A note on “Some inequalities in inner product spaces related to the generalized triangle inequality” by S.S. Dragomir et al.

In this note, we present an affirmative answer to a question presented in the paper “Some inequalities in inner product spaces related to the generalized triangle inequality” by S.S. Dragomir et al. [Appl. Math. Comput. 217 (18) (2011) 7462-7468].     

Morrey spaces and fractional integral operators

The present paper is devoted to the boundedness of fractional integral operators in Morrey spaces defined on quasimetric measure spaces. In particular, Sobolev, trace and weighted inequalities with power weights for potential operators are established. In the case when measure satisfies the doubling condition the derived conditions are simultaneously necessary and sufficient for appropriate inequalities.     

Characterizations of Sobolev inequalities on metric spaces

We discuss Maz'ya type isocapacitary characterizations of Sobolev inequalities on metric measure spaces.     

Reverses of the continuous triangle inequality for Bochner integral in complex Hilbert spaces

Some reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in complex Hilbert spaces are given. Applications for complex-valued functions are provided as well.     

On Pólya–Szegö reverse of Schwarz's inequality

We present some new results on the Cauchy–Schwarz inequality in inner product spaces, where four vectors are involved. This naturally extends Pólya–Szegö reverse of Schwarz's inequality onto complex inner product spaces. Applications to the famous Hadamard's inequality about determinants and the triangle inequality for norms are given.     

A bilinear inequality

We establish an inequality for symmetric bilinear forms involving both the norm and the inner product of vectors. We use the inequality to convert known inequalities in real Hilbert spaces such as classical Hilbert's inequality to sharper inequalities.     

Q-正则Loewner空间中的拟对称映射

本文在Q-正则Loewner空间中用环模不等式刻划了拟对称映射.另外,在 Q-维Ahlfors-David正则空间中建立了拟对称映射作用下的Grotzsch-Teichmuller型 模不等式,它是通过伸张系数的积分平均来表示.     

加权Orlicz空间上的Littlewood Paley算子

该文证明了一个与Littlewood Paley 算子有关的不等式，并由此导出Littlewood Paley 算子在加权Orlicz 空间和Morrey空间的有界特征。     

Two-Weighted Inequalities for Integral Operators in Lorentz Spaces Defined on Homogeneous Groups

The optimal sufficient conditions are found for weights, which guarantee the validity of two-weighted inequalities for singular integrals in the Lorentz spaces defined on homogeneous groups. In some particular case, the found conditions are necessary for the corresponding inequalities to be valid. Also, the necessary and sufficient conditions are found for pairs of weights, which provide the validity of two-weighted inequalities for the generalized Hardy operator in the Lorentz spaces defined on homogeneous groups.     

On Trudinger–Moser type inequalities involving Sobolev–Lorentz spaces

Generalizations of the Trudinger–Moser inequality to Sobolev–Lorentz spaces with weights are considered. The weights in these spaces allow for the addition of certain lower order terms in the exponential integral. We prove an explicit relation between the weights and the lower order terms; furthermore, we show that the resulting inequalities are sharp, and that there are related phenomena of concentration–compactness.      

自反Banach空间中广义混合双线性变分不等式解的存在性

介绍了自反B anach空间中一类广义混合双线性型变分不等式,利用极大极小不等式和辅助变分原理技术证明了这类混合双线性型变分不等式解的存在性与唯一性.     

On a reverse Heinz–Kato–Furuta inequality

In the set up of Minkowski spaces, the Schwarz inequality holds with the reverse inequality sign. As a consequence, the same occurs with the triangle inequality. In this note, extensions of this indefinite version of the Schwarz inequality are presented. Namely, a reverse Heinz–Kato–Furuta inequality valid for timelike vectors is included and related inequalities that also hold with the reverse sign are investigated.     

Herz–Morrey spaces of variable exponent,Riesz potential operator and duality

Our aim in this paper is to deal with the boundedness of the Hardy–Littlewood maximal operator on Herz–Morrey spaces and to establish Sobolev’s inequalities for Riesz potentials of functions in Herz–Morrey spaces. Further, we discuss the associate spaces among Herz–Morrey spaces.     

Real interpolation method,Lorentz spaces and refined Sobolev inequalities

In this article we give a straightforward proof of refined inequalities between Lorentz spaces and Besov spaces and we generalize previous results of H. Bahouri and A. Cohen [2]. Our approach is based in the characterization of Lorentz spaces as real interpolation spaces. We will also study the sharpness and optimality of these inequalities.     

Interpolation inequality of logarithmic type in abstract Besov spaces and an application to semilinear evolution equations

An abstract version of Besov spaces is introduced by using the resolvent of nonnegative operators. Interpolation inequalities with respect to abstract Besov spaces and generalized Lorentz spaces are obtained. These inequalities provide a generalization of Sobolev inequalities of logarithmic type. Uniqueness problems to abstract semilinear evolution equations are also discussed (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An elementary proof of the triangle inequality for the Wasserstein metric

We give an elementary proof for the triangle inequality of the -Wasserstein metric for probability measures on separable metric spaces. Unlike known approaches, our proof does not rely on the disintegration theorem in its full generality; therefore the additional assumption that the underlying space is Radon can be omitted. We also supply a proof, not depending on disintegration, that the Wasserstein metric is complete on Polish spaces.

     

Frechet空间的几个重要不等式

本文给出了Frechet空间中的几个重要不等式，它们是Hilbert空间中的著名极化恒等式在Frechet空间中的情形．推广了Banach空间的许多不等式，且在许多领域中有着各种各样的应用．利用这些不等式，可将许多结果从Banach空间推广到Frechet空间．     

Sharp generalized Trudinger inequalities via truncation

We prove that the generalized Trudinger inequalities into exponential and double exponential Orlicz spaces improve to inequalities on Orlicz-Lorentz spaces provided they are stable under truncation.     

Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications

In this paper, using the discrete Calderon reproducing formula on spaces of homogeneous type obtained by the author, we obtain the Plancherel-Pôlya type inequalities on spaces of homogeneous type. These inequalities give new characterizations of the Besov spaces and the Triebel-Lizorkin spaces on spaces of homogeneous type introduced earlier by the author and E. T. Sawyer and also allow us to generalize these spaces to the case where . Moreover, using these inequalities, we can easily show that the Littlewood-Paley -function and -function are equivalent on spaces of homogeneous type, which gives a new characterization of the Hardy spaces on spaces of homogeneous type introduced by Macias and Segovia.

     

一个三角形几何不等式的推广

首先从第3届国际数学奥林匹克IMO竞赛命题中一个三角形几何不等式出发,将问题推广到对更一般的三角几何不等式及多边形几何不等式的研究.然后利用凸函数的Jensen不等式,得到更一般的三角形几何不等式及圆外切多边形几何不等式,推广了原命题.