Poincar´e Inequality Statistical estimation of the Poincar´e Constant Future Work? A historical perspective Poincar´e inequalities in the modern framework Application of Poincar´e inequalities Poincar´e inequality for bounded open convex set in Rn Theorem (H.Poincar´e 1890) For Ω open bounded convex set of Rd, f smooth from Ω¯ to R ...May 9, 2017 · Prove the Poincare inequality: for any u ∈ H10(0, 1) u ∈ H 0 1 ( 0, 1) ∫1 0 u2dx ≤ c∫1 0 (u′)2dx ∫ 0 1 u 2 d x ≤ c ∫ 0 1 ( u ′) 2 d x. for some constant c > 0 c > 0. Hint: Write u(x) =∫x 0 u′(s)ds u ( x) = ∫ 0 x u ′ ( s) d s, then square this identity. Proof: Let u(x) =∫x 0 u′(s)ds ⇒ |u(x)| ≤∫x 0 |u(s)|ds u ... Poincare Inequality implies Equivalent Norms. I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following in the book (page 266.) "In view of the Poincare Inequality, on W1,p0 (U) W 0 1, p ( U) the norm ||DU ...How does income inequality affect real workers? SmartAsset's study of annual earnings found that management-level workers make 5 times more than workers... By almost any measure, income inequality in the United States has grown steadily ove...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) ⊂ ΩExtensions 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 ...MATHEMATICS OF COMPUTATION Volume 80, Number 273, January 2011, Pages 119–140 S 0025-5718(2010)02296-3 Article electronically published on July 8, 2010If μ satisfies the inequality SG(C) on Rd then (1.3) can be rewritten in a more pleasant way: for all subset A of (Rd)n with μn(A)≥1/2, ∀h≥0 μn A+ √ hB2 +hB1 ≥1 −e−hL (1.4) with a constant L depending on C and the dimension d. The archetypic example of a measure satisfying the classical Poincaré inequality is the exponential ...Using our results concerning embeddings combined with a generalization of a result of Heinonen and Koskela, we show that Orlicz-Sobolev extension domains satisfy the measure density condition. In the case of Hajłasz-Orlicz-Sobolev spaces, it follows that the measure density condition, or the validity of certain Orlicz-Poincaré inequalities ...Thus 1/λ1 1 / λ 1 is the best constant in the Poincaré inequality since the infimum is achieved by the solution to the Dirichlet problem. Now, the crucial feature of this is that for a ball, namely Ω = B(0, r) Ω = B ( 0, r), we can explicitly compute the eigenfunctions and eigenvalues of the Laplacian by using the classical PDE technique ...The following is the well known Poincaré inequality for $H_0^1(\Omega)$: Suppose that $\Omega$ is an open set in $\mathbb{R}^n$ that is bounded. Then there is a ... 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 ≥Poincare (Wirtinger) Inequality vanishing on subset of boundary? 0. Explaining the Proof of Schwarz Inequality for Scalar Product in a Vector Space. 1. Explain Proof of Convergence of Matrix when Spectral Radius Less than 1. 1. Question about the proof of the Poincaré inequality. 1.A Poincaré inequality states that the variance of an admissible function is controlled by the homogeneous norm. In the case of Loop spaces, it was observed by L. Gross that the homogeneous norm alone may not control the norm and a potential term involving the end value of the Brownian bridge is introduced. Aida, on the other hand, introduced a ...We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, n -dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary. …By Theorem 1.4 [1], we show that if there exists a Lyapunov function V ( x) satisfying the drift condition, then μ satisfies a L 2 Poincaré inequality with constant C P = 1 λ ( 1 + b κ R), where κ R is the L2 Poincaré constant of μ restricted to the ball B (0,R). Given a smooth function g, we know that V a r μ ( g) ≤ ∫ ( g − c) 2 ...A GENERALIZED POINCARE INEQUALITY FOR GAUSSIAN MEASURES WILLIAM BECKNER (Communicated by J. Marshall Ash) ABSTRACT. New inequalities are obtained which interpolate in a sharp way be-tween the Poincare inequality and the logarithmic Sobolev inequality for both Gaussian measure and spherical surface measure.Theorem 1. The Poincare inequality (0.1) kf fBk Lp (B) C(n; p)krfkLp(B); B Rn; f 2 C1(R n); where B is Euclidean ball, 1 < n and p = np=(n p), implies (0.2) Z jf jBj B Z fBjpdx c(n; p)diam(B)p jrfjpdx; jBj B Rn; f 2 C1(R n); where B is Euclidean ball and 1 < n. Proof. By the interpolation inequality, we get (0.3) kf fBkp kf fBkp kf fBk1 ;This paper deduces exponential matrix concentration from a Poincare inequality via a short, conceptual argument. Among other examples, this theory applies to matrix-valued functions of a uniformly log-concave random vector. The proof relies on the subadditivity of Poincare inequalities and a chain rule inequality for the trace of the matrix Dirichlet form. It also uses a symmetrization ...The 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. The aim of this paper is to prove a Poincare type \(p-q\) inequality in a homogeneous space \((\mathbb {R}^N, d, \mu ) \) estimating weighted Lebesgue norm …Every graph of bounded degree endowed with the counting measure satisfies a local version of Lp-Poincaré inequality, p ∈ [1, ∞]. We show that on graphs which are trees the Poincaré constant grows at least exponentially with the radius of balls. On the other hand, we prove that, surprisingly, trees endowed with a flow measure support a global version of Lp-Poincaré inequality, despite ...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.See also: Poincaré Inequality. Share. Cite. Follow edited Apr 13, 2017 at 12:21. Community Bot. 1. answered Jul 11, 2014 at 20:23. user147263 user147263 $\endgroup$ ... Poincare Inequality on compact Riemannian manifold. 0. Integration by parts on compact, non-orientable Riemannian manifold with boundary.We also discuss exponential integrability under Poincaré inequalities and its consequence to sharp diameter upper bounds on spectral gaps. AB - We present a simple proof based on modified logarithmic Sobolev inequalities, of Talagrand's concentration inequality for the exponential distribution. We actually observe that every measure satisfying ...This algebraic property is at the core of all Korn-type inequalities, it means that derivatives of \(D^au\) are in the span of the derivatives of \(D^s u\).Note that the Schwarz Theorem also implies \(D^a\,\nabla =0\) which is central in the construction of the De Rham complex. \(\textcircled {3}\) The rigidity constants, as defined in (), () and (), measure the defects of axisymmetry of the ...1.1. Results. In this work, we establish a general Poincaré type inequality on submanifolds of suitable Riemannian ambient spaces. Using such estimate and additional mild conditions we obtain rigidity results for hypersurfaces of space forms and of suitable Einstein manifolds, as we briefly describe in the following.In functional analysis, the term "Poincaré-Friedrichs inequality" is a term used to describe inequalities which are qualitatively similar to the classical Poincaré Inequality and/or Friedrichs inequalities. Sometimes referred to as inequalities of Poincaré-Friedrichs type, such expressions play important roles in the theories of partial differential equations and function spaces, often ...Download a PDF of the paper titled Poincar\'e Inequality Meets Brezis--Van Schaftingen--Yung Formula on Metric Measure Spaces, by Feng Dai and 3 other authors3 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 ...of finite area, the analytic Poincare inequality (1.5) is equivalent to (1.6) for p = 2. Therefore, the Axler-Shields question is answered by our main results: Theorem 1. If D c Rd is a Holder domain, then D is a p-Poincare domain for all p > d. Theorem 2. If a domain D is a Holder domain, then (1.7) f kp(xo, x)dx < 0 D for all p < 00.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.We characterize complete RNP-differentiability spaces as those spaces which are rectifiable in terms of doubling metric measure spaces satisfying some local (1, p)-Poincaré inequalities. This gives a full characterization of spaces admitting a strong form of a differentiability structure in the sense of Cheeger, and provides a partial converse to his theorem. The proof is based on a new ...It is known that this inequality is valid for bounded John domains if w ∈ Ap (see [DD]). As we will see, this result can be extended for more general weights. For example, for a class of weights introduced in [FKS] where the authors consider the classic Poincaré inequality in weighted norms, (1.6) kϕ−ϕΩ,wkLp w(Ω) ≤ Ck∇ϕkLp w(Ω)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.Stack 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.. Visit Stack ExchangeIndeed, such estimates are directly related to well-known inequalities from pure mathematics (e.g logarithmic Sobolev and Poincáre inequalities). In probability theory, Brascamp–Lieb and Poincaré inequalities are two very important concentration inequalities, which give upper bounds on variance of function of random variables.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.Article Poincaré and log-Sobolev inequalities for mixtures André Schlichting1,† 1 RWTH Aachen, Institut für Geometrie und Praktische Mathematik; [email protected] Abstract: This work studies mixtures of probability measures on Rn and gives bounds on the Poincaré and the log-Sobolev constant of two-component mixtures provided that each component satisfiesConsequently, inequality (4.2) holds for all functions u in the Sobolev space W1,p ( B ). Inequality (4.2) is often called the Sobolev-Poincaré inequality, and it will be proved momentarily. Before that, let us derive a weaker inequality (4.4) from inequality (4.2) as follows. By inserting the measure of the ball B into the integrals, we find ...In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré.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 smooth functions in a Hilbert space. But a standard derivation of theFor other inequalities named after Wirtinger, see Wirtinger's inequality.. In the mathematical field of analysis, the Wirtinger inequality is an important inequality for functions of a single variable, named after Wilhelm Wirtinger.It was used by Adolf Hurwitz in 1901 to give a new proof of the isoperimetric inequality for curves in the plane. A variety of closely related results are today ...In this paper we unify and improve some of the results of Bourgain, Brezis and Mironescu and the weighted Poincaré-Sobolev estimate by Fabes, Kenig and Serapioni. More precisely, we get weighted counterparts of the Poincaré-Sobolev type inequality and also of the Hardy type inequality in the fractional case under some mild natural restrictions. A main feature of the results we obtain is the ...Thus 1/λ1 1 / λ 1 is the best constant in the Poincaré inequality since the infimum is achieved by the solution to the Dirichlet problem. Now, the crucial feature of this is that for a ball, namely Ω = B(0, r) Ω = B ( 0, r), we can explicitly compute the eigenfunctions and eigenvalues of the Laplacian by using the classical PDE technique ...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 ...We study weighted Poincaré and Poincaré-Sobolev type inequalities with an explicit analysis on the dependence on the Ap constants of the involved weights. We obtain inequalities of the form ( 1 w(Q) ∫ Q |f − fQ|w ) 1 q ≤ Cw`(Q) ( 1 w(Q) ∫ Q |∇f |w ) 1 p , with different quantitative estimates for both the exponent q and the constant Cw. We will derive those estimates together with ...free functional inequalities, namely, the free transportation and Log-Sobolev inequalities. AsintheclassicalcasethePoincar´eisimpliedbytheothers. This investigation is driven by a nice lemma of Haagerup which relates logarith- ... THE ONE DIMENSIONAL FREE POINCARE INEQUALITY 4813´ ...Our understanding of the interplay between Poincare inequalities, Sobolev inequalities and the geometry of the underlying space has changed considerably in recent years. These changes have simultaneously provided new insights into the classical theory and allowed much of that theory to be extended to a wide variety of different settings. This paper reviews some of these new results and ...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 … ExpandThe 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 AnalysisThe only reference for inequalities of Poincare type on punctured domains I could find was Lieb–Seiringer–Yngvason (Ann. Math 2003) arXiv link. I suspect the Poincaré inequality on punctured domains in the way it is asked above might be false. If it is false, then I would like to understand is what sort of functions admit the second ...A Poincare Inequality on Loop Spaces´ Xin Chen, Xue-Mei Li and Bo Wu Mathemtics Institute University of Warwick Coventry CV4 7AL, U.K. November 9, 2018 Abstract We investigate properties of measures in infinite dimension al spaces in terms of Poincare´ inequalities. A Poincare´ inequality states that the L2 vari-Poincare--Friedrichs inequalities for piecewise H1 functions are established and can be applied to classical nonconforming finite element methods, mortar methods, and discontinuous Galerkin methods. Poincare--Friedrichs inequalities for piecewise H1 functions are established. They can be applied to classical nonconforming finite element methods, mortar methods, and discontinuous Galerkin methods.Download a PDF of the paper titled Poincar\'e Inequality Meets Brezis--Van Schaftingen--Yung Formula on Metric Measure Spaces, by Feng Dai and 3 other authorsLipschitz Domain. Dyadic Cube. Bound Lipschitz Domain. Common Face. Uniform Domain. We show that fractional (p, p)-Poincaré inequalities and even fractional Sobolev-Poincaré inequalities hold for bounded John domains, and especially for bounded Lipschitz domains. We also prove sharp fractional (1,p)-Poincaré inequalities for s-John domains.In different from Sobolev’s inequality, the geometry of domain is essential for Poincare inequality. Quite a number of results on weighted Poincare inequality are available e.g. in [9, 17, 27, 36]. We cite [8, 17, 33] for further continuation of those results. For a weighted capacity characterization of this inequalities see, .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 ...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 ... 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 ...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? ...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 …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.his Poincare inequality discussed previously [private communication]. The conclusion of Theorem 4 is analogous to the conclusion of the John-Nirenberg theorem for functions of bounded mean oscillation. I would like to thank Gerhard Huisken, Neil Trudinger, Bill Ziemer, and particularly Leon Simon, for helpful comments and discussions. NOTATION.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 ≥POINCARE INEQUALITIES 5 of a Sobolev function uis, up to a dimensional constant, the minimal that can be inserted to the Poincar e inequality. This is proved along with the …The 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. Dec 30, 2017 · While studying two seemingly irrelevant subjects, probability theory and partial differential equations (PDEs),I ran into a somewhat surprising overlap:the Poincaré inequality.On one hand, it is not out of the ordinary for analysis based subjects to share inequalities such as Cauchy-Schwarz and Hölder;on the other hand, the two forms ofPoincaré inequality have quite different applications. 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.his Poincare inequality discussed previously [private communication]. The conclusion of Theorem 4 is analogous to the conclusion of the John-Nirenberg theorem for functions of bounded mean oscillation. I would like to thank Gerhard Huisken, Neil Trudinger, Bill Ziemer, and particularly Leon Simon, for helpful comments and discussions. NOTATION. 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) ⊂ ΩTheorem 1. The Poincare inequality (0.1) kf fBk Lp (B) C(n; p)krfkLp(B); B Rn; f 2 C1(R n); where B is Euclidean ball, 1 < n and p = np=(n p), implies (0.2) Z jf jBj B Z fBjpdx c(n; p)diam(B)p jrfjpdx; jBj B Rn; f 2 C1(R n); where B is Euclidean ball and 1 < n. Proof. By the interpolation inequality, we get (0.3) kf fBkp kf fBkp kf fBk1 ;$\begingroup$ @Jeff: Thank you for your comment. What's in my mind is actually the mixed Dirichlet-Neumann boundary problem: an elliptic equation with zero on one portion of the boundary and zero normal derivative on the rest of the portion.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 …Reverse Poincare inequalities, Isoperimetry, and Riesz transforms in Carnot groups. Fabrice Baudoin, Michel Bonnefont. We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness ...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.Every graph of bounded degree endowed with the counting measure satisfies a local version of Lp-Poincaré inequality, p ∈ [1, ∞]. We show that on graphs which are trees the Poincaré constant grows at least exponentially with the radius of balls. On the other hand, we prove that, surprisingly, trees endowed with a flow measure support a global version of Lp-Poincaré inequality, despite ...In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French 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 of the calculus of …Poincar e inequalities and geometric bounds themodern era : Lichnerowicz’s bound (1958) (M;g)compact Riemannian manifold normalized Riemannian volume elementIn this paper we establish necessary and sufficient conditions for weighted Orlicz-Poincaré inequalities in dimension one. Our theorems generalize the main results of Chua and Wheeden, who established necessary and sufficient conditions for weighted $(q,p)$ Poincaré inequalities. We give an example of a weight satisfying sufficient conditions for a $(Φ, p)$ Orlicz-Poincaré inequality where ...Abstract. In order to describe L2 -convergence rates slower than exponential, the weak Poincaré inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the corresponding weak Poincaré inequality can be determined by each other. Conditions for the weak Poincaré inequality to hold are presented, which are easy ...As an important intermediate step in order to get our results we get the validity of a Poincaré inequality with respect to the natural weighted measure on any translator and we prove that any end of a translator must have infinite weighted volume. Similar tools can be obtained for properly immersed self-expanders permitting to get topological ...In this paper, we study the sharp Poincaré inequality and the Sobolev inequalities in the higher-order Lorentz-Sobolev spaces in the hyperbolic spaces. These results generalize the ones obtained in Nguyen VH (J Math Anal Appl, 490(1):124197, 2020) to the higher-order derivatives and seem to be new in the context of the Lorentz-Sobolev spaces defined in the hyperbolic spaces.HARDY-POINCARE, RELLICH AND UNCERTAINTY PRINCIPLE INEQUALITIES ON RIEMANNIAN MANIFOLDS ISMAIL ΚΟΜΒΕ AND MURAD OZAYDIN ABSTRACT. We continue our previous study of improved Hardy, Rellich and uncertainty principle inequalities on a Riemannian manifold M, started in our earlier paper from 2009. In the present paper we prove new weightedinequality (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) ⊂ ΩWe demonstrate $\Omega$ is a John domain if a $(\phi_\frac{n}{s}, \phi)$-Poincaré inequality holds. Subjects: Functional Analysis (math.FA) Cite as: arXiv:2305.04016 [math.FA] (or arXiv:2305.04016v1 [math.FA] for this version) Submission history From: Tian Liang [v1] Sat, 6 May 2023 11:18:17 UTC (20 KB) Full-text links: Download: ...Inequalities related to Gaussian concentration In the sequel, (X ,d) is a polish space. A probability measure µ on X e, Stack Exchange network consists of 183 Q&A communities including Stack, Regarding this point, a parabolic Poincaré type inequality for u in the framework of Orlicz spa, The strong Orlicz-Poincaré inequality coincides with the one, The Bill & Melinda Gates Foundation, based in Seattle, Washington, was launched in 2000 by Bill and Melinda, ThisMarkovchainisirreducibleandreversible,thustheoperatorKdefinedby [K˚](x) = X y2X K(x;y)˚(y) isaself-adjointcontracti, $\begingroup$ In general, computing the exact value of the Poincare-Friedrichs constant is quite challenging and , inequality with constant κR and a L1 Poincar´e inequa, My thoughts/ideas: I looked at the case that v ( x) = , Theorem 2.4 of [16] also derives concentration inequalities from , Consider the PDE. ∂tu = Lu ∂ t u = L u. where L = Δ + ∇V ⋅ ∇ L = Δ + , These are quite different things. On one hand, an hourglass-shaped s, We then establish a comparison procedure with the we, Jules Henri Poincaré (UK: / ˈ p w æ̃ k ɑːr eɪ /, US: / ˌ p w æ̃ k, Some generalized Poincaré inequalities and applications , MATRIX POINCARE INEQUALITIES AND CONCENTRATION 3´ its scalar counter, Remark 1.10. The inequality (1.6) can be viewed as an implici, We show that unbounded John domains (and even a large.