A r -Weighted Poincaré-Type Inequalities for Differential Forms in Some Domains

We prove local weighted integral inequalities for differential forms. Then byusing the local results, we prove global weighted integral inequalities for differential forms in L _s (μ)-averaging domains and in John domains, respectively, which can be considered as generalizations of the classical Poincaré-type inequality.     

Variational Formulas of Poincaré-type Inequalities for Birth-Death Processes

Abstract In author’s one previous paper, the same topic was studied for one dimensional diffusions. As a continuation, this paper studies the discrete case, that is the birth-death processes. The explicit criteria for the inequalities, the variational formulas and explicit bounds of the corresponding constants in the inequalities are presented. As typical applications, the Nash inequalities and logarithmic Sobolev inequalities are examined. Research supported in part by NSFC (No. 10121101), 973 Project and RFDP     

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 Poincaré 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).

     

Poincar and Weak Poincar 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 Poincar and weak Poincar inequalities are established, while another Poincar type inequality is disproved under some reasonable assumptions.     

Poincaré-Type inequalities and reconstruction of Paley-Wiener functions on manifolds

The main goal of the article is to show that Paley-Wiener functions ƒ ∈ L ₂(M) of a fixed band width to on a Riemannian manifold of bounded geometry M completely determined and can be reconstructed from a set of numbers Φ_i (ƒ), i ∈ ℕwhere Φ_i is a countable sequence of weighted integrals over a collection of “small” and “densely” distributed compact subsets. In particular, Φ_i, i ∈ ℕ,can be a sequence of weighted Dirac measures δ_xi, x_i ∈M. It is shown that Paley-Wiener functions on M can be reconstructed as uniform limits of certain variational average spline functions. To obtain these results we establish certain inequalities which are generalizations of the Poincaré-Wirtingen and Plancherel-Polya inequalities. Our approach to the problem and most of our results are new even in the one-dimensional case.     

Criteria for Super-and Weak-Poincaré Inequalities of Ergodic Birth-Death Processes

Criteria for the super-Poincaré inequality and the weak-Poincaré inequality about ergodic birth-death processes are presented. Our work further completes ten criteria for birth-death processes presented in Table 1.4 (p. 15) of Prof. Mu-Fa Chen's book "Eigenvalues, Inequalities and Ergodic Theory" (Springer, London, 2005). As a byproduct, we conclude that only ergodic birth-death processes on finite state space satisfy the Nash inequality with index 0 ν≤ 2.     

Poincaré and Logarithmic Sobolev Inequalities for Nearly Radial Measures

Acta Mathematica Sinica, English Series - Poincaré inequality has been studied by Bobkov for radial measures, but few are known about the logarithmic Sobolev inequality in the radial case. We...     

Computational Aspect for Function-Valued Padé-Type Approximation

The computational problems of two special determinants are investigated. Those tion for computing Fredholm integral equation of the second kind. The main tool to be used in this paper is the well-known Schur complement theorem.     

Riesz-Fejér Inequalities for Harmonic Functions

Potential Analysis - In this article, we prove the Riesz - Fejér inequality for complex-valued harmonic functions in the harmonic Hardy space hp for all p &gt;?1. The result is sharp...     

Poincaré-Hopf inequalities for periodic orbits

The interplay between the dynamics of a nonsingular Morse-Smale flow on a smooth, closed, n-dimensional manifold, M, and the topology of M, was exhibited in Franks (Comment Math Helv 53(2):279?C294, 1978), Smale (Bull Am Math Soc 66:43?C49, 1960), by means of a collection of inequalities, which we refer to as Morse-Smale inequalities. These inequalities relate the number of closed orbits of each index to the Betti numbers of M. These well-known inequalities provide the necessary conditions for a given dynamical data in the form of a specified number of closed orbits of a given index to be realized as a nonsingular Morse-Smale flow on M. In this article we provide two inequalities, hereby referred to as Poincaré-Hopf inequalities for periodic orbits, which imposes constraints on the dynamics of periodic orbits without reference to the Betti numbers of the manifold M. The main theorem establishes the necessity and sufficiency of the Poincaré-Hopf inequalities in order for the Morse-Smale inequalities to hold.     

Harnack Inequalities for some Lévy Processes

In this paper we prove Harnack inequality for nonnegative functions which are harmonic with respect to random walks in ℝ ^d . We give several examples when the scale invariant Harnack inequality does not hold. For any α ∈ (0,2) we also prove the Harnack inequality for nonnegative harmonic functions with respect to a symmetric Lévy process in ℝ ^d with a Lévy density given by $c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}$c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}, where 0 ≤ j(r) ≤ cr ^{− d − α} , ∀ r > 1, for some constant c. Finally, we establish the Harnack inequality for nonnegative harmonic functions with respect to a subordinate Brownian motion with subordinator with Laplace exponent ϕ(λ) = λ ^α/2ℓ(λ), λ > 0, where ℓ is a slowly varying function at infinity and α ∈ (0,2).     

Duality and the Poincaré–Hopf Inequalities

In this article we obtain a duality result for an n-manifold N with boundary ∂N = N + ⊔N ⁻ a disjoint union, where N ⁺ and N ⁻ are arbitrarily chosen parts in ∂N and need not be compact. This duality result is used to generalize the Poincaré–Hopf inequalities in a non-compact setting.     

Chebyshev-Type Polynomials Arising in Poincaré Limit Inequalities

Mathematical Notes -     

A Generalization of Poincaré and Log-Sobolev Inequalities

A generalized Beckner-type inequality interpolating the Poincaré and the log-Sobolev inequalities is studied. This inequality possesses the additivity property and characterizes certain exponential convergence of the corresponding Markov semi-group. A correspondence between this inequality and the so-called F-Sobolev inequality is presented, with the known criteria of the latter applying also to the former. In particular, an important result of Lataa and Oleszkiewicz is generalized.     

Maximal Inequalities for Fractional Lévy and Related Processes

In this article we study processes that are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding “convoluted martingale” is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.     

On Poincaré-Friedrichs Type Inequalities for the Broken Sobolev Space ${\rm W}^{2,1}$

We are concerned with the derivation of Poincaré-Friedrichs type inequalities in the broken Sobolev space $W^{2,1}$($Ω$; $\mathcal{T}_h$) with respect to a geometrically conforming, simplicial triagulation $\mathcal{T}_h$ of a bounded Lipschitz domain $Ω$ in $\mathbb{R}^d$ , $d$ $∈$ $\mathbb{N}$. Such inequalities are of interest in the numerical analysis of nonconforming finite element discretizations such as ${\rm C}^0$ Discontinuous Galerkin (${\rm C}^0$${\rm DG}$) approximations of minimization problems in the Sobolev space $W^{2,1}$($Ω$), or more generally, in the Banach space $BV^2$($Ω$) of functions of bounded second order total variation. As an application, we consider a ${\rm C}^0$${\rm DG}$ approximation of a minimization problem in$BV^2$($Ω$) which is useful for texture analysis and management in image restoration.     

Bilinear Sobolev–Poincaré Inequalities and Leibniz-Type Rules

The dual purpose of this article is to establish bilinear Poincaré-type estimates associated with an approximation of the identity and to explore the connections between bilinear pseudo-differential operators and bilinear potential-type operators. The common underlying theme in both topics is their applications to Leibniz-type rules in Sobolev and Campanato–Morrey spaces under Sobolev scaling.     

A generalization of Poincaré's theorem for recurrence systems

A new proof and a genuine generalization to systems of first order equations is given from Poincaré classical theorem on ratio asymptotics of solutions of higher order recurrence equations. The asymptotic behavior of a fundamental system of solutions is obtained.     

Borel-Carathéodory and Fan Type Inequalities for Hilbert Space Bicontractions

This paper deals with some operator versions of the classical Borel–Carathéodory and Ky Fan theorems for analytic functions of complex variable. Our results refer to the functional calculus with analytic functions on the unit bidisk induced by strict bicontractions on a complex Hilbert space.