Probability distance inequalities on Riemannian manifolds and path spaces

We construct Otto-Villani's coupling for general reversible diffusion processes on a Riemannian manifold. As an application, some new estimates are obtained for Wasserstein distances by using a Sobolev-Poincaré type inequality introduced by Lata?a and Oleszkiewicz. The corresponding concentration estimates of the measure are presented. Finally, our main result is applied to obtain the transportation cost inequalities on the path space with respect to both of the L²-distance and the intrinsic distance. In particular, Talagrand's inequality holds on the path space over a compact manifold.     

Super Poincaré Inequalities,Orlicz Norms and Essential Spectrum

We prove some results about the super Poincaré inequality (SPI) and its relation to the spectrum of an operator: we show that it can be alternatively written with Orlicz norms instead of L ¹ norms, and we use this to give an alternative proof that a bound on the bottom of the essential spectrum implies a SPI. Finally, we apply these ideas to give a spectral proof of the log Sobolev inequality for the Gaussian measure.     

p-Poincaré inequality versus ∞-Poincaré inequality: some counterexamples

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 demonstrate by examples that an analogous geometric characterization for finite p is not possible. The examples we give are metric measure spaces which are doubling and support an ∞-Poincaré inequality, but support no finite p-Poincaré inequality. In particular, these examples show that one cannot expect a self-improving property for ∞-Poincaré inequality in the spirit of Keith–Zhong (Ann Math 167(2):575–599, 2008). We also show that the persistence of Poincaré inequality under measured Gromov–Hausdorff limits fails for ∞-Poincaré inequality.     

A generalized real-valued measure of the inequality associated with a fuzzy random variable

Fuzzy random variables have been introduced by Puri and Ralescu as an extension of random sets. In this paper, we first introduce a real-valued generalized measure of the “relative variation” (or inequality) associated with a fuzzy random variable. This measure is inspired in Csiszár's f-divergence, and extends to fuzzy random variables many well-known inequality indices. To guarantee certain relevant properties of this measure, we have to distinguish two main families of measures which will be characterized. Then, the fundamental properties are derived, and an outstanding measure in each family is separately examined on the basis of an additive decomposition property and an additive decomposability one. Finally, two examples illustrate the application of the study in this paper.     

Square area integral estimates for subharmonic functions in NTA domains

In this paper, we prove a good-λ inequality between the nontangential maximal function and the square area integral of a subharmonic functionu in a bounded NTA domainD inR ⁿ . We achieve this by showing that a weighted Riesz measure ofu is a Carleson measure, with the Carleson norm bounded by a constant independent ofu. As consequences of the good-λ inequality, we obtain McConnell-Uchiyama's inequality and an analogue of Murai-Uchiyama's inequality for subharmonic functions inD.     

Heat kernel analysis on semi-infinite Lie groups

This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the Lp norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds in this setting.     

A class of nonconforming quadrilateral finite elements for incompressible flow

This paper focuses on the low-order nonconforming rectangular and quadrilateral finite elements approximation of incompressible flow.Beyond the previous research works,we propose a general strategy to construct the basis functions.Under several specific constraints,the optimal error estimates are obtained,i.e.,the first order accuracy of the velocities in H1-norm and the pressure in L2-norm,as well as the second order accuracy of the velocities in L2-norm.Besides,we clarify the differences between rectangular and quadrilateral finite element approximation.In addition,we give several examples to verify the validity of our error estimates.     

Uniform Sublevel Radon-like Inequalities

This paper is concerned with establishing uniform weighted L ^p –L ^q estimates for a class of operators generalizing both Radon-like operators and sublevel set operators. Such estimates are shown to hold under general circumstances whenever a sublevel-type inequality is satisfied by certain associated measures (the inequality is of the sort studied by Oberlin (Math. Proc. Camb. Philos. Soc. 129(3):517–526, 2000), relating measures of parallelepipeds to powers of their Euclidean volumes). These ideas lead to previously unknown, weighted affine-invariant estimates for Radon-like operators as well as new L ^p -improving estimates for degenerate Radon-like operators with folding canonical relations which satisfy an additional curvature condition of Greenleaf and Seeger (J. Reine Angew. Math. 455:35–56, 1994) for FIOs (building on the ideas of Sogge (Invent. Math. 104(2):349–376, 1991) and Mockenhaupt et al. (J. Am. Math. Soc. 6(1):65–130, 1993)); these new estimates fall outside the range of estimates which are known to hold in the generality of the FIO context.     

Weak Poincaré Inequalities, Decay of Markov Semigroups and Concentration of Measures

For a reasonable class of weak Poincaré inequalities, the decay of the corresponding Markov semigroups obtained earlier by Röckner and the first named author is improved by removing an extra L ²–norm. Next, a concentration estimate of the reference measure is presented for the weak Poincaré inequality, which is sharp as illustrated by some examples of one–dimensional diffusion processes.     

Functional inequalities on abstract Hilbert spaces and applications

We study the essential spectrum and the semigroup property for self-adjoint operators on abstract Hilbert spaces by using functional inequalities. Some known results obtained on the L ² -space w.r.t. a measure space are generalized. The functional inequality is also used to study non-symmetric semigroups. Mathematics Subject Classification (2000): 49R20, 58F19.Research supported in part by NNSFC (10121101, 10025105), TRAPOYT and the 973-Project.     

Integrability of optimal mappings

In this paper, we study the integrability of optimal mappings T taking a probability measure μ to another measure g · μ. We assume that T minimizes the cost function c and μ satisfies some special inequalities related to c (the infimum-convolution inequality or the logarithmic c-Sobolev inequality). The results obtained are applied to the analysis of measures of the form exp(?|x|^α).     

Dimension-Independent Estimates for Heat Operators and Harmonic Functions

We establish dimension-independent estimates related to heat operators e ^tL on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply these results to two examples of interest: when L is the Laplace–Beltrami operator on a Riemannian manifold with Ricci curvature bounded from below, and when L is an invariant subelliptic operator of Hörmander type on a Lie group. In the former example, we also obtain pointwise bounds on harmonic and subharmonic functions, while in the latter example, we obtain pointwise bounds on harmonic functions when a generalized curvature-dimension inequality is satisfied.     

Fractional Navier-Stokes equations and a Hölder-type inequality in a sum of singular spaces

In this paper we study the local well-posedness of the fractional Navier-Stokes system with initial data belonging to a sum of two pseudomeasure-type spaces denoted by PM^a,b:=PM^a+PM^b. The proof requires showing a Hölder-type inequality in PM^a,b, as well as establishing estimates of the semigroup generated by the fractional power of Laplacian (−Δ)^γ on these spaces.     

Norming sets and integration with respect to vector measures

Let v be a countably additive measure defined on a measurable space (Ω, Σ) and taking values in a Banach space X. Let f : Ω → ? be a measurable function. In order to check the integrability (respectively, weak integrability) of f with respect to v it is sometimes enough to test on a norming set Λ ⊂ X^*. In this paper we show that this is the case when A is a James boundary for BX^* (respectively, Λ is weak^*-thick). Some examples and applications are given as well.     

Pressure projection stabilized finite element method for Stokes problem with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with the Stokes operator. The H¹ and L² error estimates for the velocity and the L² error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.     

Gap in the essential spectrum of an elliptic formally self-adjoint system of differential equations

For a formally self-adjoint elliptic system of partial differential equations with periodic coefficients in the space ℝ ⁿ , we show that, by conferring contrast properties to the coefficients of differential operators, one can open a gap in the essential spectrum of the system. We suggest a method based on the derivation of an asymptotically sharp generalized Korn inequality and the use of the maximin principle; this method applies to perforated media as well as to periodic layered and quasi-cylindrical waveguides.     

Functions of bounded variation on “good” metric spaces

In this paper we give a natural definition of Banach space valued BV functions defined on complete metric spaces endowed with a doubling measure (for the sake of simplicity we will say doubling metric spaces) supporting a Poincaré inequality (see Definition 2.5 below). The definition is given starting from Lipschitz functions and taking closure with respect to a suitable convergence; more precisely, we define a total variation functional for every Lipschitz function; then we take the lower semicontinuous envelope with respect to the L¹ topology and define the BV space as the domain of finiteness of the envelope. The main problem of this definition is the proof that the total variation of any BV function is a measure; the techniques used to prove this fact are typical of Γ-convergence and relaxation. In Section 4 we define the sets of finite perimeter, obtaining a Coarea formula and an Isoperimetric inequality. In the last section of this paper we also compare our definition of BV functions with some definitions already existing in particular classes of doubling metric spaces, such as Weighted spaces, Ahlfors-regular spaces and Carnot–Carathéodory spaces.     

Integral properties of the classical warping function of a simply connected domain

Let u(z,G) be the classical warping function of a simply connected domain G. We prove that the L ^p -norms of the warping function with different exponents are related by a sharp isoperimetric inequality, including the functional u(G) = sup _x∈G u(x, G). A particular case of our result is the classical Payne inequality for the torsional rigidity of a domain. On the basis of the warping function, we construct a new physical functional possessing the isoperimetric monotonicity property. For a class of integrals depending on the warping function, we also obtain a priori estimates in terms of the L ^p -norms of the warping function as well as the functional u(G). In the proof, we use the estimation technique on level lines proposed by Payne.     

On the convergence of the proximal algorithm for nonsmooth functions involving analytic features

We study the convergence of the proximal algorithm applied to nonsmooth functions that satisfy the ?jasiewicz inequality around their generalized critical points. Typical examples of functions complying with these conditions are continuous semialgebraic or subanalytic functions. Following ?jasiewicz’s original idea, we prove that any bounded sequence generated by the proximal algorithm converges to some generalized critical point. We also obtain convergence rate results which are related to the flatness of the function by means of ?jasiewicz exponents. Apart from the sharp and elliptic cases which yield finite or geometric convergence, the decay estimates that are derived are of the type O(k ^?s ), where s ∈ (0, + ∞) depends on the flatness of the function.     

Hypoelliptic heat kernel inequalities on the Heisenberg group

We study the existence of “L^p-type” gradient estimates for the heat kernel of the natural hypoelliptic “Laplacian” on the real three-dimensional Heisenberg Lie group. Using Malliavin calculus methods, we verify that these estimates hold in the case p>1. The gradient estimate for p=2 implies a corresponding Poincaré inequality for the heat kernel. The gradient estimate for p=1 is still open; if proved, this estimate would imply a logarithmic Sobolev inequality for the heat kernel.