Integral inequalities related to the Tchebychev semigroup

We study the semigroup (P _t ) _t≥0 generated by the operator

Hardy's inequalities for Hermite and Laguerre expansions

Hardy's inequalities are proved for higher-dimensional Hermite and special Hermite expansions of functions in Hardy spaces. Inequalities for multiple Laguerre expansions are also deduced.

     

Schrödinger equation and the oscillatory semigroup for the Hermite operator

We discuss the regularity of the oscillatory semigroup e^itH, where H=-Δ+|x|² is the n-dimensional Hermite operator. The main result is a Strichartz-type estimate for the oscillatory semigroup e^itH in terms of the mixed L^p spaces. The result can be interpreted as the regularity of solution to the Schrödinger equation with potential V(x)=|x|².     

The oscillation of the Poisson semigroup associated to parabolic Hermite operator

Let O(P_τ~L) be the oscillation of the Possion semigroup associated with the parabolic Hermite operator L = ?_t-?+|x|~2. We show that O(P_τ~L) is bounded from L~p(R~(n+1))into itself for 1 p ∞, bounded from L~1(R~(n+1)) into weak-L~1(R~(n+1)) and bounded from L_c~∞(R~(n+1)) into BMO(R~(n+1)). In the case p = ∞ we show that the range of the image of the operator O(P_τ~L) is strictly smaller than the range of a general singular operator.     

Ornstein-Uhlenbeck半群证明不等式的一些应用

按照Ornstein-Uhlenbeck的思想方法,用Ornstein-Uhlenbeck半群和Ornstein-Uhlenbeck算子的一些重要性质,对Brascamp-Lieb不等式、高斯对数Sobolev不等式、逆Bobkov等周不等式等几个重要的几何与分析不等式给出了另一证明.     

Error inequalities for quintic and biquintic discrete Hermite interpolation

In this paper we shall develop a class of discrete Hermite interpolates in one and two independent variables. Further, we offer explicit error bounds in ?_∞ norm for the quintic and biquintic discrete Hermite interpolates. Some numerical examples are included to illustrate the results obtained.     

Sharp inequalities and regularity of heat semigroup on infinite dimensional spaces

This paper discusses the regularity of the heat semigroup P_t generated from the abstract Wiener measure p_t on an abstract Wiener space (H, B). It is shown that if f ϵ L^x(p_t(x, dy)) for some α > 1, then P_tf(x) is infinitely H-differentiable and DⁿP_tf(x) is a symmetric n-linear Hilbert-Schmidt operator in Lⁿ(H). A compact formula of DⁿP_tf(x) and the estimation of ∥DⁿP_tf(x)∥_HS are also given. In the course of proving the above results, we obtain some sharp integral inequalities.     

On the search for weighted inequalities for operators related to the Ornstein-Uhlenbeck semigroup

In this paper, for each given $ we characterize the weights v for which the centered maximal function with respect to the gaussian measure and the Ornstein-Uhlenbeck maximal operator are well defined for every function in and their means converge almost everywhere. In doing so, we find that this condition is also necessary and sufficient for the existence of a weight u such that the operators are bounded from into We approach the poblem by proving some vector valued inequalities. As a byproduct we obtain the strong type (1,1) for the “global” part of the centered maximal function. Received May 18, 1999 / Revised December 9, 1999 Published online July 20, 2000     

Two-weight norm inequalities for the Cesàro means of Hermite expansions

An accurate estimate is obtained of the Cesàro kernel for Hermite expansions. This is used to prove two-weight norm inequalities for Cesàro means of Hermite polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and ``unweighted' cases. An almost everywhere convergence result is obtained as a corollary. It is also shown that the conditions used to prove norm boundedness of the means and most of the conditions used to prove the boundedness of the Cesàro supremum of the means are necessary.

     

Algebras of almost periodic functions with Bohr-Fourier spectrum in a semigroup: Hermite property and its applications

It is proved that the unital Banach algebra of almost periodic functions of several variables with Bohr-Fourier spectrum in a given additive semigroup is an Hermite ring. The same property holds for the Wiener algebra of functions that in addition have absolutely convergent Bohr-Fourier series. As applications of the Hermite property of these algebras, we study factorizations of Wiener-Hopf type of rectangular matrix functions and the Toeplitz corona problem in the context of almost periodic functions of several variables.     

Analytic inequalities,isoperimetric inequalities and logarithmic Sobolev inequalities

There is a simple equivalence between isoperimetric inequalities in Riemannian manifolds and certain analytic inequalities on the same manifold, more extensive than the familiar equivalence of the classical isoperimetric inequality in Euclidean space and the associated Sobolev inequality. By an isoperimetric inequality in this connection we mean any inequality involving the Riemannian volume and Riemannian surface measure of a subset α and its boundary, respectively. We exploit the equivalence to give log-Sobolev inequalities for Riemannian manifolds. Some applications to Schrödinger equations are also given.     

Variational inequalities, coincidence theory, and minimax inequalities

New fixed-point theorems in Fréchet spaces are used to establish new variational inequalities, coincidence results, analytic alternatives, and minimax inequalities.     

Two-point symmetrization and convexity

We prove a conjecture of R. Schneider: the spherical caps are the only spherically convex bodies of the sphere which remain spherically convex after any two-point symmetrization. More generally, we study the relationships between convexity and two-point symmetrization in the Euclidean space and on the sphere. Received: 4 April 2003