On several new Hilbert-type inequalities involving means operators

In this paper, we establish several new Hilbert-type inequalities with a homogeneous kernel, involving arithmetic, geometric, and harmonic mean operators in both integral and discrete case. Such inequalities are derived by virtue of some recent results regarding general Hilbert-type inequalities and some well-known classical inequalities. We also prove that the constant factors appearing in established inequalities are the best possible. As an application, we consider some particular settings and compare our results with previously known from the literature.     

Norm inequalities for the spectral spread of Hermitian operators

In this work, we introduce a new measure for the dispersion of the spectral scale of a Hermitian (self-adjoint) operator acting on a separable infinite-dimensional Hilbert space that we call spectral spread. Then, we obtain some submajorization inequalities involving the spectral spread of self-adjoint operators, that are related to Tao's inequalities for anti-diagonal blocks of positive operators, Kittaneh's commutator inequalities for positive operators and also related to the arithmetic–geometric mean inequality. In turn, these submajorization relations imply inequalities for unitarily invariant norms (in the compact case).     

Spectral radius inequalities for Hilbert space operators

We prove several spectral radius inequalities for sums, products, and commutators of Hilbert space operators. Pinching inequalities for the spectral radius are also obtained.

     

Refinements of the Heinz operator inequalities

In this article, we study the Heinz inequalities for two positive operators. We refine the ordering relations among the Heinz means with different parameters and obtain some improvements of the Heinz operator inequalities.     

Norm inequalities in operator ideals

In this paper we introduce a new technique for proving norm inequalities in operator ideals with a unitarily invariant norm. Among the well-known inequalities which can be proved with this technique are the Löwner-Heinz inequality, inequalities relating various operator means and the Corach-Porta-Recht inequality. We prove two general inequalities and from them we derive several inequalities by specialization, many of them new. We also show how some inequalities, known to be valid for matrices or bounded operators, can be extended with this technique to normed ideals in C^∗-algebras, in particular to the noncommutative Lp-spaces of a semi-finite von Neumann algebra.     

A transfer principle for inequalities in vector lattices

A transfer principle from inequalities with inner products to inequalities containing positive semidefinite symmetric bilinear operators with values in a vector lattices is proved. Some applications are also given.     

Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes

We prove weighted norm inequalities for pseudodifferential operators with amplitudes which are only measurable in the spatial variables. The result is sharp, even for smooth amplitudes. Nevertheless, in the case when the amplitude contains the oscillatory factor ξ?ei|ξ|^1−ρ, the result can be substantially improved. We extend the Lp-boundedness of pseudo-pseudodifferential operators to certain weights. End-point results are obtained when the amplitude is either smooth or satisfies a homogeneity condition in the frequency variable. Our weighted norm inequalities also yield the boundedness of commutators of these pseudodifferential operators with functions of bounded mean oscillation.     

Existence of anti-periodic solutions for hemivariational inequalities

J. Y. Park and T. G. Ha [Nonlinear Anal., 2008, 68: 747-767; 2009, 71: 3203-3217] investigated the existence of anti-periodic solutions for hemivariational inequalities with a pseudomonotone operator. In this note, we point out that the methods used there are not suitable for the proof of the existence of anti-periodic solutions for hemivariational inequalities and we shall give a straightforward approach to handle these problems. The main tools in our study are the maximal monotone property of the derivative operator with antiperiodic conditions and the surjectivity result for L-pseudomonotone operators.     

Jackson-type and Bernstein-type inequalities for multipliers on Herz-type Hardy spaces

We establish Jackson-type and Bernstein-type inequalities for multipliers on Herz-type Hardy spaces. These inequalities can be applied to some important operators in Fourier analysis, such as the Bochner-Riesz multiplier over the critical index, the generalized Bochner-Riesz mean and the generalized Able-Poisson operator. This work was supported by Key Academic Discipline of Zhejiang Province of China and National Natural Science Foundation of China (Grant Nos. 10571014, 10631080, 10671019)     

Normal operators and inequalities in norm ideals

In this work we characterize normal invertible operators via inequalities with unitarily invariant norm of elementary operators.     

Fractional Sobolev-type inequalities

Here we present univariate Sobolev-type fractional inequalities involving fractional derivatives of Canavati, Riemann–Liouville and Caputo types. The results are general L _p inequalities forward and converse on a closed interval. We give an application to a fractional ODE. We present also the mean Sobolev-type fractional inequalities.     

Hermite-Hadamard’s type inequalities for operator convex functions

Some Hermite-Hadamard’s type inequalities for operator convex functions of selfadjoint operators in Hilbert spaces are given. Applications for particular cases of interest are also provided.     

Non-compact generalized variational inequalities for quasi-monotone and hemi-continuous operators with applications

Some results are obtained for non-compact cases in topological vector spaces for the existence problem of solutions for some set-valued variational inequalities with quasi-monotone and lower hemi-continuous operators, and with quasi-semi-monotone and upper hemi-continuous operators. Some applications are given in non-reflexive Banach spaces for these existence problems of solutions and for perturbation problems for these set-valued variational inequalities with quasi-monotone and quasi-semi-monotone operators. This revised version was published online in June 2006 with corrections to the Cover Date.     

多线性Fourier乘子算子的加权估计——献给陆善镇教授75华诞

本文考虑多线性Fourier乘子算子在加权Lebesgue空间的乘积空间上的性质,利用多线性Fourier乘子算子的核估计以及多线性奇异积分算子的加权理论,建立多线性Fourier乘子算子的（关于多重Ap/r（R^mn）权函数以及关于一般权函数的）两个加权估计.     

Flows and functional inequalities for fractional operators

This paper collects results concerning global rates and large time asymptotics of a fractional fast diffusion equation on the Euclidean space, which is deeply related with a family of fractional Gagliardo–Nirenberg–Sobolev inequalities. Generically, self-similar solutions are not optimal for the Gagliardo–Nirenberg–Sobolev inequalities, in strong contrast with usual standard fast diffusion equations based on non-fractional operators. Various aspects of the stability of the self-similar solutions and of the entropy methods like carré du champ and Rényi entropy powers methods are investigated and raise a number of open problems.     

Variational inequalities for monotone operators in nonreflexive Banach spaces

The purpose of this paper is to study the existence problem of solutions and perturbation problem for some kind of variational inequalities with monotone operators in nonreflexive Banach spaces, and to obtain some results.     

Two-weight norm inequalities for commutators of potential type integral operators

We give a condition which is sufficient for the two-weight (p,q) inequalities for commutators of potential type integral operators.     

Refined convex Sobolev inequalities

This paper is devoted to refinements of convex Sobolev inequalities in the case of power law relative entropies: a nonlinear entropy-entropy production relation improves the known inequalities of this type. The corresponding generalized Poincaré-type inequalities with weights are derived. Optimal constants are compared to the usual Poincaré constant.     

Cea's error estimate for strongly monotone variational inequalities

Céa's approximation lemma is extended to variational inequalities which are defined by strongly monotone operators in closed convex subsets of linear normed spaces. This abstract error estimate is applied to the finite element discretization of a nonlinear elliptic two-sided obstacle problem providing an asymptotic error estimate for a smooth enough solution.