The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space

Sharp Trace Hardy-Sobolev-Maz'ya Inequalities and the Fractional Laplacian

In this work we establish trace Hardy and trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy-Sobolev-Maz'ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space our results cover the full range of the exponent $s \in (0,1)$ of the fractional Laplacians. We answer in particular an open problem raised by Frank and Seiringer \cite{FS}.

Hardy-Sobolev插值不等式

Hardy-Sobolev interpolation inequalities

本文建立了一族改进Hardy不等式并覆盖Sobolev临界指数的插值估计，确定了端点情形下径向函数的所有最优函数，并提出了非限制最优函数的未解决问题。

A unified divergent approach to Hardy-Poincaré inequalities in classical and variable Sobolev spaces

We present a unified strategy to derive Hardy-Poincaré inequalities on bounded and unbounded domains. The approach allows proving a general Hardy-Poincaré inequality from which the classical Poincaré and Hardy inequalities immediately follow. The idea also applies to the more general context of variable exponent Sobolev spaces. The argument, concise and constructive, does not require a priori knowledge of compactness results and retrieves geometric information on the best constants.

涉及广义哈代算子的索伯列夫范数等价性

Equivalence of Sobolev norms involving generalized Hardy operators

本文研究带哈代位势的分数阶薛定谔算子所生成的齐次索伯列夫空间与通常齐次索伯列夫空间的等价性，得到广义及反向哈代不等式，并推广了非分数阶情形下的近期结果。

Improved Hardy-Sobolev inequalities

The main result includes features of a Hardy-type inequality and an inequality of either Sobolev or Gagliardo-Nirenberg type. It is inspired by the method of proof of a recent improved Sobolev inequality derived by M. Ledoux which brings out the connection between Sobolev embeddings and heat kernel bounds. Here Ledoux's technique is applied to the operator $L:= {\bf{x}} \cdot \nabla$ and the analysis requires the determination of the operator semigroup $\{e^{-tL^{*}L}\}_{t>0}$ and its properties.

对数 Hardy–Littlewood–Sobolev 不等式的稳定性及其在 Keller–Segel 系统中的应用

Stability for the logarithmic Hardy-Littlewood-Sobolev Inequality with application to the Keller-Segel system

本文利用对偶方法建立对数 Hardy–Littlewood–Sobolev 不等式的最优稳定性结果，并据此估计临界质量 Keller–Segel 系统的平衡趋近速率。