Maryam Khosravi, Alemeh Sheikhhosseini
Some improvements of Young inequality and its reverse for positive numbers with Kontrovich constant are given. Using these inequalities some operator versions and Hilbert-Schmidt norm versions for matrices are proved.
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We prove the classical Hausdorff-Young inequality for Orlicz spaces on compact homogeneous manifolds.
We give a new proof of the sharp form of Young's inequality for convolutions, first proved by Beckner [Be] and Brascamp-Lieb [BL]. The latter also proved a sharp reverse inequality in the case of exponents less than $1$. Our proof is simpler and gives Young's inequality and its converse altogether.
Jensen's operator inequality for convexifiable functions is obtained. This result contains classical Jensen's operator inequality as a particular case. As a consequence, a new refinement and a reverse of Young's inequality is given.
In this paper we obtain some new refinements and reverses of Young's operator inequality. Extensions for convex functions of operators are also provided.
In this note, some refinements of Young inequality and its reverse for positive numbers are proved and using these inequalities some operator versions and Hilbert-Schmidt norm versions for matrices of these inequalities are obtained.
In this paper, sharp results on operator Young's inequality are obtained. We first obtain sharp multiplicative refinements and reverses for the operator Young's inequality. Secondly, we give an additive result, which improves a well-known inequality due to Tominaga. We also provide some estimates for $A{{\sharp}_{v}}B-A{{\nabla }_{v}}B$ in which $v\notin \left[ 0,1 \right]$.
In this paper, we generalize Young's inequality for locally compact quantum groups and obtain some results for extremal pairs of Young's inequality and extremal functions of Hausdorff-Young inequality.
Two reverse inequalities for Young's inequality were shown by M. Tominaga, using Specht ratio. In this short paper, we show alternative reverse inequalities for Young's inequality without using Specht ratio.
In this paper, we give a new inequality for convex functions of real variables, and we apply this inequality to obtain considerable generalizations, refinements, and reverses of the Young and Heinz inequalities for positive scalars. Applications to unitarily invariant norm inequalities involving positive semidefinite matrices are also given.
Young's inequality for Banach function spaces and its application to the maximal regularity estimate
该研究给出了Young不等式成立的Banach函数空间的充要条件，并将其应用于相关Besov空间上热方程的极大正则性估计。
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.
We prove several results about the best constants in the Hausdorff-Young inequality for noncommutative groups. In particular, we establish a sharp local central version for compact Lie groups, and extend known results for the Heisenberg group. In addition, we prove a universal lower bound to the best constant for general Lie groups.
In this paper we extend classical Titchmarsh theorems on the Fourier transform of H$\ddot{\text{o}}$lder-Lipschitz functions to the setting of harmonic $NA$ groups, which relate smoothness properties of functions to the growth and integrability of their Fourier transform. We prove a Fourier multiplier theorem for $L^2$-H$\ddot{\text{o}}$lder-Lipschitz spaces on Harmonic $NA$ groups. We also derive conditions and a characterisation of Dini-Lipschitz classes on Harmonic $NA$ groups in terms of the behaviour of their Fourier transform. Then, we shift our attention to the spherical analysis on Har…
In this note, we establish a novel maximal inequality of the 2D Young integral $\int_a^b\int_c^d FdG$ in terms of the $(p,q)$-bivariation norms of the section functions $x\mapsto F(x,y)$ and $y\mapsto F(x,y)$ where $G:[a,b]\times [c,d]\rightarrow \mathbb{R}$ is a controlled path satisfying finite $(p,q)$-variation conditions. The proof is reminiscent from the Young's original ideas \cite{young1} in defining two-parameter integrals in terms of $(p,q)$-finite bivariations. Our result complements the standard maximal inequality established by Towghi \cite{towghi1} in terms of joint variations. We…
Noncommutative sharp Hausdorff-Young inequality
本文在量子欧几里得空间上证明了尖锐Hausdorff-Young不等式，并推导出Weyl变换和海森堡群的相应常数，方法基于算子演算和Gabor变换相关的新型算子值流，同时定义了非交换卷积并给出Young不等式的部分结果。
It has been shown in "On the Hausdorff-Young theorem for commutative hypergroups" by Sina Degenfeld-Schonburg, that one can extend the domain of Fourier transform of a commutative hypergroup $K$ to $L^p(K)$ for $1\leq p \leq 2$, and the Hausdorff-Young inequality holds true for these cases. In this article, we examine the structure of non-zero functions in $L^p(K)$ for which equality is attained in the Hausdorff-Young inequality, for $1<p<2$, and further provide a characterization for the basic uncertainty principle for commutative hypergroups with non-trivial centre.
We show that the sharp Young's inequality for convolutions first obtained by Bechner and Brascamp-Lieb can be derived from the monotone in time evolution of a Lyapunov functional of the convolution of two solutions to the heat equation, with different diffusion coefficients, first introduced by Bennett and Bez. Our proof is based on a suitable adaptation of an old idea of Stam and Blachman, used to obtain Shannon's entropy power inequality.
For a semi-finite von Neumann algebra $\mathcal A$, we study the case of equality in Young's inequality of s-numbers for a pair of $τ$-measurable operators $a,b$, and we prove that equality is only possible if $|a|^p=|b|^q$. We also extend the result to unbounded operators affiliated with $\mathcal A$, and relate this problem with other symmetric norm Young inequalities.
The main purpose of this paper is to study the validity of the Hausdorff-Young inequality for vector-valued functions defined on a non-commutative compact group. The natural context for this research is that of operator spaces. This leads us to formulate a whole new theory of Fourier type and cotype for the category of operator spaces. The present paper is the first step in this program, where the basic theory is presented, the main examples are analyzed and some questions are posed.
If $a,b$ are $n\times n$ matrices, Ando proved that Young's inequality is valid for their singular values: if $p>1$ and $1/p+1/q=1$, then $$ λ_k|ab^*|\le λ_k( \frac1p |a|^p+\frac 1q |b|^q ) \, \textit{ for all }k. $$ Later, this result was extended for the singular values of a pair of compact operators acting on a Hilbert space by Erlijman, Farenick and Zeng. In this paper we prove that if $a,b$ are compact operators, then equality holds in Young's inequality if and only if $|a|^p=|b|^q$, obtaining a complete characterization of such $a,b$ in relation to other (operator norm) Young inequalitie…
The nonlinear Hausdorff-Young inequality
该研究证明了常数1的离散非线性豪斯多夫-杨不等式，通过极限论证得到连续空间的对应不等式，解决了Muscalu-Tao-Thiele一致性问题，其证明利用隐藏的希尔伯特空间结构将非线性问题转化为经典插值问题。
Let $μ$ be the Haar measure of a unimodular locally compact group $G$ and $m (G)$ as the infimum of the volumes of all open subgroups of $G$. The main result of this paper is that \begin{align*} \int_{G}^{} f \circ \left( ϕ_1 * ϕ_2 \right) \left( g \right) dg \leq \int_{\mathbb{R}}^{} f \circ \left( ϕ_1^* * ϕ_2^* \right) \left( x \right) dx \end{align*} holds for any measurable functions $ϕ_1, ϕ_2 \colon G \to \mathbb{R}_{\geq 0}$ with $μ( \mathrm{supp} \; ϕ_1 ) + μ( \mathrm{supp} \; ϕ_2 ) \leq m(G)$ and any convex function $f \colon \mathbb{R}_{\geq 0} \to \mathbb{R}$ with $f(0) = 0$. Here $ϕ…
We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if $A, B\in {\mathfrak B}(\mathcal{H})$ are positive operators and $r\geq 0$, $A\nabla_{-r}B+2r(A\nabla B-A\sharp B)\leq A\sharp_{-r}B$ and prove that equality holds if and only if $A=B$. We also establish several reverse Young-type inequalities involving trace, determinant and singular values. In particular, we show that if $A, B$ are positive definite matrices and $r\geq 0$, then …
In this paper, we study refinements of some inequalities related to Young inequality for scalar and for operator. As our main results, we show refined Young inequalities for two positive operators. This results refine the ordering relations among the arithmetic mean, the geometric mean and the harmonic mean. Finally, we give supplements for refined Young inequalities for two positive real numbers. And then we also give operator inequalities based on the supplemental inequalities.
In this paper, we show refined Young inequalities for two positive operators. Our results refine the ordering relations among the arithmetic mean, the geometric mean and the harmonic mean for two positive operators. In addition, we give two different reverse inequalities for the refined Young inequality for two positive operators.
A considerable amount of literature in the theory of inequality is devoted to the study of Jensen's and Young's inequality. This article presents a number of new inequalities involving the log-convex functions and the geometrically convex functions. As their consequences, we derive the refinements for Young's inequality and Jensen's inequality. In addition, the operator Jensen's type inequality is also developed for conditioned two functions. Utilizing these new inequalities, we investigate the operator inequalities related to the relative operator entropy.
Multiversion of the Hausdorff--Young inequality
本文研究相关联合高斯向量上经Mehler变换或噪声算子作用后的多函数不等式，统一推广实复超压缩性，并得到Hausdorff–Young等不等式的多函数版本及含噪Brascamp–Lieb刻画。
In this paper we present several applications of Cartwright-Field's inequality. Among these we found Young's inequality, Bernoulli's inequality, the inequality between the weighted power means, Hölder's inequality and Cauchy's inequality. We give also two applications related to arithmetic functions and to operator inequalities.
A generalization of Young's inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified proof of recent entropy power inequalities of Barron and Madiman, as well as of a (conjectured) generalization of the Brunn-Minkowski inequality. It is shown that the generalized Brunn-Minkowski conjecture is true for convex sets; an application of this to the law of large numbers for random sets is described.
Inequality on the optimal constant of Young's convolution inequality for locally compact groups and their closed subgroups
该研究定义了局部紧群上Young卷积不等式的最优常数，证明其对闭子群的不等式关系，并推导出一类连通李群的最优常数上界。
We give some new refinements of Heinz inequality and an improvement of the reverse Young's inequality for scalars and we use them to establish new inequalities for operators and the Hilbert-Schmidt norm of matrices. We give a uniformly and abbreviated form of the inequalities presented by Kittaneh and Mansarah, and the inequalities presented by Kai and we obtain some of their operator and matrix versions.
On some Fréchet spaces associated to the functions satisfying Mulholland inequality
本文研究了满足Mulholland条件的Young函数，并构建了关联到Banach空间的F-空间，用于插值空间的构造，同时改进了Hahn-Banach扩展性质。
Paley inequality for the Weyl transform and its applications
本文围绕Weyl变换建立多种Paley不等式，并应用于Weyl乘子的$L^p$-$L^q$有界性与Hörmander乘子定理，同时证明Hardy-Littlewood不等式并研究其向量值版本及相关关系。
In this paper, we give some refinements for the second inequality in $\frac{1}{2}\|A\| \leq w(A) \leq \|A\|$, where $A\in B(H)$. In particular, if $A$ is hyponormal by refining the Young inequality with the Kantorovich constant $K(\cdot, \cdot)$, we show that $w(A)\leq \dfrac{1}{\displaystyle {2\inf_{\| x \|=1}}ζ(x)}\| |A|+|A^{*}|\|\leq \dfrac{1}{2}\| |A|+|A^*|\|$, where $ζ(x)=K(\frac{\langle |A|x,x \rangle}{\langle |A^{*}|x,x \rangle},2)^{r},~~~r=\min\{λ,1-λ\}$ and $0\leq λ\leq 1$ . We also give a reverse for the classical numerical radius power inequality $w(A^{n})\leq w^{n}(A)$ for any oper…
In this paper, we give some reverse-types of Ando's and Hölder-McCarthy's inequalities for positive linear maps, and positive invertible operators. For our purpose, we use a recently improved Young inequality and its reverse.
We obtain several versions of the Hausdorff-Young and Hardy-Littlewood inequalities for the $(k,a)$-generalized Fourier transform recently investigated at length by Ben Saï d, Kobayashi, and Ø rsted. We also obtain a number of weighted inequalities - in particular Pitt's inequality - that have application to uncertainty principles. Specifically we obtain several analogs of the Heisenberg-Pauli-Weyl principle for $L^p$-functions, local Cowling-Price-type inequalities, Donoho-Stark-type inequalities and qualitative extensions. We finally use the Hausdorff-Young inequality as a means to obtain en…
This paper is twofold. In the first part, we present a refinement of the Rényi Entropy Power Inequality (EPI) recently obtained in \cite{BM16}. The proof largely follows the approach in \cite{DCT91} of employing Young's convolution inequalities with sharp constants. In the second part, we study the reversibility of the Rényi EPI, and confirm a conjecture in \cite{BNT15, MMX16} in two cases. Connections with various $p$-th mean bodies in convex geometry are also explored.
New Improvements to Heron and Heinz Inequality Using Matrix Techniques
本文研究矩阵均值插值与比较，将参数$\vartheta$扩展至$\mathbb{R}^+$，引入结合Kantorovich常数的Heinz不等式新标量变体并强化算子版本，提出专为半正定矩阵迹、行列式和范数设计的改进型Young不等式。
A new proof of Lee's conjecture on the Frobenius norm via the matrix Cauchy-Schwarz inequality
本文仅依靠矩阵 Cauchy-Schwarz 不等式，为 Eun-Young Lee 关于复矩阵 Frobenius 范数的不等式猜想提供了一种新证明。
The goal of this note is to show that some convolution type inequalities from Harmonic Analysis and Information Theory, such as Young's convolution inequality (with sharp constant), Nelson's hypercontractivity of the Hermite semi-group or Shannon's inequality, can be reduced to a simple geometric study of frames of $\R^2$. We shall derive directly entropic inequalities, which were recently proved to be dual to the Brascamp-Lieb convolution type inequalities.
Classical inequalities for all Fourier matrix coefficients of $\mathrm{SL}(2,\mathbb{R})$ and their applications
本文为 $\mathrm{SL}(2,\mathbb{R})$ 上的 $(l, n)$ 型函数建立了三个基础 Fourier 不等式，据此证明了 Fourier 乘子的 $L^p$-$L^q$ 有界性，并将其应用于热核估计与非线性方程的全局适定性研究。
Fourier theoretic inequalities for inclusion of simple C*-algebras
本文研究了简单C*-代数包含关系中的非交换傅里叶理论不等式，证明了Hirschman-Beckner和Donoho-Stark不确定性原理，并推广了Jiang等人之前的结果。
Multiple positive solutions for a double phase system with singular nonlinearity
本文研究含奇异和混合非线性项的双相系统，通过构造凹函数和利用Nehari流形，证明了具有非负能量的多个正解的存在性，并发现了鞍点解。
The Pólya-Szegő inequality for smoothing rearrangements
本文证明适用于所有光滑化重排的一般 Pólya-Szegő 不等式，涵盖多类函数并建立各向异性版本，将既有基本形式统一于同一框架。
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