Sobolev trace inequality on $W^{s,q}(\mathbb{R}^n)$

Beckner共形对数索伯列夫不等式的尖锐稳定性常数的达成

Attainment of the sharp stability constant of Beckner conformal log-Sobolev inequality

本文解决了对数索伯列夫不等式稳定性极小元存在性的未决问题，证明了n≥2时Sⁿ上Beckner共形对数索伯列夫不等式的尖锐L²稳定性常数可达成，且该常数小于2log2/n，还建立了分数阶索伯列夫不等式稳定性极小元序列的收敛性。

Remarks on a Sobolev-Hardy inequality

We compute the optimal constant for a generalized Hardy-Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a nonlinear elliptic equation arising in astrophysics.

Optimal decay of extremals for the fractional Sobolev inequality

We obtain the sharp asymptotic behavior at infinity of extremal functions for the fractional critical Sobolev embedding.

On the Dirichlet Problem Generated by the Maz'ya--Sobolev Inequality

We discuss the attainability of sharp constants for the Maz'ya--Sobolev inequalities in wedges, "perturbed" wedges and bounded domains.

On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality

We show existence of minimizers for the Hardy-Sobolev-Maz'ya inequality in $R^{m+n}\setminus\R^n$ for $m=1$ and $n>2$ or for $m>2$ and $n>0$.

The second best constant for the Hardy-Sobolev inequality on manifolds

We consider the second best constant in the Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested with the existence of extremal functions for this inequality. This problem was tackled by Djadli-Druet [5] for Sobolev inequalities. Here, we establish the corresponding result for the singular case. In addition, we perform a blow-up analysis of solutions Hardy-Sobolev equations of minimizing type. This yields informations on the value of the second best constant in the related Riemannian functional inequality.

Gradient stability for the Sobolev inequality: the case $p\geq 2$

We prove a strong form of the quantitative Sobolev inequality in $\mathbb{R}^n$ for $p\geq 2$, where the deficit of a function $u\in \dot W^{1,p} $ controls $\| \nabla u -\nabla v\|_{L^p}$ for an extremal function $v$ in the Sobolev inequality.

On Trace Theorems and Poincare inequality for one-dimensional Sobolev spaces

In these notes, we present versions of trace theorems for Sobolev spaces over an interval in the real line, and also a one-dimensional version of the well-known Poincare inequality.

The effect of curvature in fractional Hardy--Sobolev inequality with the Spectral Dirichlet Laplacian

We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.

The role of the mean curvature in a Hardy-Sobolev trace inequality

The Hardy-Sobolev trace inequality can be obtained via Harmonic extensions on the half-space of the Stein and Weiss weighted Hardy-Littlewood-Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy-Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.

Another approach to the equality case of the sharp logarithmic Sobolev inequality

In this note, we characterize the equality case of the sharp $L^2$-Euclidean logarithmic Sobolev inequality with monomial weights, exploiting the idea by Bobkov and Ledoux \cite{Bob}. Our approach is new even in the unweighted case. Also, we show that the same strategy yields the equality case of the sharp $L^p$-Euclidean logarithmic Sobolev inequality for $1 < p < \infty$ with arbitrary norm on ${\mathbb R}^n$, if the inequality is unweighted.

A note on the Moser-Trudinger inequality in Sobolev-Slobodeckij spaces in dimension one

We discuss some recent results by Parini and Ruf on a Moser-Trudinger type inequality in the setting of Sobolev-Slobodeckij spaces in dimension one. We push further their analysis considering the inequality on the whole $\mathbb{R}$ and we give an answer to one of their open questions.

Asymptotic behavior of extremal functions to an inequality involving Hardy potential and critical Sobolev exponent

In this paper, we study the asymptotic behavior of radial extremal functions to an inequality involving Hardy potential and critical Sobolev exponent. Based on the asymptotic behavior at the origin and the infinity, we shall deduce a strict inequality between two best constants. Finally, as an application of this strict inequality, we consider the existence of nontrivial solution of a quasilinear Brezis-Nirenberg type problem with Hardy potential and critical Sobolev exponent.

Complete classification and nondegeneracy of minimizers for the fractional Hardy-Sobolev inequality, and applications

We study linear and non-linear equations related to the fractional Hardy--Sobolev inequality. We prove nondegeneracy of ground state solutions to the basic equation and investigate existence and qualitative properties, including symmetry of solutions to some perturbed equations.

关于一个Sobolev型不等式及其极小化子

On a Sobolev-type inequality and its minimizers

本文建立了全空间上加权Sobolev空间的临界Sobolev型不等式，研究了相关变分问题极值函数的存在性，并以此证明了一类与多调和算子相关的临界半线性椭圆方程弱解的存在性。

非局部Gagliardo-Nirenberg-Sobolev型不等式

Nonlocal Gagliardo-Nirenberg-Sobolev type inequality

本文在非局部Sobolev空间中建立Gagliardo-Nirenberg-Sobolev型不等式，导出Sobolev嵌入及紧嵌入，并揭示其与Poincaré型不等式的互反关系，经典分数阶情形作为特例。

锥中 Sobolev 不等式的稳定性

Stability for the Sobolev inequality in cones

本文针对锥区域建立 Bianchi-Egnell 型定量 Sobolev 不等式，在极小元非退化条件下证明稳定性，并对经典气泡情形给出更精确结果，同时指出局部估计无法得到最优常数。

On the integral systems related to Hardy-Littlewood-Sobolev inequality

We prove all the maximizers of the sharp Hardy-Littlewood-Sobolev inequality are smooth. More generally, we show all the nonnegative critical functions are smooth, radial with respect to some points and strictly decreasing in the radial direction. In particular, this resolves all the cases left open by previous works of Chen, Li and Ou on the corresponding integral systems.

Sharp Stability of Log-Sobolev and Moser-Onofri inequalities on the Sphere

In this paper, we are concerned with the stability problem for endpoint conformally invariant cases of the Sobolev inequality on the sphere $\mathbb{S}^n$. Namely, we will establish the stability for Beckner's log-Sobolev inequality and Beckner's Moser-Onofri inequality on the sphere. We also prove that the sharp constant of global stability for the log-Sobolev inequality on the sphere $\mathbb{S}^n$ must be strictly smaller than the sharp constant of local stability for the same inequality. Furthermore, we also derive the non-existence of the global stability for Moser-Onofri inequality on th…

On a parabolic logarithmic Sobolev inequality

In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) $BMO$ norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) $BMO$ norm. More precisely we give an upper bound for the $L^{\infty}$ norm of a function in terms of its parabolic $BMO$ norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time …

A Hardy-Littlewood-Sobolev type inequality for variable exponents and applications to quasilinear Choquard equations involving variable exponent

In this work, we have proved a version of the Hardy-Littlewood-Sobolev inequality for variable exponents. After we use the variational method to establish the existence of solution for a class of Choquard equations involving the $p(x)$-Laplacian operator.

精确的毛细作用Sobolev不等式和Moser-Trudinger不等式在凸域外

Sharp capillary Sobolev inequality and Moser-Trudinger inequality outside convex domain

本文在凸域外建立了精确的毛细作用Sobolev不等式和Moser-Trudinger不等式，方法基于构造毛细作用Pálya-Szegö排列不等式，还考虑了毛细作用Talenti比较原理和Bossel-Daners不等式。

Higher order Sobolev trace inequalities on balls revisited

Inspired by a recent sharp Sobolev trace inequality of order four on the balls $\mathbb B^{n+1}$ found by Ache and Chang [AC15], we propose a slightly different approach to reprove Ache-Chang's trace inequality. To illustrate this approach, we reprove the classical Sobolev trace inequality of order two on $\mathbb B^{n+1}$ and provide sharp Sobolev trace inequalities of orders six and eight on $\mathbb B^{n+1}$. As the limiting case of the Sobolev trace inequality, a Lebedev-Milin type inequality of order up to eight is also considered.

Sharp Hardy-Littlewood-Sobolev inequality on the upper half space

There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $λ=n-α$ (that is for the case of $α>n$). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via…

凸集外的加权等周不等式与Sobolev不等式

The weighted isoperimetric inequality and Sobolev inequality outside convex sets

利用λ_w-ABP方法建立凸集外的加权毛细等周不等式，并基于此发展毛细Schwarz对称化技术，证明加权Pólya-Szegö原理和尖锐加权毛细Sobolev不等式，推广了半空间上的加权Sobolev不等式。

Reversed Hardy-Littlewood-Sobolev inequality on Heisenberg group $\mathbb{H}^n$ and CR sphere $\mathbb{S}^{2n+1}$

This paper is mainly devoted to the study of the reversed Hardy-Littlewood-Sobolev (HLS) inequality on Heisenberg group $\mathbb{H}^n$ and CR sphere $\mathbb{S}^{2n+1}$. First, we establish the roughly reversed HLS inequality and give a explicitly lower bound for the sharp constant. Then, the existence of the extremal functions with sharp constant is proved by subcritical approach and some compactness techniques. Our method is rearrangement free and can be applied to study the classical HLS inequality and other similar inequalities.

带权临界Sobolev不等式中涉及多项谐振算子的极小值与最佳常数

Minimizers and best constants for a weighted critical Sobolev inequality involving the polyharmonic operator

本文研究带权临界Sobolev不等式中多项谐振算子的最佳常数，通过建立正则性和分类结果来解决相关方程。

关于Hessian Hardy-Sobolev不等式及相关变分问题

On the Hessian Hardy-Sobolev Inequality and Related Variational Problems

本文通过Hessian泛函的下降梯度流证明了Hessian积分的Hardy-Sobolev不等式，将其应用于相关变分问题解的存在性与正则性研究，扩展了Hessian方程的变分理论。

Generalized logarithmic Hardy-Littlewood-Sobolev inequality

This paper is devoted to logarithmic Hardy-Littlewood-Sobolev inequalities in the two-dimensional Euclidean space, in presence of an external potential with logarithmic growth. The coupling with the potential introduces a new parameter, with two regimes. The attractive regime reflects the standard logarithmic Hardy-Littlewood-Sobolev inequality. The second regime corresponds to a reverse inequality, with the opposite sign in the convolution term, that allows us to bound the free energy of a drift-diffusion-Poisson system from below. Our method is based on an extension of an entropy method prop…

An Isoperimetric Problem With Density and the Hardy Sobolev Inequality in $\mathbb{R}^2$

We prove, using elementary methods of complex analysis, the following generalization of the isoperimetric inequality: if $p\in\re$, $Ω\subset\re^2$ then the inequality $$ \left(\frac{|Ω|}π\right)^{\frac{p+1}{2}}\leq\frac{1}{2π}\int_{\delomega}|x|^pdσ(x) $$ holds true under appropriate assumptions on $Ω$ and $p.$ This solves an open problem arising in the context of isoperimetric problems with density and poses some new ones (for instance generalizations to $\re^n$). We prove the equivalence with a Hardy-Sobolev inequality, giving the best constant, and generalize thereby the equivalence betwee…

An Extended Discrete Hardy-Littlewood-Sobolev Inequality

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case: μ=n. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: μ=n and p=q, by limiting the inequality on a finite domain. The best constant in the inequality and its corresponding solution, the optimizer, are studied. First, we obtain a sharp estimate for the best constant. Then for the optimizer, we prove the uniqueness and a symmetry property. This is achieved by proving that the corresponding Euler-Lagrange equation has a unique nontrivial nonnegative critic…

Adams-Moser-Trudinger inequality in the cartesian product of Sobolev spaces and its applications

The main aim of this article is to study non-singular version of Moser-Trudinger and Adams-Moser-Trudinger inequalities and the singular version of Moser-Trudinger equality in the Cartesian product of Sobolev spaces. As an application of these inequalities, we study a system of Kirchhoff equations with exponential non-linearity of Choquard type.

An isoperimetric inequality for extremal Sobolev functions

Let D be a bounded domain in n-dimensional Euclidean space, where n>2, and let 1<p< (2n)/(n-2). We prove a reverse-Holder inequality for functions realizing equality in the Sobolev inequality, which finds a lower bound for their (p-1)-norm in terms of their p-norm. This inequality is sharp, and it is an equality if and only if the domain is a round ball. Our result generalizes a theorem of Payne and Rayner and our proof relies on integral rearrangements and an analysis of the ODE corresponding to the radial case.

Caffarelli-Kohn-Nirenberg超球面上的锐利Sobolev不等式

A sharp Sobolev inequality on the Caffarelli-Kohn-Nirenberg hyperbolic space

本文研究了Caffarelli-Kohn-Nirenberg不等式在超球面上的锐利性，发现当有效维度$n$在[3,4)区间时，不等式可以得到改进。

Density estimates for a nonlocal variational model via the Sobolev inequality

We consider the minimizers of the energy $$ \|u\|_{H^s(Ω)}^2+\int_ΩW(u)\,dx,$$ with $s \in (0,1/2)$, where $\|u\|_{H^s(Ω)}$ denotes the total contribution from $Ω$ in the $H^s$ norm of $u$, and $W$ is a double-well potential. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer $u$ in a large ball $B_R$ occupy a volume comparable with the volume of $B_R$.

海森堡群上Sobolev不等式带维度依赖常数的渐近锐利稳定性

Asymptotically sharp stability of Sobolev inequalities on the Heisenberg group with dimension-dependent constants

本文利用CR Yamabe流克服CR背景下重排不等式缺失的困难，建立了海森堡群上Sobolev不等式及特殊共形指标HLS不等式带维度依赖常数的最优稳定性，并提出了可推广至分数阶情形的无重排方法。

Sharp Logarithmic Sobolev and related inequalities with monomial weights

We derive a sharp Logarithmic Sobolev inequality with monomial weights starting from a sharp Sobolev inequality with monomial weights. Several related inequalities such as Shannon type and Heisenberg's uncertain type are also derived. A characterization of the equality case for the Logarithmic Sobolev inequality is given when the exponents of the monomial weights are all zero or integers. Such a proof is new even in the unweighted case.

分数Sobolev对数不等式

Fractional Sobolev logarithmic inequalities

本文提出了一种分数Sobolev对数不等式，通过插值不等式和变分问题，扩展了经典Sobolev不等式到非局部框架，并提供了新的分析工具。

关于区域中分数阶Orlicz-Sobolev半范数的Pólya-Szegö不等式的一个注记

A note on Pólya-Szegö inequality for fractional Orlicz-Sobolev seminorm in domains

本文证明了对称径向递减重排会增大区域中的分数阶Orlicz-Sobolev半范数，从而将Li-Wang的Pólya-Szegö型结果推广到比幂次更一般的Orlicz增长情形。

带权重的分数Hardy-Sobolev和Hardy-Sobolev-Maz'ya不等式及其奇点在平坦子流形上的应用

Weighted fractional Hardy-Sobolev and Hardy-Sobolev-Maz'ya inequalities with singularities on flat submanifold

本文研究了带权重的分数Hardy-Sobolev和Hardy-Sobolev-Maz'ya不等式，探讨了奇点在平坦子流形上的应用，并证明了相关不等式在特定条件下的不成立性。

Classification of solutions of an equation related to a conformal log Sobolev inequality

We classify all finite energy solutions of an equation which arises as the Euler--Lagrange equation of a conformally invariant logarithmic Sobolev inequality on the sphere due to Beckner. Our proof uses an extension of the method of moving spheres from $\mathbb R^n$ to $\mathbb S^n$ and a classification result of Li and Zhu. Along the way we prove a small volume maximum principle and a strong maximum principle for the underlying operator which is closely related to the logarithmic Laplacian.

Extremal functions for the second-order Sobolev inequality on groups of polynomial growth

In this paper, we prove the second-order Sobolev inequalities on Cayley graphs of groups of polynomial growth. We use the discrete Concentration-Compactness principle to prove the existence of extremal functions for best constants in supercritical cases. As applications, we get the existence of positive ground state solutions to the $p$-biharmonic equations and the Lane-Emden systems.

分数阶Sobolev空间中具有光滑子流形奇点的Trudinger-Moser型不等式

Trudinger-Moser type inequality in fractional Sobolev space with singularity on smooth submanifold

研究分数阶Sobolev空间中在光滑子流形上具有奇点的Trudinger-Moser型不等式，通过分数阶Hardy不等式证明并展示常数尖锐性，同时建立奇点集上消失的分数阶Sobolev空间等价性。

The logarithmic Sobolev inequality for a submanifold in Euclidean space

We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.

A Fractional Hardy-Sobolev-Maz'ya Inequality on the Upper Halfspace

We prove several Sobolev inequalities, which are then used to establish a fractional Hardy-Sobolev- Maz'ya inequality on the upper halfspace.

整数格上的精确Gagliardo-Nirenberg不等式与对数Sobolev不等式

Sharp Gagliardo-Nirenberg inequality and logarithmic Sobolev inequality on integer lattices

本文在整数格上获得了精确的Gagliardo-Nirenberg不等式并刻画了其刚性，进而作为推论推导出整数格上精确的对数Sobolev不等式。

对数Sobolev不等式的稳定性

Stability for the logarithmic Sobolev inequality

该论文聚焦高斯对数Sobolev不等式的稳定性问题，推导得出了显式的稳定性常数，为相关不等式的稳定性分析提供了量化结果。

三球上的四阶尖锐 Sobolev 迹不等式

A sharp Sobolev trace inequality of order four on three-balls

本文研究三球上的四阶尖锐 Sobolev 迹不等式，并揭示其与二维球面上三阶尖锐 Sobolev 不等式的等价性。

Instability results for the logarithmic Sobolev inequality and its application to related inequalities

We show that there are no general stability results for the logarithmic Sobolev inequality in terms of the Wasserstein distances and $L^{p}(dγ)$ distance for $p>1$. To this end, we construct a sequence of centered probability measures such that the deficit of the logarithmic Sobolev inequality converges to zero but the relative entropy and the moments do not, which leads to instability for the logarithmic Sobolev inequality. As an application, we prove instability results for Talagrand's transportation inequality and the Beckner--Hirschman inequality.

Minimizers for the fractional Sobolev inequality on domains

We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains.

A note on the sublinear Sobolev inequality

In this note we show that the classical Sobolev inequality cannot in unrestricted form hold for exponents $p \in (0,1)$.

Sobolev不等式可达性的一个简单证明

A simple proof of attainability for the Sobolev inequality

本文通过截断基本解的表示公式，为Sobolev不等式极小值的存在性提供了一个简单证明。

The Best Constant, the Nonexistence of Extremal Functions and Related Results for an Improved Hardy-Sobolev Inequality

We present the best constant and the existence of extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in $\mathbb{R}^N$. We also discuss the connection of the related functional spaces and as a result we obtain some Caffarelli - Kohn - Nirenberg inequalities. Our starting point is the existence of a minimizer for the Bliss' inequality and the indirect dependence of the Hardy inequality at the origin.

Time-dependent Sobolev inequality along the Ricci flow

In this article we get a time-dependent Sobolev inequality along the Ricci flow which generalizes the earlier results of Zhang, Ye, Hsu. As an application of the time-dependent Sobolev inequality, we also get a growth of the ratio of bob-collapsing along the Ricci flow.