Symmetrization and sharp Sobolev inequalities in metric spaces
Jan Kalis, Mario Milman
本文给出了度量空间上Sobolev空间的一类尖锐Sobolev不等式.特别地，我们得到了Hörmander向量场的新的Sobolev嵌入和Faber-Krahn估计.
We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for Hörmander vector fields.
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Pointwise Symmetrization Inequalities for Sobolev functions and applications
我们发展了一种获得新的对称化不等式的方法，为研究Sobolev不等式、浓度不等式和椭圆型方程解的可积性提供了一个统一的框架
Isoperimetry and symmetrization for logarithmic Sobolev inequalities
利用等周和对称化方法，我们提供了一个统一的框架来研究经典的和对数的Sobolev不等式。特别地，我们得到了新的高斯对称化不等式，并将它们与对数Sobolev不等式联系起来。我们的方法非常通用，可以很容易地适应更一般的情况。
Self Improving Sobolev-Poincare Inequalities, Truncation and Symmetrization
在前一篇文章中，我们发展了一种新的方法来获得$W_{0}^{1,1}(Ω)$中函数的Sobolev型对称化不等式。在本文中，我们将我们的方法推广到Sobolev函数，不消失的边界。
On the gradient of Schwarz symmetrization of functions in Sobolev spaces
设S是在定义域边界不为零的函数的Sobolev或Orlicz-索伯列夫空间.给出了S中非负函数的球面重排（Schwartz对称化）仍属于S的充分条件.这些结果是通过相关的等周不等式得到的，并在一定程度上推广了一个著名的Polya-Szego定理.我们还证明了任意函数在S中的重排是局部的。
Sobolev Inequalities: Symmetrization and Self Improvement via Truncation
给出了Sobolev型对称化不等式的一种新的证明方法。我们的方法导致新的不等式和相当大的简化理论的索伯列夫空间的嵌入重排不变空间的基础上。
Addendum to Isoperimetry and Symmetrization for Logarithmic Sobolev inequalities
我们给出了完整的细节上的Polya-Szego原则，在我们的论文中的注释1中提到的替代配方“Isoperimetry and Symmetriization for Logarithmic Sobolev inequalities”。我们还提供了一个替代的证明，在同一文件中的结果。
Isoperimetry and Symmetrization for Sobolev spaces on metric spaces
利用等周方法得到了新的对称化不等式，为研究度量空间中的Sobolev不等式提供了一个统一的框架。应用包括浓度不等式，以及度量版本的Pó% lya-Szegö和Faber-Krahn原则。
Fractional Sobolev Inequalities: Symmetrization, Isoperimetry and Interpolation
利用Peetre $K-$泛函和等周轮廓得到了度量空间中新的振动不等式.所提供的应用包括分数Sobolev不等式和Morrey-Sobolev嵌入定理在不同情况下的详细研究。特别是，我们包括高斯措施以及高斯和指数之间的概率措施的详细研究。我们展示了一种反向Polya-Szego原理，它允许我们使用对称化不等式从有界性获得连续性作为自我改进。我们的方法也允许精确估计广义Sobolev和Besov空间的增长包络的度量空间。我们还考虑嵌入到$BMO$和他们的连接到Sobolev嵌入。
We obtain symmetrization inequalities on probability metric spaces with convex isoperimetric profile which incorporate in their formulation the isoperimetric estimator and that can be applied to provide a unified treatment of sharp Sobolev-Poincaré and Nash inequalities.
We obtain symmetrization inequalities in the context of Fractional Hajlasz-Sobolev spaces in the setting of rearrangement invariant spaces and prove that for a large class of measures our symmetrization inequalities are equivalent to the lower bound of the measure.
In this paper, we develop a theory of symmetrization on the one dimensional integer lattice. More precisely, we associate a radially decreasing function $u^*$ with a function $u$ defined on the integers and prove the corresponding Polya-Szegö inequality. Along the way we also prove the weighted Polya-Szegö inequality for the decreasing rearrangement on the half-line, i.e., non-negative integers. As a consequence, we prove the discrete weighted Hardy's inequality with the weight $n^α$ for $1 < α\leq 2$.
Anisotropic symmetrization, convex bodies, and isoperimetric inequalities
本文利用各向异性等周不等式、Sobolev 函数精细性质及 Brunn-Minkowski 理论，给出各向异性 Pólya-Szegö 型不等式无需逼近论证的新证明，并实现对极值函数的刻画。
Characterizations of Sobolev inequalities on metric spaces
在非常一般的度量测度空间中给出了Sobolev不等式的等容性刻画.
It is known that classical Hardy and Sobolev inequalities hold when the exponent $p$ and the dimension $N$ satisfy $p < N < \infty$. In this note, we consider two limits of Hardy and Sobolev inequalities as $p \nearrow N$ and $N \nearrow \infty$ in some sense.
The purpose of this text is twofold. We present a review of the existing stability results for Sobolev, Hardy-Littlewood-Sobolev (HLS) and related inequalities. We also contribute to the topic with some observations on constructive stability estimates for (HLS).
On a class of weighted anisotropic Sobolev inequalities
本文受Caffarelli和Cordoba在相变分析中的工作的启发，证明了新的加权各向异性Sobolev型不等式，即不同导数具有不同权函数的Sobolev型不等式.我们正在处理的不等式，也密切相关的加权Sobolev不等式Grushin型运营商，权重不一定是Muckenhoupt。例如，我们认为这里索伯列夫不等式有限圆柱，权重是不同的权力的距离函数从顶部和底部的圆柱。我们还证明了类似的不等式在更一般的情况下，其中的重量是距离函数从一个较高的余维部分的边界。
Sobolev inequalities for nonlinear Dirichlet forms
本文研究了非线性Dirichlet形式中Sobolev不等式与等容量不等式之间的等价关系。
A theory of Sobolev inequalities in arbitrary open sets of Euclidean space is established. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities involve constants independent of the geometry of the domain, and exhibit the same critical exponents as in the classical inequalities on regular domains. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set.
Stability results for Sobolev, logarithmic Sobolev, and related inequalities
本文综述Sobolev型泛函不等式显式稳定性估计这一30年未决问题，介绍新方法并给出获得显式估计的若干途径。
We prove logarithmic Sobolev inequalities for semi-direct product operators (see definition in Section 1). We apply our main results to examples of operators and provide some applications to ultracontractive bounds of semigroups. Hardy's inequalities are proved and use to study some operators of Grushin's type.
On the best constant of Hardy-Sobolev Inequalities
我们得到了涉及到原点距离的Hardy-Sobolev不等式的一个锐常数。这个不等式等价于一个极限Caffarelli-Kohn-Nirenberg不等式。在三维空间中，在某些情况下，锐常数与最佳索伯列夫常数一致。
We find a new sharp trace Gagliardo-Nirenberg-Sobolev inequality on convex cones, aswell as a weighted sharp trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell-Brascamp-Lieb inequality, coming from the Brunn-Minkowski theory.
In this note, we establish a $L^p-$version of the Poincaré--Sobolev inequalities in the hyperbolic spaces $\mathbb H^n$. The interest of this result is that it relates both the Poincaré (or Hardy) inequality and the Sobolev inequality with the sharp constant in $\mathbb H^n$. Our approach is based on the comparison of the $L^p-$norm of gradient of the symmetric decreasing rearrangement of a function in both the hyperbolic space and the Euclidean space, and the sharp Sobolev inequalities in Euclidean spaces. This approach also gives the proof of the Poincaré--Gagliardo--Nirenberg and Poincaré--…
Sharp Hardy-Littlewood-Sobolev Inequalities on Octonionic Heisenberg Group
本文是继我们的工作[CLZ 13]之后的第二篇，研究Heisenberg型群上的Hardy-Littlewood-Sobolev不等式.第一个重要的突破是弗兰克和利布在[FL 12]中取得的。本文对八元数海森堡群也得到了类似的结果。
Frank and Lieb gave a new, rearrangement-free, proof of the sharp Hardy-Littlewood-Sobolev inequalities by exploiting their conformal covariance. Using this they gave new proofs of sharp Sobolev inequalities for the embeddings $W^{k,2}(\mathbb{R}^n)\hookrightarrow L^{\frac{2n}{n-2k}}(\mathbb{R}^n)$. We show that their argument gives a direct proof of the latter inequalities without passing through Hardy-Littlewood-Sobolev inequalities, and, moreover, a new proof of a sharp fully nonlinear Sobolev inequality involving the $σ_2$-curvature. Our argument relies on nice commutator identities deduce…
We derive the sharp Moser-Trudinger-Onofri inequalities on the standard $n$-sphere and CR $(2n+1)$- sphere as the limit of the sharp fractional Sobolev inequalities for all $n\ge 1$. On the $2$-sphere and $4$-sphere, this was established recently by S.-Y. Chang and F. Wang. Our proof uses an alternative and elementary argument.
By using quasi-Banach techniques as key ingredient we prove Poincaré- and Sobolev- type inequalities for $m$-subharmonic functions with finite $(p,m)$-energy. A consequence of the Sobolev type inequality is a partial confirmation of Błocki's integrability conjecture for $m$-subharmonic functions.
Sharp Hardy inequalities for Sobolev-Bregman forms
本文研究了Sobolev-Bregman形式中半空间和凸域的精确分数Hardy不等式，扩展了Bogdan和Dyda以及Loss和Sloane的结果。
We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic …
We generalize in this article the classical Sobolev's and Sobolev's trace inequalities on the Grand Lebesgue Spaces under monomial weight instead the classical Lebesgue or grand Lebesgue Spaces. We will distinguish the classical Sobolev's inequality and the so-called trace Sobolev's inequality.
Sobolev inequalities, rearrangements, isoperimetry and interpolation spaces
利用重排不等式刻画了度量空间中的Poincaré不等式
We define Euler-Hilbert-Sobolev spaces and obtain embedding results on homogeneous groups using Euler operators, which are homogeneous differential operators of order zero. Sharp remainder terms of $L^{p}$ and weighted Sobolev type and Sobolev-Rellich inequalities on homogeneous groups are given. Most inequalities are obtained with best constants. As consequences, we obtain analogues of the generalised classical Sobolev type and Sobolev-Rellich inequalities. We also discuss applications of logarithmic Hardy inequalities to Sobolev-Lorentz-Zygmund spaces. The obtained results are new already in…
Quantitative stability of Sobolev inequalities on compact Riemannian manifolds
本文研究紧致黎曼流形上Sobolev不等式的定量稳定性，证明近临界函数在$W^{1,2}$意义下接近极值函数集合，并涵盖次临界情形及退化现象。
We show that the first order Sobolev spaces on cuspidal symmetric domains can be characterized via pointwise inequalities. In particular, they coincide with the Hajlasz-Sobolev spaces.
We use the characterization of weak type inequalities via Garsia-Rodemich conditions to show self improving properties of Poincaré-Sobolev inequalities in a very general context.
Necessary Conditions for Fractional Hardy-Sobolev's Inequalities
本文在多维情形下得到了所谓分数阶Hardy-Sobolev不等式成立的若干必要条件。我们也给出了一些例子来说明这些不等式的尖锐性。
On capacitary strong type inequalities for Orlicz-Sobolev functions
证明了Orlicz-索伯列夫空间中函数的容性强型不等式.作为一个应用，我们考虑电容平均值和它们的极限。
We prove a trace Hardy type inequality with the best constant on the polyhedral convex cones which generalizes recent results of Alvino et al. and of Tzirakis on the upper half space. We also prove some trace Hardy-Sobolev-Maz'ya type inequalities which generalize the recent results of Filippas et al.. In applications, we derive some Hardy type inequalities and Hardy-Sobolev-Maz'ya type inequalities for fractional Laplacian. Finally, we prove the logarithmic Sobolev trace inequalities and logarithmic Hardy trace inequalities on the upper half spaces. The best constants in these inequalities ar…
Towards a unified theory of Sobolev inequalities
我们讨论了我们的工作点态不等式的梯度连接的等周轮廓与一个给定的几何。我们展示了如何可以用来统一某些方面的索伯列夫不等式理论。特别是，我们讨论了我们最近的论文分数阶不等式，Coulhon型不等式，转移和无量纲不等式和我们即将开展的工作，尖锐的高阶Sobolev不等式，可以通过迭代得到。
Generalized affine Hardy-Littlewood-Sobolev inequalities
该研究建立了涉及两个不同函数、强于经典形式的仿射Hardy-Littlewood-Sobolev不等式，并证明了对数凹函数下该新不等式的反向不等式。
Complex and Quaternionic Analogues of Busemann's Random Simplex and Intersection Inequalities
本文利用对称化下的单调性，将 Busemann 的随机单纯形与相交不等式推广至复和四元数向量空间，并证明标准 Steiner 对称化不具备该单调性。
We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn--Sobolev and Korn--Hardy inequalities on stratified homogeneous groups.
We derive some anisotropic Sobolev inequalities in $\mathbb{R}^{n}$ with a monomial weight in the general setting of rearrangement invariant spaces. Our starting point is to obtain an integral oscillation inequality in multiplicative form.
We establish point wise inequalities for Sobolev functions on a wider class of outward cuspidal domains. It is a generalization of an earlier result by the author and his collaborators
We present a class of modified logarithmic Sobolev inequality, interpolating between Poincaré and logarithmic Sobolev inequalities, suitable for measures of the type $\exp(-|x|^\al)$ or more complex $\exp(-|x|^\al\log^β(2+|x|))$ ($\al\in]1,2[$ and $\be\in\dR$) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincaré inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities.
We establish sharp affine weighted $L^p$ Sobolev type inequalities by using the $L_p$ Busemann-Petty centroid inequality proved by Lutwak, Yang and Zhang. Our approach consists in combining in a convenient way the latter one with a suitable family of sharp weighted $L^p$ Sobolev type inequalities obtained by Nguyen and allows to characterize all extremizers in some cases. The new inequalities don't rely on any euclidean geometric structure.
The study of Sobolev inequalities can be divided in two cases: p = 1 and 1 < p < +$\infty$. In the case p = 1 we study here a relaxed version of refined Sobolev inequalities. When p > 1, using as base space classical Lorentz spaces associated to a weight from the Arino-Muckenhoupt class Bp, we will study Gagliardo-Nirenberg inequalities. As a by-product we will also consider Morrey-Sobolev inequalities. These arguments can be generalized to many different frameworks, in particular the proofs are given in the setting of stratified Lie groups.
A family of sharp $L^p$ Sobolev inequalities is established by averaging the length of $i$-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical $L^p$ Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them -- the affine $L^p$ Sobolev inequality of Lutwak, Yang, and Zhang. When $p = 1$, the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extrem…
In this short paper, we establish a range of Caffarelli-Kohn-Nirenberg and weighted $L^{p}$-Sobolev type inequalities on stratified Lie groups. All inequalities are obtained with sharp constants. Moreover, the equivalence of the Sobolev type inequality and Hardy inequality is shown in the $L^{2}$-case.
Morrey--Sobolev inequalities with power weights on the half-space
该研究建立了n维半空间上带边界距离幂权的Sobolev空间及n维双曲空间上Sobolev空间的Morrey--Sobolev不等式，且所有估计在相差乘法常数意义下最优。
In this paper, we study the sharp Poincaré inequality and the Sobolev inequalities in the higher order Lorentz--Sobolev spaces in the hyperbolic spaces. These results generalize the ones obtained in \cite{Nguyen2020a} to the higher order derivatives and seem to be new in the context of the Lorentz--Sobolev spaces defined in the hyperbolic spaces.
New weighted Riesz-type pointwise inequalities and applications to generalized Sobolev estimates
本文提出新的加权里斯型点wise不等式，并将其应用于广义索伯列夫估计中。
In this paper we focus our attention on an embedding result for a weighted Sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function $F$. Starting from this type of inequalities we prove some refined Hardy-type inequalities.
Logarithmic Sobolev Trace Inequalities
证明了高斯空间中的一个对数Sobolev迹不等式，并研究了充分正则域上加权索伯列夫空间W^{1，p}（Ω，γ）中的迹算子.我们展示的例子，以显示结果的清晰度。PDE的应用也被考虑。
In this paper we show different inequalities for fractional order differential operators. In particular, the Sobolev, Hardy, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg type inequalities for the Caputo, Riemann-Liouville and Hadamard derivatives are obtained. In addition, we show some applications of these inequalities.
Hardy-Sobolev-Rellich, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups
该研究在一般连通李群的次椭圆微分算子框架下，建立了多类几何不等式及其临界形式，覆盖不同拓扑与模性情形，并推导了对应的不确定型原理。
In this paper we describe the Euler semigroup $\{e^{-t\mathbb{E}^{*}\mathbb{E}}\}_{t>0}$ on homogeneous Lie groups, which allows us to obtain various types of the Hardy-Sobolev and Gagliardo-Nirenberg type inequalities for the Euler operator $\mathbb{E}$. Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and $|\cdot|$-radial weighted Hardy-Sobolev type inequality are established.
Exact constant in Sobolev's and Sobolev's trace inequalities for Grand Lebesgue Spaces
本文将经典的Sobolev和Sobolev迹不等式推广到大Lebesgue空间而不是经典的Lebesgue空间上。我们将区分经典的Sobolev不等式和所谓的迹Sobolev不等式。为了简单起见，我们只考虑整个空间的情况。
In this short article we show a particular version of the Hedberg inequality which can be used to derive, in a very simple manner, functional inequalities involving Sobolev and Besov spaces in the general setting of Lebesgue spaces of variable exponents and in the framework of Orlicz spaces.
Integral isoperimetric transference and dimensionless Sobolev inequalities
引入高斯积分等周迁移的概念，并证明了如何应用它得到一类新的常数与维数无关的尖锐Sobolev-Poincaré不等式.在单位$n-$维立方体上的$L^{q}$空间的特殊情况下，我们的结果推广了最近在\cite{FKS}中利用外推得到的不等式.
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