidity. Cheng and Zhou [5] generalized one result of [17]. Li and the first author in [10] recently refined the main results due to Cheng and Zhou ...

GLOBAL SENSITIVITY ANALYSIS AND POINCARE INEQUALITIES´ 6-8 JULY 2022 TOULOUSE Contents 1. Introduction 2 2. The diffusion operator associated to the measure 3 2.1. Link with a diffusion operator 3 2.2. The spectrum and the semi-group of the diffusion operator 4 2.3. The Poincar´e inequality, the spectral gap and the convergence of the

GLOBAL SENSITIVITY ANALYSIS AND POINCARE INEQUALITIES´ 6-8 JULY 2022 TOULOUSE Contents 1. Introduction 2 2. The diffusion operator associated to the measure 3 2.1. Link with a diffusion operator 3 2.2. The spectrum and the semi-group of the diffusion operator 4 2.3. The Poincar´e inequality, the spectral gap and the convergence of theLp for all k, and hence the Poincar e inequality must fail in R. 3 Poincar e Inequality in Rn for n 2 Even though the Poincar e inequality can not hold on W1;p(R), a variant of it can hold on the space W1;p(Rn) when n 2. To see why this might be true, let me rst explain why the above example does not serve as a counterexample on Rn. The doubling condition and the Poincar e inequality are relatively standard assumptions in analysis on metric measure spaces. There are several phenomena in harmonic analysis and PDEs for which a (q;p ")-Poincar e inequality for some ">0 would be a more natural assumption than a (q;p)-Poincar e inequality. This isSo basically, I have proved the Poincare's inequality for p = 1 case. That is, for u ∈ W 1, 1 ( Ω), I have | | u − u ¯ | | L 1 ≤ C | | ∇ u | | L 1. Here u ¯ is the average of u on Ω. Now I need to get the general p case, i.e., for u ∈ W 1, p ( Ω), there is | | u − u ¯ | | L p ≤ C | | ∇ u | | L p. My professor in class ...Hence the best constant of Poincare inequality is just $1/\lambda_1$? Am I correct? I think this problem has been well studied. So if you know where I can find a good reference, please kindly direct me there. Thank you! sobolev …We consider a domain $$\\varOmega \\subset \\mathbb {R}^d$$ Ω ⊂ R d equipped with a nonnegative weight w and are concerned with the question whether a Poincaré inequality holds on $$\\varOmega $$ Ω , i.e., if there exists a finite constant C independent of f such that It turns out that it is essentially sufficient that on all superlevel sets of w there hold Poincaré inequalities w.r.t ...There exists an open set of data satisfying the indicated required conditions, obtained by first choosing $\lambda_0$ greater than some constant linked with the Poincaré inequality of the manifold $(S, \sigma)$." Here, I don't really know how to use this inequality. If I could have some sort of inequality

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Moreover, if a p-logarithmic Sobolev inequality holds then the Poincaré inequality is shown to hold too, therefore the previous regularization result is valid. Finally, the weighted Sobolev-type inequality ‖ u ‖ q ⩽ C E (p) (u) (q < p) implies L q 0 - L ϱ regularization of the evolution for any ϱ < ϱ ˜, all q 0 < ϱ ˜ and an ...MATRIX POINCARE INEQUALITIES AND CONCENTRATION 3´ its scalar counterpart, establishing a matrix concentration inequality is reduced to proving a matrix Poincar´e inequality. To this aim, for a given probability measure, the main task lies in designing the appropriate Markov generator and calculating the corresponding matrix carr´e du champ ...The sharp Sobolev type inequalities in the Lorentz-Sobolev spaces in the hyperbolic spaces. Journal of Mathematical Analysis and Applications, Vol. 490, Issue. 1, p. 124197. Journal of Mathematical Analysis and Applications, Vol. 490, Issue. 1, p. 124197.in a manner analogous to the classical proof. The discrete Poincare inequality would be more work (and the constant there would depend on the boundary conditions of the difference operator). But really, I would also like this to work for e.g. centered finite differences, or finite difference kernels with higher order of approximation. Abstract. L p Poincaré inequalities for general symmetric forms are established by new Cheeger's isoperimetric constants. L p super-Poincaré inequalities are introduced to describe the ...Analogous to , higher order Poincaré inequality involving higher order derivatives also holds in \(\mathbb {H}^{N}\). In this context, a worthy reference on this inequality is [22, Lemma 2.4] where it has been shown that for k and l be non-negative integers with \(0\le l<k\) there holdsConsider the PDE. ∂tu = Lu ∂ t u = L u. where L = Δ + ∇V ⋅ ∇ L = Δ + ∇ V ⋅ ∇ is a self-adjoint operator. I read that if L L has a spectral gap λ > 0 λ > 0 then " [convergence of the initial condition to the stationary distribution us(x) =e−V(x) u s ( x) = e − V ( x)] easily follows by elementary spectral analysis, or by ...An Isoperimetric Inequality for the N-dimensional Free Membrane Problem. J. Rational Mech. Anal. 5, 633–636 (1956). MATH MathSciNet Google Scholar Download references. Author information. Authors and Affiliations. Institute for Fluid Dynamics and Applied Mathematics University of Maryland, College Park, Maryland ...Viewed 182 times. 1. The Gaussian Poincare inequality states that for a differentiable function f: Rn → R f: R n → R and d d -dimensional Gaussian X ∼ N(0, Σ) X ∼ N ( 0, Σ), then. Var(f(X)) ≤E Σ∇f(X), ∇f(X) . Var ( f ( X)) ≤ E Σ ∇ f ( X), ∇ f ( X) . I would like to know if there is an extension to multivariate functions ...Aug 15, 2022 · 1. (1) This inequality requires f f to be differentiable everywhere. (2) With that condition, the answer is the linear functions. The challenge is to prove that. (3) You might as well assume n = 1: n = 1: larger values of n n are trivial generalizations because both sides split into sums over the coordinates. This paper is devoted to investigate an interpolation inequality between the Brezis-Vázquez and Poincaré inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. ... ….

In 1999, Bobkov [ 10] has shown that any log-concave probability measure satisfies the Poincaré inequality. Here log-concave means that ν ( dx ) = e −V (x)dx where V is a convex function with values in \ (\mathbb R \cup \ {+ \infty \}\). In particular uniform measures on convex bodies are log-concave.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.I have trouble proving the following problem (Evans PDE textbook 5.10. #15). Could anyone kindly help me solving the problem? I know that I should somehow use Poincaré inequality but I still cannot...On the Gaussian Poincare inequality. Let X X be a standard normal random variable. Then, for any differentiable f: R → R f: R → R such that Ef(X)2 < ∞, E f ( X) 2 < ∞, the Gaussian Poincare inequality states that. Var(f(X)) ≤E[f′(X)2]. V a r ( f ( X)) ≤ E [ f ′ ( X) 2]. Suppose this inequality is proved for all functions that ...(i) It suffices to prove the inequality (1) for all f ∈ C∞. 0 (Ω). In this context we need the generalized H ̈ older inequality, namely, if fj ∈ Lpj(Ω), = 1, · · · , m, such that p−1 + . · · · …Weighted Poincaré inequalities. Abstract: Poincaré-type inequalities are a key tool in the analysis of partial differential equations. They play a particularly central role in the analysis of domain decomposition and multilevel iterative methods for second-order elliptic problems. When the diffusion coefficient varies within a subdomain or ...The main result of this article is that when a four-dimensional Poincaré-Einstein metric satisfies a certain point-wise curvature inequality, then g is automatically non-degenerate. We will give the inequality shortly, but first we explain the geometric importance of non-degeneracy.Cheeger, Hajlasz, and Koskela showed the importance of local Poincaré inequalities in geometry and analysis on metric spaces with doubling measures in [9, 15].In this paper, we establish a family of global Poincaré inequalities on geodesic spaces equipped with Borel measures, which satisfy a local Poincaré inequality along with certain other geometric conditions.During the past 55 years substantial progress concerning sharp constants in Poincare-type and Steklov-type inequalities has been achieved. Original results of H. Poincare, V. A. Steklov and his … ExpandRacial, gender, age and socio-economic inequalities lead to discrimination against some people everyday. These inequalities are present in such aspects as education, the workplace, politics, community and even health care. Poincare inequality, 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 …, 2.3+ billion citations. Download scientific diagram | Poincaré inequality in 2 dimensions from publication: A Quick Tutorial on DG Methods for Elliptic Problems | We recall a few basic ..., The Poincaré inequality for the domain on the sphere (see e.g. Theorem 3.21 [145]). Let u ∈ W 1 (Ω) and Ω is convex domain on the unit sphere S N -1 . Then || u − …, This example shows that the super-Poincare inequality and the Nash-type inequality can be satisfied by a generator but without ultracontractivity of the corresponding semigroup. 4.2 The Riemannian setting. Let \(M\) be a connected complete Riemannian manifold with Ricci curvature bounded below., A Poincare inequality on fractional Sobolev space. Let Ω be a bounded smooth domain. Does the following inequality hold for all u ∈ H 0 s ( Ω): where the right hand side is the H 0 s ( Ω) seminorm. H 0 s is defined as an interpolaton space of H 0 1 and L 2., Improved fractional Poincaré type inequalities on John domains 289 given r>0andx∈X, the ball centered at x with radius r is the set B(x,r):={y∈ X:d(x,y)<r}.Given a ball B⊂X, r(B) will denote its radius and x B its center. For any λ>0, λB will be the ball with same center as B and radius λr(B). A doubling metric space is a metric space (X,d) with the following (geometric), In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré., The following is the well known Poincaré inequality for H 0 1 ( Ω): Suppose that Ω is an open set in R n that is bounded in some direction. Then there is a constant C such that. ∫ Ω u 2 d x ≤ C ∫ Ω | D u | 2 d x for all u ∈ H 0 1 ( Ω). Here are my questions: Could anyone come up with an example that f ∈ H 1 ( Ω) ∖ H 0 1 ( Ω)?, norms on both sides of the inequality is quite natural and along the lines of the results for improved Poincaré inequalities involving the gradient found in [7, 8, 14, 22], we believe that the only antecedent of these weighted fractional inequalities is found in [1, Proposition 4.7], where (1.6) is obtained in a star-shaped domain in the case, This estimate only depends on the weight function of the Poincaré inequality, and yields a criterion of parabolicity of connected components at infinity in terms of the weight function. AB - We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré inequality. In the process ..., Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze. We study the validity of the Lp inequality for the Riesz transform when p > 2 and of its reverse inequality when 1 < p < 2 on complete Riemannian manifolds under the doubling property and some Poincare inequalities. MSC numbers 2000: 58J35, 42B20. View PDF on arXiv., From Poincar\'e Inequalities to Nonlinear Matrix Concentration. June 2020. This paper deduces exponential matrix concentration from a Poincar\'e inequality via a short, conceptual argument. Among ..., Title: Poincaré inequality 3/2 on the Hamming cube. Authors: Paata Ivanisvili, Alexander Volberg. Download a PDF of the paper titled Poincar\'e inequality 3/2 on the Hamming cube, by Paata Ivanisvili and 1 other authors., 3. I have a question about Poincare-Wirtinger inequality for H1(D) H 1 ( D). Let D D is an open subset of Rd R d. We define H1(D) H 1 ( D) by. H1(D) = {f ∈ L2(D, m): ∂f ∂xi ∈ L2(D, m), 1 ≤ i ≤ d}, H 1 ( D) = { f ∈ L 2 ( D, m): ∂ f ∂ x i ∈ L 2 ( D, m), 1 ≤ i ≤ d }, where ∂f/∂xi ∂ f / ∂ x i is the distributional ..., Poincaré Inequality Stephen Keith ABSTRACT. The main result of this paper is an improvement for the differentiable structure presented in Cheeger [2, Theorem 4.38] under the same assumptions of [2] that the given metric measure space admits a Poincaré inequality with a doubling mea sure. To be precise, it is shown in this paper that the ..., in a manner analogous to the classical proof. The discrete Poincare inequality would be more work (and the constant there would depend on the boundary conditions of the difference operator). But really, I would also like this to work for e.g. centered finite differences, or finite difference kernels with higher order of approximation., 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 …, DOI: 10.1214/ECP.V13-1352 Corpus ID: 18581137; A simple proof of the Poincaré inequality for a large class of probability measures @article{Bakry2008ASP, title={A simple proof of the Poincar{\'e} inequality for a large class of probability measures}, author={Dominique Bakry and Franck Barthe and Patrick Cattiaux and Arnaud Guillin}, …, Remark 1.10. The inequality (1.6) can be viewed as an implicit form of the weak Poincar e inequality. Note that setting K= 0 (which is excluded in the theorem) leads to the Poincar e inequality. The power of this result is demonstrated in the following corollary, where the celebrated Nash inequality is obtained as a simple consequence. , A NOTE ON SHARP 1-DIMENSIONAL POINCAR´E INEQUALITIES 2311 Poincar´e inequality to these subdomains with a weight which is a positive power of a nonnegative concave function. Moreover, it has recently been shown in [11] by a similar method that the best constant C in the weighted Poincar´e inequality for 1 ≤ q ≤ p<∞, f − f av Lq w (Ω ..., A NOTE ON SHARP 1-DIMENSIONAL POINCAR´E INEQUALITIES 2311 Poincar´e inequality to these subdomains with a weight which is a positive power of a nonnegative concave function. Moreover, it has recently been shown in [11] by a similar method that the best constant C in the weighted Poincar´e inequality for 1 ≤ q ≤ p<∞, f − f av Lq w (Ω ..., By choosing the functional F appropriately, (4) becomes a Poincaré inequality with weight ϕ, see Section 3. Such inequalities have been studied extensively because of their importance for the regularity theory of partial differential equations, see the exposition in [5]. 2. Proof Lemma 2. Let Ω be a finite measure space and p ≥ 1. Assume ..., The main aim of this note is to prove a sharp Poincaré-type inequality for vector-valued functions on $\mathbb{S}^2$ that naturally emerges in the context of micromagnetics of spherical thin films. On a Sharp Poincaré-Type Inequality on the 2-Sphere and its Application in Micromagnetics | SIAM Journal on Mathematical Analysis, The proof is essentially the same as the one for the Poincare inequality you stated $\endgroup$ - Quickbeam2k1. Jan 26, 2015 at 9:04 $\begingroup$ @Quickbeam2k1 Thanks for the additional comment. This is new to me - I will check it. $\endgroup$ - MathProb. Jan 26, 2015 at 20:00., We establish functional inequalities on the path space of the stochastic flow x ↦ X t x including gradient inequalities, log-Sobolev inequalities and Poincaré inequalities. These inequalities are shown to be equivalent to bounds on the horizontal Ricci operator Ric H: H → H which is defined taking the trace of the curvature tensor only over H., Sobolev’s Inequality, Poincar´e Inequality and Compactness I. Sobolev inequality and Sobolev Embeddig Theorems Theorem 1 (Sobolev’s embedding theorem). Given the bounded, open set Ω ⊂ Rn with n ≥ 3 and 1 ≤ p<n, then W1,p 0 (Ω) ⊂ L np n−p (Ω) and W1,p 0 (Ω) is continuously embedded in the space L np n−p (Ω). This means that ..., Applications include showing that the p-Poincaré inequality (with a doubling measure), for p≥1, persists through to the limit of a sequence of converging pointed metric measure spaces — this extends results of Cheeger. ... We study a generalization of classical Poincare inequalities, and study conditions that link such an inequality with ..., inequality (2.4) provides a way to quantify the ergodicity of the Markov process. As it happens, the trace Poincaré inequality is equivalent to an ordinary Poincaré inequality. We are grateful to Ramon Van Handel for this observation. Proposition 2.4 (Equivalence of Poincaré inequalities). Consider a Markov process (Zt: t ≥ 0) ⊂ Ω, During the past 55 years substantial progress concerning sharp constants in Poincare-type and Steklov-type inequalities has been achieved. Original results of H. Poincare, V. A. Steklov and his … Expand, "Poincaré Inequality." From MathWorld --A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/PoincareInequality.html Subject classifications Let Omega be an open, bounded, and connected subset of R^d for some d and let dx denote d-dimensional Lebesgue measure on R^d., What kind of Poincare inequality is that, in which the derivative lies on the left hand-side? Is $\partial_X^{-1} B$ the inverse derivative of B or what? Is there any way, one can modify the classical Poincare inequality (see Evans, PDEs, §5.8) using Fourier transform in order to obtain something similar to this?, DOI: 10.1214/ECP.V13-1352 Corpus ID: 18581137; A simple proof of the Poincaré inequality for a large class of probability measures @article{Bakry2008ASP, title={A simple proof of the Poincar{\'e} inequality for a large class of probability measures}, author={Dominique Bakry and Franck Barthe and Patrick Cattiaux and Arnaud Guillin}, journal={Electronic Communications in Probability}, year ..., Friedrichs's inequality. In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the Lp norm of a function using Lp bounds on the weak derivatives of the function and the geometry of the domain, and can be used to show that certain norms on Sobolev spaces are equivalent.