Poincare-type Inequalities Research Articles

Graph Signal Sampling and Interpolation Based on Clusters and Averages

We consider a disjoint cover (partition) of an undirected weighted finite or infinite graph G by J connected subgraphs (clusters) $$\{S_{j}\}_{j\in J}$$ and select functions $$\psi _{j}$$ on each of the clusters. For a given signal f on G the set of its weighted average values samples is defined via inner products $$\{\langle f, \psi _{j}\rangle \}_{j\in J}$$ . The main results of the paper are based on Poincare-type inequalities that we introduce and prove. These inequalities provide an estimate of the norm of the signal f on the entire graph G from sets of samples of f and its local gradient on each of the subgraphs. This allows us to establish discrete Plancherel-Polya-type inequalities (or Marcinkiewicz-Zigmund-type or frame inequalities) for signals whose gradients satisfy a Bernstein-type inequality. These results enable the development of a sampling theory for signals on undirected weighted finite or infinite graphs. For reconstruction of the signals from their samples an interpolation theory by weighted average variational splines is developed. Here by a weighted average variational spline we understand a minimizer of a discrete Sobolev norm which takes on the prescribed weighted average values on a set of clusters (in particular, just values on a subset of vertices). Although our approach is applicable to general graphs it’s especially well suited for finite and infinite graphs with multiple clusters. Such graphs are known as community graphs and they find many important applications in materials science, engineering, computer science, economics, biology, and social studies.

Estimates of the Distance to Exact Solutions of the Stokes Problem with Slip and Leak Boundary Conditions

We deduce a posteriori error estimates of functional type for the stationary Stokes problem with slip and leak boundary conditions. The derived error majorants do not contain mesh dependent constants and are valid for a wide class of energy admissible approximations that satisfy the Dirichlet boundary condition on a part of the boundary. Different forms of error majorants contain global constants associated with Poincare type inequalities or the stability (LBB) condition for the Stokes problem or constants associated with subdomains (if a domain decomposition is applied). It is proved that the majorants are guaranteed and vanish if and only if the functions entering them coincide with the respective exact solutions.

A $$\texttt {p}(\cdot )$$-Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups

We prove a \(\texttt {p}(\cdot )\)-Poincare-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups. We then establish the existence and uniqueness (up to a set of zero \(\texttt {p}(\cdot )\)-capacity) of a minimizer to the Dirichlet energy integral for the variable exponent case.

ON STABLE SOLUTIONS OF BOUNDARY REACTION-DIFFUSION EQUATIONS AND APPLICATIONS TO NONLOCAL PROBLEMS WITH NEUMANN DATA

We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincare-type inequality and classification results for stable solutions, and we apply them to the study of an associated nonlocal problem. We also establish a counterexample in the corresponding framework for the fractional Laplacian.

Square functions and the Hamming cube: duality

Square functions and the Hamming cube: duality, Discrete Analysis 2018:1, 18 pp. This paper establishes a duality between, on the one hand, continuous objects such as square functions and stochastic processes in analysis, and on the other hand, discrete phenomena on the Hamming cube $\{−1, 1\}^n$ with large $n$. More specifically, there is a correspondence between estimates for square functions on the interval $[0,1]$ and discrete gradient estimates for functions on the Hamming cube. The rough idea of a square-function estimate is to cut a function $f$ up into pieces $f_i$ and consider instead the function $Sf=(\sum_i|f_i|^2)^{1/2}$, which turns out to be advantageous in many contexts. As for discrete gradient estimates, they are measures of how much the value of a function $f$ at an average point $x$ differs from the values at the neighbours of $x$. The paper demonstrates a strong connection between two different areas of harmonic analysis that could potentially have a further significant impact on both of them. The main result establishes a Poincare-type inequality for functions on the Hamming cube. Related estimates have been known previously (due to e.g. Naor and Schechtman), but the method used here is new, and in addition to offering a unified approach to a variety of related questions it also provides improved constants in important special cases. The authors show the versatility of their approach by applying it to several well known inequalities on both sides, such as the Sobolev, log-Sobolev, and Chang-Wilson-Wolff inequalities. They also improve a well-known inequality of Beckner for functions on the Hamming cube. The proofs, and the duality, are based on convexity in the guise of Bellman functions (a variant of the Legendre transform). Roughly speaking, one can start with a square function estimate, set up an appropriate Bellman function (the authors use a construction due to B. Davis and used also by G. Wang), dualize it using a minimax procedure to obtain a new “dual” Bellman function, and pass from there to an estimate on the Hamming cube. A similar argument works in the other direction.

On Gaussian Nikolskii–Besov classes

In this note we study Nikolskii–Besov classes of functions of fractional smoothness on finitedimensional and infinite-dimensional spaces with Gaussian measures. We prove the equivalence of two characterizations of these classes: one is based on a certain nonlinear integration by parts formula and the other one is given in terms of the Ornstein–Uhlenbeck semigroup. In addition, we obtain a new Poincare-type inequality. The case of Lebesgue measure has been considered in [1] (see also [2, 3]).

On Weak Solutions to Dirac-Harmonic Equations for Differential Forms

In this paper, we consider the Dirac-harmonic equations for differential forms with Dirichlet boundary data in weak sense. Using the Hodge-decomposition theory of differential forms and Browder–Minty theory of monotone operators, the existence is obtained. According to the Poincare-type inequality of Dirac–Sobolev space, we get the higher integrability.

On the geometry of rectifiable sets with Carleson and Poincar\'{e}-type conditions

A central question in geometric measure theory is whether geometric properties of a set translate into analytical ones. In 1960, E. R. Reifenberg proved that if an $n$-dimensional subset $M$ of $\mathbb{R}^{n+k}$ is well approximated by $n$-planes at every point and at every scale, then $M$ is a locally bi-Holder image of an $n$-plane. Since then, Reifenberg's theorem has been refined in several ways in order to ensure that $M$ is a bi-Lipschitz image of an $n$-plane. In this paper, we show that a Carleson condition on the oscillation of the unit normal of an $n$-Ahlfors regular rectifiable subset $M$ of $\mathbb{R}^{n+1}$ satisfying a Poincare-type inequality is sufficient to prove that $M$ is contained inside a bi-Lipschitz image of an $n$-dimensional affine subspace of $\mathbb{R}^{n+1}$. We also show that this Poincare-type inequality encodes geometrical information about $M$, namely it implies that $M$ is quasiconvex.

Poincaré inequalities on intervals – application to sensitivity analysis

The development of global sensitivity analysis of numerical model outputs has recently raised new issues on 1-dimensional Poincare inequalities. Typically two kind of sensitivity indices are linked by a Poincare type inequality , which provide upper bounds of the most interpretable index by using the other one, cheaper to compute. This allows performing a low-cost screening of unessential variables. The efficiency of this screening then highly depends on the accuracy of the upper bounds in Poincare inequalities. The novelty in the questions concern the wide range of probability distributions involved, which are often truncated on intervals. After providing an overview of the existing knowledge and techniques, we add some theory about Poincare constants on intervals, with improvements for symmetric intervals. Then we exploit the spectral interpretation for computing exact value of Poincare constants of any admissible distribution on a given interval. We give semi-analytical results for some frequent distributions (truncated exponential, triangular, truncated normal), and present a numerical method in the general case. Finally, an application is made to a hydrological problem, showing the benefits of the new results in Poincare inequalities to sensitivity analysis.

Reverse Poincaré-type inequalities for the weak solution of system of partial differential inequalities

In this work we develop the square integral estimates for the functions of n variables which are subharmonic with respect to some variables and for other remaining variables are superharmonic. It is in a sense a generalization of reverse Poincare type inequalities for the difference of superharmonic functions developed in (J. Inequal. Appl. 2015: doi:10.1186/s13660-015-0916-9, 2015).

Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary

It is known that by dualizing the Bochner–Lichnerowicz–Weitzenbock formula, one obtains Poincare-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry–Emery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). When the manifold has a boundary, an appropriate generalization of the Reilly formula may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Brascamp–Lieb-type inequalities on the manifold. All previously known inequalities of Lichnerowicz, Brascamp–Lieb, Bobkov–Ledoux, and Veysseire are recovered, extended to the Riemannian setting and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are obtained. Our framework allows to encompass the entire class of Borell’s convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative generalized dimension.

Riemannian metrics on ℝ n and Sobolev-type Inequalities

Poincare-type estimates for a logarithmically concave measure μ on a convex set Ω are obtained. For this purpose, Ω is endowed with a Riemannian metric g in which the Riemannian manifold with measure (Ω, g, μ) has nonnegative Bakry–Emery tensor and, as a corollary, satisfies the Brascamp–Lieb inequality. Several natural classes of metrics (such as Hessian and conformal metrics) are considered; each of these metrics gives new weighted Poincare, Hardy, or log-Sobolev type inequalities and other results.

Approximation by Hölder Functions in Besov and Triebel–Lizorkin Spaces

In this paper, we show that Besov and Triebel–Lizorkin functions can be approximated by a Holder continuous function both in the Lusin sense and in norm. The results are proved in metric measure spaces for Hajlasz–Besov and Hajlasz–Triebel–Lizorkin functions defined by a pointwise inequality. We also prove new inequalities for medians, including a Poincare type inequality, which we use in the proof of the main result.

Lipschitz and BMO norm inequalities for the composite operator on differential forms

In this paper, we obtain Poincare-type inequalities for the composite operator acting on differential forms and establish the $L^{p}$ , Lipschitz, and BMO norm estimates. We also give the weighted versions of the comparison theorems for the $L^{p}$ , Lipschitz, and BMO norms.

Isoperimetric inequality and the poincare inequality for distributions of polynomials on convex compact set

An isoperimetric inequality and a Poincare-type inequality are proved for probability measures on the line that are the images of a uniform distribution on a convex compact subset of Rn under polynomial mappings of fixed degree d.

Estimates of the Distance to the Exact Solution of Parabolic Problems Based on Local Poincaré Type Inequalities

The goal of the paper is to derive two-sided bounds of the distance between the exact solution of the evolutionary reaction–diffusion problem with mixed Dirichlet–Robin boundary conditions and any function in the admissible energy space. The derivation is based upon special transformations of the integral identity that defines the generalized solution. To obtain estimates with easily computable local constants, the classical Poincare inequalities and Poincare type inequalities for functions with zero mean boundary traces are exploited. The corresponding constants were estimated earlier. Bounds of the distance to the exact solution contain only these constants associated with subdomains. It is proved that the bounds are equivalent to the energy norm of the error. Bibliography: 15 titles.

Isoperimetric inequalities on weighted manifolds with boundary

It is well known that Poincare-type inequalities on Riemannian manifolds with measure satisfying the generalized Bakry–Emery condition can be obtained by using the Bochner–Lichnerowicz–Weitzenbock formula. In the case of manifolds with boundary, a suitable generalization is Reilly’s formula. New Poincaretype inequalities for manifolds with measure are obtained by systematically using this formula combined with various conditions on the boundary of the manifold and boundary conditions for elliptic equations. Among other results, a generalization of Colesanti’s inequality, proved earlier in Euclidean space, is presented. It implies a generalization of Brunn–Minkowski-type inequalities for manifolds. A new evolution equation for surfaces on Riemannian manifolds is studied, which determines the Minkowski addition of convex sets in the Euclidean case. The proposed approach covers a large class of convex measures, including measures with heavy tails, which correspond to negative analytic dimension.

Interpolation of functions based on Poincare type inequalities for functions with zero mean boundary traces

AbstractWe consider interpolation operators for scalar and vector valued functions based on mean values (or on mean values of normal components) defined on certain amount of faces of a mesh consisting of the Lipschitz subdomains (cells). The main result establishes sufficient conditions, which guarantee that a function u ∈ H

A note on zero-sets of fractional sobolev functions with negative power of integrability

We extend a Poincare-type inequality for functions with large zero-sets by Jiang and Lin to fractional Sobolev spaces. As a consequence, we obtain a Hausdorff dimension estimate on the size of zero sets for fractional Sobolev functions whose inverse is integrable. Also, for a suboptimal Hausdorff dimension estimate, we give a completely elementary proof based on a pointwise Poincare-style inequality.

Poincaré and weak Poincaré inequalities for the mixed Poisson measure

By using the Mecke identity, we study a class of birth-death type Dirichlet forms associated with the mixed Poisson measure. Both Poincare and weak Poincare inequalities are established, while another Poincare type inequality is disproved under some reasonable assumptions.