Operator extensions of Hua’s inequality

We give an extension of Hua’s inequality in pre-Hilbert C^∗-modules without using convexity or the classical Hua’s inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert C^∗-modules, another extension of Hua’s inequality is obtained. We also present an operator Hua’s inequality, which is equivalent to operator convexity of given continuous real function.     

Extensions of Heinz-Kato-Furuta inequality

We give an extension of Lin's recent improvement of a generalized Schwarz inequality, which is based on the Heinz-Kato-Furuta inequality. As a consequence, we can sharpen the Heinz-Kato-Furuta inequality.

     

Regularity for Certain Nonlinear Parabolic Systems

Abstract

By means of an inequality of Poincaré type, a weak Harnack inequality for the gradient of a solution and an integral inequality of Campanato type, it is shown that a solution to certain degenerate parabolic system is locally Hölder continuous. The system is a generalization of p-Laplacian system. Using a difference quotient method and Moser type iteration it is then proved that the gradient of a solution is locally bounded. Finally using the iteration and scaling it is shown that the gradient of the solution satisfies a Campanato type integral inequality and is locally Hölder continuous.     

On a dual Hardy-Hilbert’s inequality and its generalization

Summary This paper deals with a dual Hardy-Hilbert’s inequality with a best constant factor involving the beta function, which is an extension of the Hilbert’s inequality with the form of (p,q)-parameter. We also consider its extended form and an equivalent inequality.     

Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator

 We establish, by simple semigroup arguments, a Lévy–Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature. Oblatum 19-VI-1995     

Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝⁿ. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested and left as an open problem in [4]. Many variants will be investigated in a companion article [10].     

A modified log-Harnack inequality and asymptotically strong Feller property

We introduce a new Harnack type inequality, which is a modification of the log-Harnack inequality established by R?ckner and Wang and prove that it implies the asymptotically strong Feller property (ASF). This inequality generalizes the criterion for ASF introduced by Hairer and Mattingly. As an example, we show by an asymptotic coupling that the 2D stochastic Navier-Stokes equation driven by highly degenerate but essentially elliptic noise satisfies our modified log-Harnack inequality.     

Stability of some versions of the Prékopa–Leindler inequality

Two consequences of the stability version of the one dimensional Prékopa–Leindler inequality are presented. One is the stability version of the Blaschke–Santaló inequality, and the other is a stability version of the Prékopa– Leindler inequality for even functions in higher dimensions, where a recent stability version of the Brunn–Minkowski inequality is also used in an essential way.     

Existence of extremal functions for a Hardy-Sobolev inequality

We present the best constant and the extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in RN.     

Mixed width-integrals of convex bodies

The mixed width-integrals are defined and shown to have properties similar to those of the mixed volumes of Minkowski. An inequality is established for the mixed width-integrals analogous to the Fenchel-Aleksandrov inequality for the mixed volumes. An isoperimetric inequality (involving the mixed width-integrals) is presented which generalizes an inequality recently obtained by Chakerian and Heil. Strengthened versions of this general inequality are obtained by introducing indexed mixed width-integrals. This leads to an isoperimetric inequality similar to Busemann’s inequality involving concurrent cross-sections of convex bodies.     

An Equivalent Condition Between Poincaré Inequality and <Emphasis Type="Italic">T</Emphasis><Subscript>2</Subscript>-Transportation Cost Inequality

In this paper, We give an equivalent condition between Poincaré inequality and T ₂-transportation inequality, and by this result we find a series of measures to enhance the claim that Log-Sobolev inequality is stronger than T ₂-transportation cost inequality.     

Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator

We establish, by simple semigroup arguments, a Lévy-Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian, isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature.Oblatum 19-VI-1995     

On Hausdorff measure and an inequality due to Maz’ya

We give an “elementary” proof of an inequality due to Maz’ya. As a prerequisite we prove an approximation property for the Hausdorff measure. We also comment on the relations between Maz’ya’s inequality, the isoperimetric inequality, and the Sobolev inequality.     

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.     

On the rate of change of the sharp constant in the Sobolev–Poincaré inequality

The rate of change of the sharp constant in the Sobolev–Poincaré or Friedrichs inequality is estimated for a Euclidean domain that moves outward. The key ingredients are a Hadamard variation formula and an inequality that reverses the usual Hölder inequality.     

A lower‐epiperimetric inequality for area‐minimizing surfaces

The epiperimetric inequality introduced by E. R. Reifenberg in [3] gives a rate of decay at a point for the decreasing k‐density of area of an area‐minimizing integral k‐cycle. While dilating the cycle at that point, this rate of decay holds once the configuration is close to a tangent cone configuration and above the limiting density corresponding to that configuration. This is why we propose to call the Reifenberg epiperimetric inequality an upper‐epiperimetric inequality. A direct consequence of this upper‐epiperimetric inequality is the statement that any point possesses a unique tangent cone. The upper‐epiperimetric inequality was proven by B. White in [5] for area‐minimizing 2‐cycles in ?ⁿ. In the present paper we introduce the notion of a lower‐epiperimetric inequality. This inequality gives this time a rate of decay for the decreasing k‐density of area of an area‐minimizing integral k‐cycle, while dilating the cycle at a point once the configuration is close to a tangent cone configuration and below the limiting density corresponding to that configuration. Our main result in the present paper is to prove the lower‐epiperimetric inequality for area‐minimizing 2‐cycles in ?ⁿ. As a consequence of this inequality we prove the “splitting before tilting” phenomenon for calibrated 2‐rectifiable cycles, which plays a crucial role in the proof of the regularity of 1‐1 integral currents in [4]. © 2004 Wiley Periodicals, Inc.     

Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality

We give a functional version of the affine isoperimetric inequality for log-concave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality for entropy. A linearization of this inequality gives an inverse inequality to the Poincaré inequality for the Gaussian measure.     

Meritocratic and monopoly inequality: A computer simulation of income distribution

This paper explores the properties of a model of the distribution of income in which individual income is proportional to a multiplicative function of previous income, ability, chance, a ceiling factor determined by competition among members of an income class for resources held by members of other classes, and an additive factor summarizing effects of altruism and minimal subsistence. The behavior of the model is investigated by computer simulation for combinations of values of three model parameters representing the tendency of income to grow exponentially (the Monopoly effect), the weight of the ability factor (the meritocracy effect), and the weight of the ceiling factor resulting from competitive interactions. Steady state income distributions generated by the model are characterized by measures of income inequality, exchange mobility, elite stability, and meritocracy. Results suggest that for constant Monopoly effect, the effect of the meritocracy parameter on various aggregate outcomes is nonlinear, with a range over which greater returns to ability produce lower inequality, lower exchange mobility, greater elite stability and meritocracy, for constant returns to ability, a greater Monopoly effect generally produces greater inequality, more exchange mobility, less stability of the elite, and lower meritocracy. Results also reveal a nonlinear relationship between exchange mobility and inequality, with mobility decreasing to a minimum and then increasing again as inequality increases; a nonlinear but monotonic negative relationship between elite stability and inequality, with greater inequality, associated with less stability, and a nonlinear relationship between meritocracy and inequality, with meritocracy increasing at first with inequality at low inequality levels, reaching a maximum and then decreasing as inequality increases further. These findings are interpreted in relation to major stratification trends in the course of sociocultural evolution.     

Orlicz-Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms

Abstract

In this paper, we extend the Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms to an Orlicz setting and an Orlicz-Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms is established. The new Orlicz-Brun-Minkowski inequality in special case yields the L_p-Brunn-Minkowski inequality for the radial mixed Blaschke-Minkowski homomorphisms and the mixed intersection bodies, respectively.     

A note on the weighted Hilbert's inequality

A finite Hilbert transformation associated with a polynomial is the analogue of a Hilbert transformation associated with an entire function which is a generalization of the classical Hilbert transformation. The weighted Hilbert inequality, which has applications in analytic number theory, is closely related to the finite Hilbert transformation associated with a polynomial. In this note, we study a relation between the finite Hilbert transformation and the weighted Hilbert's inequality. A question is posed about the finite Hilbert transformation, of which an affirmative answer implies the weighted Hilbert inequality.