Weighted isoperimetric inequalities in warped product manifolds

We prove some sharp isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We also relate them to inequalities involving the higher order mean-curvature integrals. Applications include some sharp eigenvalue estimates, Pólya-Szegö inequality, Faber-Krahn inequality, Sobolev inequality and some sharp geometric inequalities in some warped product spaces.     

Improved Hölder and reverse Hölder inequalities for Gaussian random vectors

We propose algebraic criteria that yield sharp Hölder types of inequalities for the product of functions of Gaussian random vectors with arbitrary covariance structure. While our lower inequality appears to be new, we prove that the upper inequality gives an equivalent formulation for the geometric Brascamp–Lieb inequality for Gaussian measures. As an application, we retrieve the Gaussian hypercontractivity as well as its reverse and we present a generalization of the sharp Young and reverse Young inequalities. From the latter, we recover several known inequalities in the literature including the Prékopa–Leindler and Barthe inequalities.     

Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolev-type imbeddings

In 1972 the author proved the so-called conductor and capacitary inequalities for the Dirichlet-type integrals of a function on a Euclidean domain. Both were used to derive necessary and sufficient conditions for Sobolev-type inequalities involving arbitrary domains and measures.The present article contains new conductor inequalities for nonnegative functionals acting on functions defined on topological spaces. Sharp capacitary inequalities, stronger than the classical Sobolev inequality, with the best constant and the sharp form of the Yudovich inequality (Soviet Math. Dokl. 2 (1961) 746) due to Moser (Indiana Math. J. 20 (1971) 1077) are found.     

Weak Sharp Solutions for Nonsmooth Variational Inequalities

In this paper, we study the weak sharp solutions for nonsmooth variational inequalities and give a characterization in terms of error bound. Some characterizations of solution set of nonsmooth variational inequalities are presented. Under certain conditions, we prove that the sequence generated by an algorithm for finding a solution of nonsmooth variational inequalities terminates after a finite number of iterates provided that the solutions set of a nonsmooth variational inequality is weakly sharp. We also study the finite termination property of the gradient projection method for solving nonsmooth variational inequalities under weak sharpness of the solution set.     

Sharp local embedding inequalities

In this paper we establish local versions of the Onofri and sharp Sobolev inequalities. Such local inequalities enable us to give a more direct and simpler proof of the Onofri inequality on ??², as well as an alternative proof of sharp Sobolev inequalities on ??ⁿ (for n ≥ 3). © 2005 Wiley Periodicals, Inc.     

Two isoperimetric inequalities for the Sobolev constant

In this note, we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first inequality is a variation on the classical Schwarz Lemma from complex analysis, similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini, and Ransford, while the second generalizes an isoperimetric inequality for the first eigenfunction of the Laplacian due to Payne and Rayner.     

两个严格的加权Ostrowski型不等式

使用Cauchy积分不等式和Grüss不等式的变式得到两个严格的加权Ostrowski型不等式.     

On sharp triangle inequalities in Banach spaces

We shall present a couple of norm inequalities which will much improve the sharp triangle inequality with n elements and its reverse inequality in a Banach space shown recently by the last three authors.     

Inequalities for Fixed Points of Holomorphic Functions

Cowen and Pommerenke obtained inequalities for fixed pointsof holomorphic and univalent functions in [1]. There was onlyone case where their inequality was not the best possible. Byan entirely new method, we will establish a sharp improvementto this case and provide simple proofs to their main resultsin [1]     

A new modified logarithmic Sobolev inequality for Poisson point processes and several applications

By means of the martingale representation, we establish a new modified logarithmic Sobolev inequality, which covers the previous modified logarithmic Sobolev inequalities of Bobkov-Ledoux and the L ¹-logarithmic Sobolev inequality obtained in our previous work. From it we derive several sharp deviation inequalities of Talagrand's type, by following the powerful Herbst method developed recently by Ledoux and al. Moreover this new modified logarithmic Sobolev inequality is transported on the discontinuous path space with respect to the law of a Lévy process. Received: 16 June 1999 / Revised version: 13 March 2000 / Published online: 12 October 2000     

Dirichlet and Neumann problems to critical Emden–Fowler type equations

We describe recent results on attainability of sharp constants in the Sobolev inequality, the Sobolev–Poincaré inequality, the Hardy–Sobolev inequality and related inequalities. This gives us the solvability of boundary value problems to critical Emden–Fowler equations.      

Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution

Summary. We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for the exponential distribution. We actually observe that every measure satisfying a Poincaré inequality shares the same concentration phenomenon. We also discuss exponential integrability under Poincaré inequalities and its consequence to sharp diameter upper bounds on spectral gaps. Received: 10 June 1996 / In revised form: 9 August 1996     

Random Clarkson inequalities and LP versions of Grothendieck's inequality

In a recent paper KATO [3] used the LITTLEWOOD matrices to generalise CLARKSON'S inequalities. Our first aim is to indicate how KATO'S result can be deduced from a neglected version of the HAUSDORFF -YOUNG inequality which was proved by WELLS and WILLIAMS [12]. We next establish ?random CLARKSON inequalities”?. These show that the expected behaviour of matrices whose coefficients are random ± 1′s is, as one might expect, the same as the behaviour that KATO observed in the LITTLEWOOD matrices. Finally we show how sharp Lp versions of GROTHENDIECK'S inequality can be obtained by combining a KATO -like result with a theorem of BENNETT [1] on SCHUR multipliers.     

Sobolev Inequalities in Manifolds with Nonnegative Curvature

We prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael-Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic volume ratio of the ambient manifold. © 2022 Wiley Periodicals LLC.     

Sharp bounds for harmonic numbers

In the paper, we collect some inequalities and establish a sharp double inequality for bounding the n-th harmonic number.     

From best constants to critical functions

The study of sharp Sobolev inequalities starts with the notion of best constant and leads naturally to the question to know whether or not there exist extremal functions for these inequalities. We restrict ourselves in this paper to the -Sobolev inequality. Then, we extend the notion of best constant to that of critical function, and, with the help of this notion, we answer the question to know whether or not there exist extremal functions for the sharp -Sobolev inequality. Partial answers to the more general question to know whether or not an extremal function always comes with a critical function are also given. Received November 9, 1999; in final form February 21, 2000 / Published online March 12, 2001     

Harnack Inequalities and Applications for Ornstein–Uhlenbeck Semigroups with Jump

The Harnack inequality established in Röckner and Wang (J Funct Anal 203:237–261, 2003) for generalized Mehler semigroup is improved and generalized. As applications, the log-Harnack inequality, the strong Feller property, the hyper-bounded property, and some heat kernel inequalities are presented for a class of O-U type semigroups with jump. These inequalities and semigroup properties are indeed equivalent, and thus sharp, for the Gaussian case. As an application of the log-Harnack inequality, the HWI inequality is established for the Gaussian case. Perturbations with linear growth are also investigated.     

Adams inequalities on measure spaces

In 1988 Adams obtained sharp Moser–Trudinger inequalities on bounded domains of Rn. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams' results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser–Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser–Trudinger trace inequalities, sharp Adams/Moser–Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.     

A sharp analog of Young’s inequality on S<Superscript>N</Superscript> and related entropy inequalities

We prove a sharp analog of Young’s inequality on S^N, and deduce from it certain sharp entropy inequalities. The proof turns on constructing a nonlinear heat flow that drives trial functions to optimizers in a monotonic manner. This strategy also works for the generalization of Young’s inequality on R^N to more than three functions, and leads to significant new information about the optimizers and the constants.     

Weak type inequality for noncommutative differentially subordinated martingales

In the paper we focus on self-adjoint noncommutative martingales. We provide an extension of the notion of differential subordination, which is due to Burkholder in the commutative case. Then we show that there is a noncommutative analogue of the Burkholder method of proving martingale inequalities, which allows us to establish the weak type (1,1) inequality for differentially subordinated martingales. Moreover, a related sharp maximal weak type (1,1) inequality is proved. Research supported by MEN Grant 1 PO3A 012 29.