An optimal poincaré inequality for convex domains of non-negative curvature ... ~j An Optimal Poincare Inequality 273 Let k denote the expression in braces in the last line. If we sum the above in- equality over j we obtain 21 ~ f 2 dA >(Tz2/d2) ~ f 2 d a - k A ( Q ) ~. ...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 ...The Poincar ́ e inequality is an open ended condition By Stephen Keith and Xiao Zhong* Abstract Let p > 1 and let (X, d, μ) be a complete metric measure space with μ Borel and doubling that admits a (1, p)-Poincar ́ e inequality. Then there exists ε > 0 such that (X, d, μ) admits a (1, q)-Poincar ́ e inequality for every q > p−ε, quantitatively.inequality (4.2) holds for all functions u in the Sobolev space WI,P(B). Inequality (4.2) is often called the Sobolev-Poincare inequality, and it will be proved mo mentarily. 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 that (1 ) 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.As BaronVT notes, in order to do something in the frequency space, one has to translate the condition supp f ⊆ [ − R, R] there. This is what the various uncertainty inequalities do. The classical Heisenberg-Pauli-Weyl uncertainty inequality. immediately gives (1) because ‖ x f ( x) ‖ L 2 ≤ R ‖ f ‖ L 2 under your assumption.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."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.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 ...Consider a function u(x) in the standard localized Sobolev space W 1,p loc (R ) where n ≥ 2, 1 ≤ p < n. Suppose that the gradient of u(x) is globally L integrable; i.e., ∫ Rn |∇u| dx is finite. We prove a Poincaré inequality for u(x) over the entire space R. Using this inequality we prove that the function subtracting a certain constant is in the space W 1,p 0 (R ), which …We prove generalizations of the Poincaré and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is not limited to) spaces of homogeneous type with doubling measures. Several examples and applications are ...Indeed, 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.On a Poincaré inequality with weight. Let Ω Ω be a bounded convex (non-empty) open subset of Rn R n ( Ω Ω can be as smooth as you like). Is it true that there exists a constant C > 0 C > 0 such that the following holds: Assume given a probability measure ω(x)dx ω ( x) d x with ω ∈ Lp(Ω) ω ∈ L p ( Ω). Then, for any function f f in ...If Ω is a John domain, then we show that it supports a ( φn/ (n−β), φ) β -Poincaré inequality. Conversely, assume that Ω is simply connected domain when n = 2 or a bounded domain which is quasiconformally equivalent to some uniform domain when n ≥ 3. If Ω supports a ( φn/ (n−β), φ) β -Poincaré inequality, then we show that it ...We investigate links between the so-called Stein's density approach in dimension one and some functional and concentration inequalities. We show that measures having a finite first moment and a density with connected support satisfy a weighted Poincaré inequality with the weight being the Stein kernel, that indeed exists and is unique in this case. Furthermore, we prove weighted log-Sobolev ...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 functions on a Euclidean space, proved by Fuchs and Osmolovski (J ...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 ...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 ...We point out some of the differences between the consequences of p-Poincaré inequality and that of ∞-Poincaré inequality in the setting of doubling metric measure spaces. Based on the geometric characterization of ∞-Poincaré inequality given in Durand-Cartagena et al. (Mich Math J 60, 2011), we obtain a geometric property implied by the support of a p-Poincaré inequality, and ...1 The Dirichlet Poincare Inequality Theorem 1.1 If u : Br → R is a C1 function with u = 0 on ∂Br then 2 ≤ C(n)r 2 u| 2 . Br Br We will prove this for the case n = 1. Here the statement becomes r r f2 ≤ kr 2 (f )2 −r −r where f is a C1 function satisfying f(−r) = f(r) = 0. By the Fundamental Theorem of Calculus s f(s) = f (x). −r In this paper, we prove a sharp lower bound of the first (nonzero) eigenvalue of Finsler-Laplacian with the Neumann boundary condition. Equivalently, we prove an optimal anisotropic Poincaré inequality for convex domains, which generalizes the result of Payne-Weinberger. A lower bound of the first (nonzero) eigenvalue of Finsler …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.In 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 ...The weighted Poincare inequality was introduced in Blanchet et al. (2009) and Bobkov and Ledoux (2009), and using an extension of the Brascamp-Lieb inequality, is shown to hold for the class of s ...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.This paper deduces exponential matrix concentration from a Poincaré 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 Poincaré inequalities and a chain rule inequality for the trace of the matrixPoincare Inequality Meets Brezis-Van Schaftingen-Yung Formula on´ Metric Measure Spaces Feng Dai, Xiaosheng Lin, Dachun Yang*, Wen Yuan and Yangyang Zhang Abstract Let (X,ρ,µ) be a metric measure space of homogeneous type which supports a certain Poincare´ inequality. Denote by the symbol C∗ c(X) the space of all continuous func-New inequalities are obtained which interpolate in a sharp way between the Poincaré inequality and the logarithmic Sobolev inequality for both Gaussian measure and spherical surface measure. The classical Poincaré inequality provides an estimate for the first nontrivial eigenvalue of a positive self-adjoint operator that annihilates constants. For the …In 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 ... 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.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∈QWeighted Poincare Inequalities. October 2012; IMA Journal of Numerical Analysis 33(2) ... Poincaré-type inequalities are a key tool in the analysis of partial differential equations. They play a ...The Bill & Melinda Gates Foundation, based in Seattle, Washington, was launched in 2000 by Bill and Melinda Gates. The foundation is the largest private foundation in the world, with over $50 billion in assets. All lives have equal value, a...THE POINCARE INEQUALITY IS AN OPEN ENDED CONDITION´ 579 ([34]) have shown in the setting of metric measure spaces that support a dou-bling Borel regular measure …So 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 ...Poincaré-Sobolev-type inequalities indisputably play a prominent role not only in the theory of Sobolev spaces but also in a wide range of applications in analysis of partial differential equations, calculus of variations, mathematical modeling or harmonic analysis (e.g. [5, 20, 44]).These types of inequalities have been exhaustively studied for decades and have been generalized in many ...reverse poincare inequality for polynomials with vanishing boundary. 2. Equivalent definitions of Poincare inequality. Hot Network Questions Could 99942 Apophis break up due to Earth's gravity during 2029 flyby? Am I a 'repeat ESTA visitor' in US? ...p. inequality (0.1) yield. ; or. = = n : This together with Poincare. p )p n(1 p) p. (0.4) kf fBkp (C(n; p)krfkLp(B)) kf. Let us estimate the norm kf fBk1. fBk1 : given in the above …In the proof of Theorem 5.1 we need yet another result, which is a Poincaré inequality for vector fields that are tangent on the boundary of ω h (z) (see (5.1)), and with constant independent of ...therefore natural to look for higher order Poincare inequalities by using spherical harmonics and apply them to obtain new geometric inequalities, which is the goal of this paper. In general, it is well-known that on Sd 1, if Fhas mean zero, then we have the Poincare inequality (d 1) Z Sd 1 F2 Z Sd 1 jrFj2, which can be written as Z Sd 1 F F (d ...In this paper, we prove capacitary versions of the fractional Sobolev–Poincaré inequalities. We characterize localized variant of the boundary fractional Sobolev–Poincaré inequalities through uniform fatness condition of the domain in \ (\mathbb {R}^n\). Existence type results on the fractional Hardy inequality in the supercritical case ...In view of our discussion of the Dirichlet integral, we call Inequality ♦ weak Hardy inequality if ker q ={0} and weak Poincaré inequality if ker q ={0}. In the case = 0, the function α becomes a constant and Inequality ♦is referred to as Hardy inequality if ker q ={0}, respectively Poincaré inequality if ker q ={0}.Hardy's inequality is proved with the same choice of ψ that gave Hilbert's inequality. One interesting consequence should be mentioned. Suppose f(z) = Σa n z n is analytic in |z| < 1. If Σ|a n | < ∞, then f has a continuous extension to |z| ≤ 1, but the converse is false (see Exercise 7).Hardy's inequality shows, however, that if f′ ∈ H 1 (or equivalently, in light of Theorem 3.11 ...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 ...$\begingroup$ In general, computing the exact value of the Poincare-Friedrichs constant is quite challenging and is only known for some domains. I can't quite seem to find any relevant articles on the Google right now, but I'll report back if I do find something $\endgroup$The strong Orlicz-Poincaré inequality coincides with the ones considered by Heikkinen and Tuominen in, for example, [Hei10,HT10,Tuo04,Tuo07]. The inequalities of Feng-Yu Wang [Wan08] are of a ...This is Poincare's inequality: $... Stack Exchange Network 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.Below is the proof of Poincaré's inequality for open, convex sets. It is taken from "An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs" by Giaquinta and Martinazzi.THE POINCARE INEQUALITY IS AN OPEN ENDED CONDITION´ 577 Corollary 1.0.2. Let p>1 and let w be a p-admissible weight in Rn, n ≥ 1. Then there exists ε>0 such that w is q-admissible for every q>p−ε, quantitatively. For complete Riemannian manifolds, Saloﬀ-Coste ([41], [42]) established 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. In functional analysis, the Friedrichs inequality says that there exists a constant C such that int_Omegag^2(x)dx<=Cint_Omega|del g(x)|^2dx for all functions g in the Sobolev space H_0^1(Omega) consisting of those functions in L^2(Omega) having zero trace on the ...1 Answer Sorted by: 9 In the first inequality, integrate with respect to x x 1 from 0 0 to L L. Since the right hand side is independent of x1 x 1 you end up with ∫L 0 |u(x1,x′)|2dx1 ≤ L2∫L 0 |∇u(s,x′)|2ds. ∫ 0 L | u ( x 1, x ′) | 2 d x 1 ≤ L 2 ∫ 0 L | ∇ u ( s, x ′) | 2 d s. This is the inequality you apply to derive the second one.Abstract. We give a proof of the Poincare inequality in W-1,W-p (Omega) with a constant that is independent of Omega is an element of U, where U is a set of uniformly bounded and uniformly ...1 Answer. Sorted by: 5. You can duplicate the usual proof of Hardy type inequalities to the discrete case. Suppose {qn} { q n } is an eventually 0 sequence (you can weaken this to limn→∞ n1/2qn = 0 lim n → ∞ n 1 / 2 q n = 0 ). Then by telescoping you have (all sums are over n ≥ 0 n ≥ 0)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 ...The Poincar ́ e inequality is an open ended condition By Stephen Keith and Xiao Zhong* Abstract Let p > 1 and let (X, d, μ) be a complete metric measure space with μ Borel and doubling that admits a (1, p)-Poincar ́ e inequality. Then there exists ε > 0 such that (X, d, μ) admits a (1, q)-Poincar ́ e inequality for every q > p−ε, quantitatively.p. inequality (0.1) yield. ; or. = = n : This together with Poincare. p )p n(1 p) p. (0.4) kf fBkp (C(n; p)krfkLp(B)) kf. Let us estimate the norm kf fBk1. fBk1 : given in the above …In this work, we study the Poincaré inequality in Sobolev spaces with variable exponent. As a consequence of this result we show the equivalent norms over such cones. ... Poincare type inequalities for variable exponents. J. Inequalities Pure Appl. Math., 2008; Rázkosnik, Sobolev embedding with variable exponent, II, Math. Nachr. 2002;Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincaré Inequality. Alireza Mousavi-Hosseini, Tyler K. Farghly, Ye He, Krishna Balasubramanian ...2 Answers. where fΩ =∫Ω f f Ω = ∫ Ω f is the mean of f f. This is exactly your first inequality, but I think (1) captures the meaning better. The weighted Poincaré inequality would be. where fΩ,w =∫Ω fw f Ω, w = ∫ Ω f w is the weighted mean of f f. Again, this is what you have but written in a more natural way. On the weighted fractional Poincare-type inequalities. R. Hurri-Syrjanen, Fernando L'opez-Garc'ia. Mathematics. 2017; Weighted fractional Poincar\'e-type inequalities are proved on John domains whenever the weights defined on the domain are depending on the distance to the boundary and to an arbitrary compact set in …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) ⊂ ΩFor 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 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 type inequality is one of the main theorems that we expect to be satisfied (and meaningful) for abstract spaces. The Poincare inequality means, roughly speaking, that the ZAnorm of a function can be controlled by the ZAnorm of its derivative (up to a universal constant). It is well-known that the Poincare inequality implies the Sobolev 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. Degenerate Poincaré-Sobolev inequalities. We study weighted Poincaré and Poincaré-Sobolev type inequalities with an explicit analysis on the dependence on the A_p constants of the involved weights. We obtain inequalities of the form \left (\frac {1} {w (Q)}\int_Q|f-f_Q|^ {q}w\right )^\frac {1} {q}\le C_w\ell (Q)\left (\frac {1} {w (Q)}\int_Q ...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.Poincaré Inequality Add to Mendeley Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains Mikhail Borsuk, Vladimir Kondratiev, in North-Holland Mathematical Library, 2006 2.2 The Poincaré inequality Theorem 2.9 The Poincaré inequality for the domain in ℝ N (see e.g. (7.45) [129] ).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 …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.Aug 1, 2022 · mod03lec07 The Gaussian-Poincare inequality. NPTEL - Indian Institute of Science, Bengaluru. 180 08 : 52. Poincaré Conjecture - Numberphile. Numberphile. 2 ... where y1 y 1 is so large that f(y1,x2, …,xn) = 0 f ( y 1, x 2, …, x n) = 0. So you get ∥f∥∞ ≤ diamU∥∇f∥∞ ‖ f ‖ ∞ ≤ diam U ‖ ∇ f ‖ ∞. Your proof works. Otherwise, I would just prove the inequality directly. From what I wrote above, by Holder's inequality you get.Jan 1, 2021 · 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, . for 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 ...Once one has found such a "thick" family of curves, the deduction of important Sobolev and Poincaré inequalities is an abstract procedure in which the Euclidean structure no longer plays a role. See Full ... Annales de l'Institut Henri Poincare (C) Non Linear Analysis. BMO, integrability, Harnack and Caccioppoli inequalities for quasiminimizers.The case q = np/(n−p) requires the Sobolev inequality explic-itly for the proof, and thus the inequality can be called the Poincar´e-Sobolev inequality in this case. The domain Ω is required to have the “cone property” (see, e.g., [2]); i.e., each point of Ω is the vertex of a spherical cone with ﬁxed height and angle, which is ...In 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 ... reverse poincare inequality for polynomials with vanishing boundary. 2. Equivalent definitions of Poincare inequality. Hot Network Questions Could 99942 Apophis break up due to Earth's gravity during 2029 flyby? Am I a 'repeat ESTA visitor' in US? ...Abstract. We study the equation ut − Δu = uP − μ |∇ u | q, t ≥ 0 in a general (possibly unbounded) domain Ω ⊂ ℝN. When q ≥ p, we show a close connection between the Poincaré inequality and the boundedness of the solutions. To be more precise, if q > p (or q = p and μ large enough), we prove global existence of all solutions ...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.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 caseAnnali 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.数学中，庞加莱不等式（英語： Poincaré inequality ）是索伯列夫空间理论中的一个结果，由法国 数学家 昂利·庞加莱命名。 这个不等式说明了一个函数的行为可以用这个函数的变化率的行为和它的定义域的几何性质来控制。 也就是说，已知函数的变化率和定义域的情况下，可以对函数的上界作出估计。The Poincaré inequality (8.1.1), or its Banach-space-valued counterpart (8.1.41), gives control over the mean oscillation of a function in terms of the p -means of its upper gradient. In many classical situations, for example in Euclidean space ℝ n, various Sobolev-Poincaré inequalities demonstrate that one can similarly control the q ...We prove a fractional version of Poincaré inequalities in the context of R n endowed with a fairly general measure. Namely we prove a control of an L 2 norm by a non-local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of ...Lp 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. 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. On the weighted fractional Poincare-type inequalities. R. Hurri-Syrjanen, Fernando L'opez-Garc'ia. Mathematics. 2017; Weighted fractional Poincar\'e-type inequalities are proved on John domains whenever the weights defined on the domain are depending on the distance to the boundary and to an arbitrary compact set in …Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincaré inequality, have been handy for estimating the approximation errors in different scales. The purpose of this paper is to study a generalized Poincaré inequality, where the measurement function is of subsampled type ...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 .... Almost/su ciently good connectivity equivalentHelp Center Detailed answers to any questions you might have Me 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 ...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 ≥ May 8, 2002 · The case q = np/(n−p) requires the Sobolev 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. Poincaré inequality such as (5) on the cube...

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