Weighted Poincare Inequalities Research Articles

The Gursky–Streets equations

Gursky-Streets introduced a formal Riemannian metric on the space of conformal metrics in a fixed conformal class of a compact Riemannian four-manifold in the context of the $\sigma_2$-Yamabe problem. The geodesic equation of Gursky-Streets' metric is a fully nonlinear degenerate elliptic equation and Gursky-Streets have proved uniform $C^{0, 1}$ regularity for a perturbed equation. Gursky-Streets apply the results and parabolic smoothing of Guan-Wang flow to show that the solution of $\sigma_2$-Yamabe problem is unique. A key ingredient is the convexity of Chang-Yang's $\cF$-functional along the (smooth) geodesic, in view of Gursky-Streets metric and a weighted Poincare inequality of B. Andrews on manifolds with positive Ricci curvature. In this paper we establish uniform $C^{1, 1}$ regularity of the Gursky-Streets' equation. As an application, we can establish strictly the geometric structure in terms of Gursky-Streets' metric, in particular the convexity of $\cF$-functional along $C^{1, 1}$ geodesic. This in particular gives a straightforward proof of the uniqueness of solutions of $\sigma_2$-Yamabe problem.

Existence of Stein kernels under a spectral gap, and discrepancy bounds

Nous prouvons l’existence de noyaux de Stein pour les mesures de probabilites sur $\mathbb{R}^{d}$ satisfaisant une inegalite de Poincare, et obtenons des bornes sur la discrepance de Stein de telles mesures. Des applications au theoreme central limite sont donnees, dont une nouvelle borne sur la vitesse de convergence en distance de Kantorovitch–Wasserstein $W_{2}$ avec un taux et une dependance en la dimension optimales. Comme corollaire, nous obtenons une version quantitative d’une borne sur la constante de Poincare de mesures de probabilites satisfaisant une contrainte sur le moment d’ordre 2. Les resultats sont plus generalement valides dans le cadre de mesures verifiant une inegalite de Poincare a poids inversee. La preuve est basee sur des arguments simples d’analyse fonctionnelle. De plus, nous demontrons deux proprietes generales sur la discrepance de Stein, valide des lors qu’un noyau de Stein existe : la discrepance de Stein est strictement decroissante le long du TCL, et elle controle le moment d’ordre 3 d’un vecteur aleatoire.

Matrix weighted Poincare inequalities and applications to degenerate elliptic systems

We prove Poincar\'e and Sobolev inequalities in matrix A${}_p$ weighted spaces. We then use these Poincar\'e inequalities to prove existence and regularity results for degenerate systems of elliptic equations whose degeneracy is governed by a matrix A${}_p$ weight. Such results parallel earlier results by Fabes, Kenig, and Serapioni for a single degenerate equation governed by a scalar A${}_p$ weight. In addition, we prove Cacciopoli and reverse H\"older inequalities for weak solutions of the degenerate systems. As a means to prove the Poincar\'e inequalities we prove that the Riesz potential and fractional maximal function operators are bounded on matrix weighted $L^p$ spaces and go on to develop an entire matrix A${}_{p, q}$ theory.

Vanishing theorems for complete Riemannian manifolds with nonnegative scalar curvature

In this paper, we show several vanishing theorems for harmonic p-forms on compact manifolds, and $$L^q$$ harmonic p-forms on complete noncompact manifolds with nonnegative scalar curvature, under various pointwise or integral curvature conditions. These conditions involve the positive Yamabe invariant, the weighted Poincare inequality and the pure curvature tensor.

English

We deal with complete submanifolds with weighted Poincare inequality. By assuming the submanifold is δ-stable or has sufficiently small total curvature, we establish two vanishing theorems for L p harmonic 1-forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitorio.

A Sawyer-type sufficient condition for the weighted Poincaré inequality

In this paper we prove the Poincare-type weighted inequality $$\begin{aligned} \Vert v^{1/q} f \Vert _{L^q(\Omega )} \le C \Vert \omega ^{1/p} \nabla f \Vert _{L^p(\Omega )}, \quad q\ge p>1, \end{aligned}$$ for a locally Lipschitz function f with a weighted mean equal to zero over a convex bounded domain $$\Omega $$ ; here the weights v, $$\omega $$ are positive measurable functions which satisfy a certain compatibility condition. This result is a generalization of the well-known weighted Poincare inequality to the case of more general weights in the sense that we do not use the traditional conditions of high summability $$v,\, \omega ^{-\frac{1}{p-1}}\in L^{r,loc}$$ with $$r>1$$ for $$q=p$$ or the reverse doubling condition on the function v for $$q>p$$ . In other words, a Sawyer type sufficient condition on weight functions is established.

Some qualitative properties of Finsler [formula omitted]-Laplacian

The Finsler p-Laplacian is the class of nonlinear differential operators given by ΔH,pu≔div(H(∇u)p−1∇ηH(∇u))where p>1, H:Rn→[0,∞) is a convex function which is in C1(Rn∖{0}) and is positively homogeneous of degree 1. In this article we provide a comparison principle, weighted Poincare Inequality, Liouville Theorem and Hardy type inequality for the Finsler p-Laplacian.

Vanishing theorems for $$L^{2}$$ L 2 harmonic forms on complete Riemannian manifolds

This paper contains some vanishing theorems for $$L^{2}$$ harmonic forms on complete Riemannian manifolds with a weighted Poincare inequality and a certain lower bound of the curvature. The results are in the spirit of Li-Wang and Lam, but without assumptions of sign and growth rate of the weight function, so they can be applied to complete stable hypersurfaces.

Weighted Poincaré Inequalities for Non-local Dirichlet Forms

Let V be a locally bounded measurable function on $${\mathbb {R}}^d$$ such that $$\mu _V(\mathrm{d}x)=C_V \mathrm{e}^{-V(x)}\,\mathrm{d}x$$ is a probability measure. Explicit criteria are presented for weighted Poincare inequalities of the following non-local Dirichlet form $$\begin{aligned} \hat{D}_{\rho ,V}(f,f)=\iint _{\{|x-y|>1\}}(f(y)-f(x))^2\rho (|y-x|)\,\mathrm{d}y\, \mu _V(\mathrm{d}x). \end{aligned}$$ Taking $$\rho (r)={\mathrm{e}^{-\delta r}}{r^{-(d+\alpha )}}$$ with $$0<\alpha <2$$ and $$\delta \geqslant 0$$ , we get new conclusions for (exponentially) tempered fractional Dirichlet forms, which not only complete our recent work (Chen and Wang in Stoch Process Their Appl 124:123–153, 2014; Wang and Wang in J Theor Probab 28:423–448, 2015), but also improve the main result in Mouhot et al. (J Math Pures Appl 95:72–84, 2011).

Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications

We develop a constructive piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The main ingredients to derive optimal error estimates for an averaged Taylor polynomial are a suitable weighted Poincare inequality, a cancellation property and a simple induction argument. We also construct a quasi-interpolation operator, built on local averages over stars, which is well defined for functions in $$L^1$$L1. We derive optimal error estimates for any polynomial degree on simplicial shape regular meshes. On rectangular meshes, these estimates are valid under the condition that neighboring elements have comparable size, which yields optimal anisotropic error estimates over $$n$$n-rectangular domains. The interpolation theory extends to cases when the error and function regularity require different weights. We conclude with three applications: nonuniform elliptic boundary value problems, elliptic problems with singular sources, and fractional powers of elliptic operators.

On the backward in time problem for the thermoelasticity with two temperatures

This paper is devoted to the study of the existence, uniqueness, continuous dependence and spatial behaviour of the solutions for the backward in time problem determined by the Type III with two temperatures thermoelastodynamic theory. We first show the existence, uniqueness and continuous dependence of the solutions. Instability of the solutions for the Type II with two temperatures theory is proved later. For the one-dimensional Type III with two temperatures theory, the exponential instability is also pointed-out. We also analyze the spatial behaviour of the solutions. By means of the exponentially weighted Poincare inequality, we are able to obtain a function that defines a measure on the solutions and, therefore, we obtain the usual exponential type alternative for the solutions of the problem defined in a semi-infinite cylinder.

Weighted Poincare inequalities

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 within a coarse grid element, then condition number bounds for these methods based on standard Poincaré inequalities may be overly pessimistic. In this paper, we present new results on weighted Poincaré-type inequalities for very general classes of coefficients that lead to sharper bounds independent of any possible large variation in the coefficients. The main requirement on the coefficients is some form of quasi-monotonicity that we will carefully describe and analyse. The Poincaré constants depend on the topology and the geometry of regions of relatively high and/or low coefficient values, and we shall study these dependencies in detail. Applications of the inequalities in the analysis of domain decomposition and multigrid methods can be found in Pechstein & Scheichl (2011, Numer. Math., 118) and Scheichl et al. (2012, SIAM J. Numer. Anal., 50).

Harmonic forms on manifolds with weighted Poincaré inequality

Let M be a complete noncompact manifold. We prove vanishing and finiteness results for harmonic p-forms on M, assuming both the curvature operator lower bound and the weighted Poincare inequality on M.

Sobolev interpolation inequalities on generalized John domains

We obtain weighted Sobolev interpolation inequalities on generalized John domains that include John domains (bounded or unbounded) for δ-doubling measures satisfying a weighted Poincare inequality. These measures include ones arising from power weights d(x, ∂Ω) α and need not be doubling. As an application, we extend the Sobolev interpolation inequalities obtained by Caffarelli, Kohn and Nirenberg. We extend these inequalities to product spaces and give some applications on products Ω 1 × Ω 2 of John domains for A p (ℝ n x ℝ m ) weights and power weights of the type w(x, y) = dist(x, G 1 ) α dist(y, G 2 ) β , where G 1 C ∂Ω 1 and G 2 C ∂Ω 2 . For certain cases, we obtain sharp conditions.

Weighted Poincare inequalities on symmetric convex domains

Let α ≥ 0, β ∈ R, 1 ≤ p ≤ q < ∞ with 1 - n / p + n / q , 1 - n + β / p + n + α / q ≥ 0. Let Ω be a bounded convex domain in ℝ n that is symmetric with respect to its center. Define p(x) = dist(x, Ω c ) = inf{|x - γ|: γ ∈ Ω c } and ρ α (E) = ∫ E ρ(x) α dx. Let f be a Lipschitz continuous function on Ω and f Ω,ρ α = ∫ Ω f(x)ρ(x) α dx/ρ α (Ω). We obtain the following weighted Poincare inequality: ∥f - f Ω,ρ α ∥L q ρα (Ω) ≤ Cη β/p-α/q |Ω| 1/q-1/p diam(Ω) 1-β/p+α/q ∥∇f∥ L p ρ β (Ω) where η is the eccentricity of Ω and C is a constant depending only on p, q, α, β, and the dimension n. Moreover, the exponent of η is sharp.

Some weighted Poincare inequalities

Using stable solutions of suitable PDEs, we obtain some weighted Poincare inequalities, with applications to the Laplace operator in the Euclidean space, to the real part of the Laplace-Kohn operator in the Heisenberg group, to the sub-laplacian in the Engel group, to the Franchi-Grushin-Lanconelli operators, and the p-laplacian in the Euclidean space.

Weighted Poincaré inequality and heat kernel estimates for finite range jump processes

It is well-known that there is a deep interplay between analysis and probability theory. For example, for a Markovian infinitesimal generator $${\mathcal{L}}$$ , the transition density function p(t, x, y) of the Markov process associated with $${\mathcal{L}}$$ (if it exists) is the fundamental solution (or heat kernel) of $${\mathcal{L}}$$ . A fundamental problem in analysis and in probability theory is to obtain sharp estimates of p(t, x, y). In this paper, we consider a class of non-local (integro-differential) operators $${\mathcal{L}}$$ on $${\mathbb{R}^d}$$ of the form $$\mathcal{L}u(x) = \lim\limits_{{\varepsilon \downarrow 0}} \int\limits_{\{y\in \mathbb {R}^d: \, |y-x| > \varepsilon\}} (u(y)-u(x)) J(x, y) dy,$$ where $${\displaystyle J(x, y)= \frac{c (x, y)}{|x-y|^{d+\alpha}} {\bf 1}_{\{|x-y| \leq \kappa\}}}$$ for some constant $${\kappa > 0}$$ and a measurable symmetric function c(x, y) that is bounded between two positive constants. Associated with such a non-local operator $${\mathcal{L}}$$ is an $${\mathbb{R}^d}$$ -valued symmetric jump process of finite range with jumping kernel J(x, y). We establish sharp two-sided heat kernel estimate and derive parabolic Harnack principle for them. Along the way, some new heat kernel estimates are obtained for more general finite range jump processes that were studied in (Barlow et al. in Trans Am Math Soc, 2008). One of our key tools is a new form of weighted Poincare inequality of fractional order, which corresponds to the one established by Jerison in (Duke Math J 53(2):503–523, 1986) for differential operators. Using Meyer’s construction of adding new jumps, we also obtain various a priori estimates such as Holder continuity estimates for parabolic functions of jump processes (not necessarily of finite range) where only a very mild integrability condition is assumed for large jumps. To establish these results, we employ methods from both probability theory and analysis extensively.

Poincaré inequalities in weighted Sobolev spaces

The weighted Poincare inequalities in weighted Sobolev spaces are discussed, and the necessary and sufficient conditions for them to hold are given. That is, the Poincare inequalities hold if, and only if, the ball measure of non-compactness of the natural embedding of weighted Sobolev spaces is less than 1.

A weighted Poincar� inequality and removable singularities for harmonic maps

We give a sufficient condition on a closed subset Σ⊂Rn for the weighted Poincare inequality (1.5) below to be valid. As an application, we prove that, for any 2≤p<n and any such closed subset Σ⊂Rn, if u∈C1(Ω ∖ Σ, N) ∩ W1,p (Ω, N) is a stationary p-harmonic map such that ∫Ω|Du|p(x) dx is sufficiently small, then u∈C1(Ω, N). This extends previously known removal singularity theorems for p-harmonic maps.

Sharp conditions for 1-dimensional weighted Poincare inequalities

Sharp conditions for 1-dimensional weighted Poincare inequalities