Norm inequalities related to the matrix geometric mean

Ando-Hiai and Golden-Thomspon inequalities

The original Ando-Hiai and Golden-Thompson inequalities present comparisons for the operator geometric mean $\sharp_v$ when $0\leq v\leq 1.$ Our main target in this article is to study these celebrated inequalities for means other than the geometric mean and for the geometric mean when $v\not\in [0,1].$

Unitarily invariant norm inequalities for some means

We introduce some symmetric homogeneous means, and then show unitarily invariant norm inequalities for them, applying the method established by Hiai and Kosaki. Our new inequalities give the tighter bounds of the logarithmic mean than the inequalities given by Hiai and Kosaki. Some properties and norm continuities in parameter for our means are also discussed.

Stability results for some geometric inequalities and their functional versions

The Blaschke Santaló inequality and the $L_p$ affine isoperimetric inequalities are major inequalities in convex geometry and they have a wide range of applications. Functional versions of the Blaschke Santaló inequality have been established over the years through many contributions. More recently and ongoing, such functional versions have been established for the $L_p$ affine isoperimetric inequalities as well. These functional versions involve notions from information theory, like entropy and divergence. We list stability versions for the geometric inequalities as well as for their function…

通过Moore-Penrose逆的Berezin数与Berezin范数不等式

Berezin number and Berezin norm inequalities via Moore-Penrose inverse

该研究利用Moore-Penrose逆推导再生核Hilbert空间上有界线性算子的Berezin数与Berezin范数不等式，改进并推广了已有结果。

Weighted Norm Inequalities for One-sided Oscillatory Integral Operators

The purpose of this paper is to establish the weighted norm inequalities of one-sided oscillatory integral operators by the aid of interpolation of operators with change of measures.

Inequalities and equalities on the joint and generalized spectral and essential spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators

We prove new inequalities and equalities for the generalized and the joint spectral radius (and their essential versions) of Hadamard (Schur) geometric means of bounded sets of positive kernel operators on Banach function spaces. In the case of nonnegative matrices that define operators on Banach sequences we obtain additional results. Our results extend results of several authors that appeared relatively recently.

Generalized Numerical Radius Inequalities for Schatten p-Norms

In this paper, we present various inequalities for generalized numerical radius of $2 \times 2$ block matrices for Schatten p-norm. Moreover, we give a refinement of the triangle inequality for the Schatten p-generalized numerical radius.

Weighted norm inequalities for fractional oscillatory integrals and applications

We set up some weighted norm inequalities for fractional oscillatory integral operators. As applications, the corresponding results for commutators formed by $BMO(\mathbb{R}^{n})$ functions and the operators are established.

Refined Heinz operator inequalities and norm inequalities

In this article we study the Heinz and Hermite-Hadamard inequalities. We derive the whole series of refinements of these inequalities involving unitarily invariant norms, which improve some recent results, known from the literature. We also prove that if $A , B, X\in M_n(\mathbb{C})$ such that $A$ and $B$ are positive definite and $f$ is an operator monotone function on $(0,\infty)$. Then \begin{equation*} |||f(A)X-Xf(B)|||\leq \max\{||f'(A)||, ||f'(B)||\} |||AX-XB|||. \end{equation*} Finally we obtain a series of refinements of the Heinz operator inequalities, which were proved by Kittaneh an…

Norm Inequalities for Inner Product Type Integral Transformers

In this paper, we give a detailed survey on norm inequalities for inner product type integral transformers. We first consider unitarily invariant norms and operator valued functions. We then give results on norm inequalities for inner product type integral transformers in terms of Landau inequality, Gr\" uss inequality. Lastly, we explore some of the applications in quantum theory.

Weighted norm inequalities for k-plane transforms

We obtain sharp inequalities for the k-plane transform, the "j-plane to k-plane" transform, and the corresponding dual transforms, acting on $L^p$ spaces with a radial power weight. The operator norms are explicitly evaluated. Some generalizations and open problems are discussed.

Weighted Gagliardo-Nirenberg Inequalities Involving BMO Norms and Measures

Global and local weighted Gagliardo-Nirenberg inequalities with doubling measures are established. These inequalities are key ingredients for the regularity theory and existence of strong solutions for strongly coupled parabolic and elliptic systems which are degenerate or singular because of the unboundedness of dependent and independent variables.

Norm related inequalities for fractional integrals

Fractional spline wavelet systems are considered in the work. Molecular structure of elements of such systems admits estimates connecting norms of fractional integrals' images and pre--images in Besov spaces.

New Reverse Inequalities for the Numerical Radius of Normal Operators in Hilbert Spaces

In this paper, more inequalities between the operator norm and its numerical radius, for the class of normal operators, are established. Some of the obtained results are based on recent reverse results for the Schwarz inequality in Hilbert spaces due to the author.

Interpolations, convexity and geometric inequalities

We survey some interplays between spectral estimates of Hörmander-type, degenerate Monge-Ampère equations and geometric inequalities related to log-concavity such as Brunn-Minkowski, Santaló or Busemann inequalities.

Weighted norm inequalities for commutators of Littlewood-Paley functions related to Schrödinger operators

Let $L=-Δ+V$ be a Schrödinger operator, where $Δ$ is the Laplacian operator on $\rz$, while the nonnegative potential $V$ belongs to certain reverse Hölder class. In this paper, we establish some weighted norm inequalities for commutators of Littlewood-Paley functions related to Schrödinger operators.

Norm inequalities related to Heinz and Heron operator means

In this article we present some new comparisons between the Heinz and Heron operator means, which improve some recent results known from the literature. We derive some refinements of these inequalities for unitarily invariant norms with the help of the contractive maps.

Weighted norm inequalities for pseudo-differential operators with smooth symbols and their commutators

We obtain weighted $L^p$ inequalities for pseudo-differential operators with smooth symbols and their commutators by using a class of new weight functions which include Muckenhoupt weight functions. Our results improve essentially some well-known results.

Inequalities involving Berezin norm and Berezin number

We obtain new inequalities involving Berezin norm and Berezin number of bounded linear operators defined on a reproducing kernel Hilbert space $\mathscr{H}.$ Among many inequalities obtained here, it is shown that if $A$ is a positive bounded linear operator on $\mathscr{H}$, then $\|A\|_{ber}=\textbf{ber}(A)$, where $\|A\|_{ber}$ and $\textbf{ber}(A)$ are the Berezin norm and Berezin number of $A$, respectively. In contrast to the numerical radius, this equality does not hold for selfadjoint operators, which highlights the necessity of studying Berezin number inequalities independently.

Weighted norm inequalities for Schrödinger type operators

Let $L=-Δ+V$ be a Schrödinger operator, where $Δ$ is the Laplacian operator on $\rz$, while nonnegative potential $V$ belongs to the reverse Hölder class. In this paper, we establish the weighted norm inequalities for some Schrödinger type operators, which include Riesz transforms and fractional integrals and their commutators. These results generalize substantially some well-known results.

p-赋范空间中的一些不等式与几何常数

Some inequalities and geometric constants in p-normed spaces

研究完全p-赋范空间中与等腰正交性相关的对称几何常数、Milman型模量及其不等式，并推广James常数与广义von Neumann-Jordan常数的关系。

算子谱几何平均与几何平均的比较

Operator Spectral Geometric Versus Geometric Mean

本文研究希尔伯特空间上正定算子的谱几何平均，推导其新不等式，给出谱几何平均与常规几何平均的显式比较，并将结果应用于Ando型及Ando-Hiai不等式，补充了算子几何平均的相关不等式理论。

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 invar…

Weighted norm inequalities for multilinear singular integral operators and applications

In this paper, weighted norm inequalities with $A_p$ weights are established for the multilinear singular integral operators whose kernels satisfy $L^{r'}$-Hörmander regularity condition. As applications, we recover a weighted estimate for the multilinear Fourier multiplier obtained by Fujita and Tomita, and obtain several new weighted estimates for the multilinear Fourier multiplier as well.

Numerical radius inequalities concerning with algebraic norms

We give an expression for a generalized numerical radius of Hilbert space operators and then apply it to obtain upper and lower bounds for the generalized numerical radius. We also establish some generalized numerical radius inequalities involving the product of two operators. Applications of our inequalities are also provided.

谱几何平均的若干不等式及其应用

Some inequalities for spectral geometric mean with applications

本文研究谱几何平均的不等式，推导了正可逆算子的Hölder型不等式、算子序关系及范数版本，证明矩阵情形的对数优超关系，得到量子Tsallis相对熵的序关系与新下界。

On a geometric inequality related to fractional integration

In this paper we consider a new kind of inequality related to fractional integration, motivated by Gressman's paper. Based on it we investigate its multilinear analogue inequalities. Combining with the Gressman's work on multilinear integral, we establish this new kind of geometric inequalities with bilinear form and multilinear form in more general settings. Moreover, in some cases we also find the best constants and optimisers for these geometric inequalities on Euclidean spaces with Lebesgue measure settings with $L^p$ bounds.

Hardy-Littlewood inequalities for norms of positive operators on sequence spaces

We consider estimates of Hardy and Littlewood for norms of operators on sequence spaces, and we apply a factorization result of Maurey to obtain improved estimates and simplified proofs for the special case of a positive operator.

A note one Sobolev type inequalities with p<1 and modified norm

A note that points out the possibility to have p<1 in Sobolev type of inequalities by a use of the momomial structure of polynomials or power series. The proof is simple: Triangle angle inequality p*>1, monomial estimate from p* to exponent p<1, then the inverse triangle inequality.

Weighted norm inequalities, spectral multipliers and Littlewood-Paley operators in the Schrödinger settings

In this paper, we establish a good-$\lz$ inequality with two parameters in the Schrödinger settings. As it's applications, we obtain weighted estimates for spectral multipliers and Littlewood-Paley operators and their commutators in the Schrödinger settings.

Jensen and Minkowski inequalities for operator means and anti-norms

Jensen inequalities for positive linear maps of Choi and Hansen-Pedersen type are established for a large class of operator/matrix means. These results are also extensions of the Minkowski determinantal inequality. To this end we develop the study of anti-norms, a notion parallel to the symmetric norms in matrix analysis, including functionals like Schatten q-norms for a parameter q<1 and the Minkowski functional. An interpolation theorem for the Schur multiplication is given in this setting. Two sections have been added to the previous version, devoted to means of severable variables and anti…

Weighted norm inequalities for Weyl multipliers and Fourier multipliers on the Heisenberg group

In this paper we prove weighted norm inequalities for Weyl multipliers satisfying Mauceri's condition. As applications of this we obtain some estimates for $L^p$ multipliers on the Heisenberg group and also show in the context of a theorem of Weis on operator valued Fourier multipliers that the R-boundedness of the derivative of the multiplier is not necessary for the boundedness of the multiplier transform.

Two-Weighted Norm Inequalities for the Double Hardy Transforms and Strong Fractional Maximal Functions in Variable Exponent Lebesgue Spaces

Two-weight norm estimates for the double Hardy transforms and strong fractional maximal functions are established in variable exponent Lebesgue spaces. Derived conditions are simultaneously necessary and sufficient in the case when the exponent of the right-hand side space is constant.

Weighted Norm Inequalities for the Opdam--Cherednik Transform

In this paper, we study several weighted norm inequalities for the Opdam--Cherednik transform. We establish different versions of the Heisenberg--Pauli--Weyl inequality for this transform. In particular, we give an extension of this inequality using weights with different exponents and present a variation of the inequality that incorporates $L^p$-norms for the Opdam--Cherednik transform. Also, we prove a version of the Hardy--Littlewood inequality for this transform. Finally, we give other variations of the Heisenberg--Pauli--Weyl inequality such as the Nash-type and Clarkson-type inequalities…

通过Moore-Penrose逆估计A-Berezin数和A-Berezin范数不等式

Estimation of $\mathrm{A}$-Berezin number and $\mathrm{A}$-Berezin norm inequalities via Moore-Penrose inverse

利用Moore-Penrose逆建立再生核希尔伯特空间上有界线性算子的A-Berezin数和范数不等式，并扩展到两个算子之和的情况，还讨论了块矩阵相关上界，完善并推广了先前结果。

Inequalities on the joint and generalized spectral and essential spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators

Let $Ψ$ and $Σ$ be bounded sets of positive kernel operators on a Banach function space $L$. We prove several refinements of the known inequalities $$ρ\left(Ψ^{\left( \frac{1}{2} \right)} \circ Σ^{\left( \frac{1}{2} \right)} \right) \le ρ(ΨΣ) ^{\frac{1}{2}} \;\; \mathrm{and}\;\; \hatρ \left(Ψ^{\left( \frac{1}{2} \right)} \circ Σ^{\left( \frac{1}{2} \right)} \right) \le \hatρ (ΨΣ) ^{\frac{1}{2}} $$ for the generalized spectral radius $ρ$ and the joint spectral radius $\hatρ$, where $Ψ^{\left( \frac{1}{2} \right)} \circ Σ^{\left( \frac{1}{2} \right)} $ denotes the Hadamard (Schur) geometric mean…

Subelliptic geometric Hardy type inequalities on half-spaces and convex domains

In this paper we present $L^2$ and $L^p$ versions of the geometric Hardy inequalities in half-spaces and convex domains on stratified (Lie) groups. As a consequence, we obtain the geometric uncertainty principles. We give examples of the obtained results for the Heisenberg and the Engel groups.

Sharp geometric inequalities for the general $p$-affine capacity

In this article, we propose the notion of the general $p$-affine capacity and prove some basic properties for the general $p$-affine capacity, such as affine invariance and monotonicity. The newly proposed general $p$-affine capacity is compared with several classical geometric quantities, e.g., the volume, the $p$-variational capacity and the $p$-integral affine surface area. Consequently, several sharp geometric inequalities for the general $p$-affine capacity are obtained. These inequalities extend and strengthen many well-known (affine) isoperimetric and (affine) isocapacitary inequalities…

Transport-majorization to analytic and geometric inequalities

We introduce a transport-majorization argument that establishes a majorization in the convex order between two densities, based on control of the gradient of a transportation map between them. As applications, we give elementary derivations of some delicate Fourier analytic inequalities, which in turn yield geometric "slicing-inequalities" in both continuous and discrete settings. As a further consequence of our investigation we prove that any strongly log-concave probability density majorizes the Gaussian density and thus the Gaussian density maximizes the Rényi and Tsallis entropies of all o…

Inequalities for the Norm and Numerical Radius of Composite Operators in Hilbert Spaces

Some new inequalities for the norm and the numerical radius of composite operators generated by a pair of operators are given.

Norm Inequalities in Operator Ideals

In this paper we introduce a new technique for proving norm inequalities in operator ideals with an 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 par…

Symmetric norms and reverse inequalities to Davis and Hansen-Pedersen characterizations of operator convexity

Some rearrangement inequalities for symmetric norms on matrices are given as well as related results for operator convex functions.

Geometric inequalities via a general comparison principle for interacting gases

The article builds on several recent advances in the Monge-Kantorovich theory of mass transport which have -- among other things -- led to new and quite natural proofs for a wide range of geometric inequalities such as the ones formulated by Brunn-Minkowski, Sobolev, Gagliardo-Nirenberg, Beckner, Gross, Talagrand, Otto-Villani and their extensions by many others. While this paper continues in this spirit, we however propose here a basic framework to which all of these inequalities belong, and a general unifying principle from which many of them follow. This basic inequality relates the relativ…

若干广义Hardy-Morrey空间中Calderón-Zygmund算子的范数不等式

Norm inequalities for Calderon-Zygmund operators in some generalized Hardy-Morrey spaces

该研究利用广义Hardy-Morrey空间的分子刻画，推导了Calderón-Zygmund算子及其交换子在这类空间中的范数不等式，为相关算子的有界性分析提供了依据。

On the constants in some inequalities for the Sobolev norms and pointwise product

We consider the Sobolev norms of the pointwise product of two functions, and estimate from above and below the constants appearing in two related inequalities.

On Symmetric Norm Inequalities And Hermitian Block-Matrices

The main purpose of this paper is to englobe some new and known types of Hermitian block-matrices $M=\begin{pmatrix} A \& X\\ {X^*} \& B\end{pmatrix}$ satisfying or not the inequality $\|M\|\le \|A+B\|$ for all symmetric norms

受限几何不等式的函数提升

Functional Liftings of Restricted Geometric Inequalities

本文提出广义上卷积方法，将受限几何不等式提升为函数不等式，并在 Gaussian、对数情形及幂零 Lie 群中给出应用。

Inequalities between $L^p$-norms for log-concave distributions

Log-concave distributions include some important distributions such as normal distribution, exponential distribution and so on. In this note, we show inequalities between two Lp-norms for log-concave distributions on the Euclidean space. These inequalities are the generalizations of the upper and lower bound of the differential entropy and are also interpreted as a kind of expansion of the inequality between two Lp-norms on the measurable set with finite measure.

定量形式的等周不等式与几何不等式：Stein方法的途径

Isoperimetric and geometric inequalities in quantitative form: Stein's method approach

该研究通过引入带斜边界条件的椭圆型PDE解决边界项处理难题，将Stein方法几何化，得到Brock-Weinstock不等式、Steklov特征值约束下等周不等式及加权与未加权周长组合的稳定性结果，均基于α-Zolotarev距离表述。