Michael G. Cowling, Alessio Martini, Detlef Müller, Javier Parcet
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.
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In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff--Young, Hardy--Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff--Young and Hardy--Littlewood inequalities and state that the Fourier transform is bounded from $L^p(\mathbb{R}^d,|\cdot|^{βp})$ into $L^q(\mathbb{R}^d,|\cdot|^{-γq})$ under certain condition on $p,q,β$ and $γ$. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and rela…
We prove the classical Hausdorff-Young inequality for Orlicz spaces on compact homogeneous manifolds.
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。
We give an "elementary" proof of an inequality due to Maz'ya. As a prerequisite we prove an approximation property for the Hausdorff measure. We also comment on the relations between Maz'ya's inequality, the isoperimetric inequality and the Sobolev inequality.
Analog version of Hausdorff--Young's theorem for quadratic Fourier transforms and boundedness of oscillatory integral operator
本文基于 Riesz--Thorin 插值定理证明了二次 Fourier 变换和线性正则变换的 Hausdorff--Young 型不等式，并研究了多项式相位振荡积分算子的有界性。
The paper studies Hausdorff-Young inequalities for certain group extensions, by use of Mackey's theory. We consider the case in which the dual action of the quotient group is free almost everywhere. This result applies to yield a Hausdorff-Young inequality for nonunimodular groups.
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.
The classical Hausdorff-Young inequalities for the Fourier transform acting between appropriate $L_p$ spaces are cornerstones of Fourier analysis. Here we extend it to weighted spaces of Besov or Sobolev type where the weight has the form $w(x)=(1+|x|^2)^{α/2}$. This note is not a paper or draft but a sketchy complement to some earlier results where we dealt with mapping properties of the Fourier transform.
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.
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。
We study Hausdorff-Young type inequalities for vector-valued Dirichlet series which allow to compare the norm of a Dirichlet series in the Hardy space $\mathcal{H}_{p} (X)$ with the $q$-norm of its coefficients. In order to obtain inequalities completely analogous to the scalar case, a Banach space must satisfy the restrictive notion of Fourier type/cotype. We show that variants of these inequalities hold for the much broader range of spaces enjoying type/cotype. We also consider Hausdorff-Young type inequalities for functions defined on the infinite torus $\mathbb{T}^{\infty}$ or the boolean …
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。
$L^p$-$L^q$ Fourier multipliers and Hausdorff-Young-Paley inequalities on Riemannian symmetric spaces of noncompact type
本文在任意秩非紧型黎曼对称空间上建立Hörmander型Lp-Lq Fourier乘子定理，推导谱乘子定理，证明Laplace-Beltrami算子函数的有界性，并给出Hausdorff-Young和Paley不等式的混合范数版本，结果适用于Damek-Ricci空间。
Reverse Hölder Class and Two-Parameter Good-$λ$ Inequality over Hausdorff Content
本文在豪斯多夫容度框架下建立双参数好λ不等式，确定反赫尔德类$\boldsymbol{\textit{RH}}_{q,\boldsymbol{\textit{δ}}}$并证明其自改进性质，得到加权Fefferman-Stein不等式等应用，拓展了好λ原理至非加性框架。
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.
This paper deals with the inequalities devoted to the comparison between the norm of a function on a compact hypergroup and the norm of its Fourier coefficients. We prove the classical Paley inequality in the setting of compact hypergroups which further gives the Hardy-Littlewood and Hausdorff-Young-Paley (Pitt) inequalities in the noncommutative context. We establish Hörmander's $L^p$-$L^q$ Fourier multiplier theorem on compact hypergroups for $1<p \leq 2 \leq q<\infty$ as an application of Hausdorff-Young-Paley inequality. We examine our results for the hypergroups constructed from the conju…
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 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…
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.
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.
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.
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.
Hausdorff content $A_p$ weights: weighted maximal inequalities and fundamental structural properties
本文研究了容积穆肯霍普特权重与Hardy-Littlewood极大算子的加权范数不等式，建立了新的权重类并验证了经典Muckenhoupt权重的若干性质。
In this paper we prove new inequalities describing the relationship between the "size" of a function on a compact homogeneous manifold and the "size" of its Fourier coefficients. These inequalities can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood on the circle. For the example case of the group SU(2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in $L^p$ on SU(2) in terms of its Fourier coefficients. We also establish Paley and Hausdorff-Young-Paley inequalities on gener…
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.
In this article, we show that in a $Q$-doubling space $(X,d,μ),$ $Q>1,$ that supports a $Q$-Poincaré inequality and satisfies a chain condition, sets of $Q$-capacity zero have generalized Hausdorff $h$-measure zero for $h(t)=\log^{1-Q-ε}(1/t).$
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 $v\notin \left[ 0,1 \right]$.
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.
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.
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。
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…
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…
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 Capacitary John-Nirenberg Inequality Revisited
本文研究平移不变 Hausdorff 容度的容量型 John-Nirenberg 不等式，揭示填充条件是弥补容量积分非线性的关键，并证明其等价于外容量与诱导 Hausdorff 容度等价，从而推广相关理论。
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.
In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, $p$-flow, for $1\leq p<\infty.$ Here we investigate the asymptotic behavior of the planar $p$-flow for $p=\infty$ in the class of smooth, origin-symmetric convex bodies. First, we prove that the $\infty$-flow evolves suitably normalized origin-symmetric solutions to the unit disk in the Hausdorff metric, modulo $SL(2).$ Second, using the $\infty$-flow and a Harnack estimate for this flow, we prove a stability version of the pl…
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。
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 …
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…
$L^{p}$-$L^{q}$ boundedness of Fourier multipliers on Fundamental domains of Lattices in $\mathbb{R}^d$
该研究在格的基本域上建立经典Hörmander条件下Fourier乘子的$L^{p}$-$L^{q}$有界性，并通过证明Hausdorff-Young、Paley及Hausdorff-Young-Paley不等式给出定量结果，进而推出Hardy-Littlewood不等式。
A note on Sobolev inequalities in the lower limit case
本文研究了欧几里得空间中光滑函数的Poincaré-Sobolev型不等式，探讨了梯度绝对值关于δ维Hausdorff内容的Choquet δ/n可积性，并推导了Sobolev空间中quasicontinuous函数的新Sobolev不等式。
We prove that if $X,Y$ are Banach spaces, $Ω$ a compact Hausdorff space and $U\hbox{\rm :} C(Ω,X)\to Y$ is a bounded linear operator, and if $U$ is a Dunford--Pettis operator the range of the representing measure $G(Σ) \subseteq DP(X,Y)$ is an uniformly Dunford--Pettis family of operators and $\|G\|$ is continuous at $\emptyset$. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space $C([0,1],X)$ with values in $c_{0}$ or $l_{p}$, ($1\leq p<\infty$) be Dunford--Pettis and/or compact operators, in which, Khinchin's inequalit…
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 study the cases of equality and prove a rigidity theorem concerning the 1-Bakry-Émery inequality. As an application, we prove the rigidity of the Gaussian isoperimetric inequality, the logarithmic Sobolev inequality and the Poincaré inequality in the setting of ${\rm RCD}(K, \infty)$ metric measure spaces. This unifies and extends to the non-smooth setting the results of Carlen-Kerce, Morgan, Bouyrie, Ohta-Takatsu, Cheng-Zhou. Examples of non-smooth spaces fitting our setting are measured-Gromov Hausdorff limits of Riemannian manifolds with uniform Ricci curvature lower bound, and Alexandro…
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。
Heat semigroups on quantum automorphism groups of finite dimensional C*-algebras
本文研究了有限维C*-代数量子自守群上的热半群，证明了其超收缩性、超收缩性和谱间隙不等式，并获得了Sobolev嵌入性质和Hausdorff-Young不等式的精确性。
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