On optimal autocorrelation inequalities on the real line

Sobolev和log-Sobolev不等式的尖锐稳定性，具有最优的维数依赖性

Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence

本文证明了具有显式常数且大维数极限行为正确的Sobolev不等式尖锐定量稳定性，并据此推导出显式与维数无关常数的Gaussian log-Sobolev不等式最优定量稳定性估计。

A remark on optimal weighted Poincaré inequalities for convex domains

We prove a sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in a class of weighted Poincaré inequalities. The key point is the study of an optimal weighted Wirtinger inequality.

有限区间与环形区域上的最优幂加权 Birman--Hardy--Rellich 型不等式

Optimal Power-Weighted Birman--Hardy--Rellich-type Inequalities on Finite Intervals and Annuli

本文在有限区间和环形区域上导出了积分与微分形式的最优幂加权 Hardy 及 Birman-Hardy-Rellich 型不等式序列，并通过极限过程恢复了 $\mathbb{R}^n\backslash\{0\}$ 上的经典 Rellich 型不等式。

Higher order Glaeser inequalities and optimal regularity of roots of real functions

We prove a higher order generalization of Glaeser inequality, according to which one can estimate the first derivative of a function in terms of the function itself, and the Holder constant of its k-th derivative. We apply these inequalities in order to obtain pointwise estimates on the derivative of the (k+alpha)-th root of a function of class C^{k} whose derivative of order k is alpha-Holder continuous. Thanks to such estimates, we prove that the root is not just absolutely continuous, but its derivative has a higher summability exponent. Some examples show that our results are optimal.

An Optimal Inequality For The Tangent Function

In this note we deal with some inequalities for the tangent function that are valid for $x$ in $(-π/2,π/2)$. These inequalities are optimal in the sense that the best values of the exponents involved are obtained.

最优离散 $p$-Hardy-Rellich-Birman 不等式

Optimal discrete $p$-Hardy-Rellich-Birman inequalities

本文提出一个抽象框架，通过高阶单调性和渐近约束构造离散半线上高阶 $p$-Laplacian 的最优下界，并导出新的最优离散 $p$-Hardy、$p$-Rellich 和 $p$-Birman 不等式。

具有最优Hardy-Rellich不等式和临界指数的Sobolev插值不等式

Sobolev interpolation inequalities with optimal Hardy-Rellich inequalities and critical exponents

本文建立了一族新的临界高阶Sobolev插值不等式，覆盖径向与非径向函数，利用尖锐Hardy-Rellich不等式的最优二次型，并通过Fourier分析和高阶Talenti比较原理克服高阶情形下的困难，推广了Dietze和Nam的一阶结果。

Ingham type inequalities in lattices

A classical theorem of Ingham extended Parseval's formula of the trigonometrical system to arbitrary families of exponentials satisfying a uniform gap condition. Later his result was extended to several dimensions, but the optimal integration domains have only been determined in very few cases. The purpose of this paper is to determine the optimal connected integration domains for all regular two-dimensional lattices.

Optimal sine and sawtooth inequalities

We determine the optimal inequality of the form $\sum_{k=1}^m a_k\sin kx\leq 1$, in the sense that $\sum_{k=1}^m a_k$ is maximal. We also solve exactly the analogous problem for the sawtooth (or signed fractional part) function. Equivalently, we solve exactly an optimization problem about equidistribution on the unit circle.

Optimal ambiguity functions and Weil's exponential sum bound

Complex-valued periodic sequences, u, constructed by Goran Bjorck, are analyzed with regard to the behavior of their discrete periodic narrow-band ambiguity functions A_p(u). The Bjorck sequences, which are defined on Z/pZ for p>2 prime, are unimodular and have zero autocorrelation on (Z/pZ)\{0}. These two properties give rise to the acronym, CAZAC, to refer to constant amplitude zero autocorrelation sequences. The bound proven is |A_p(u)| \leq 2/\sqrt{p} + 4/p outside of (0,0), and this is of optimal magnitude given the constraint that u is a CAZAC sequence. The proof requires the full power …

最优离散Hardy-Rellich-Birman不等式

Optimal discrete Hardy-Rellich-Birman inequalities

本研究给出半线上离散拉普拉斯整数次幂不等式中确定最优权的参数序列充分条件，通过具体选取得到显式最优离散Rellich与Birman权，改进了已有相关结果并证明了一则猜想。

Caffarelli-Kohn-Nirenberg identities, inequalities and their stabilities

We set up a one-parameter family of inequalities that contains both the Hardy inequalities (when the parameter is 1) and the Caffarelli-Kohn-Nirenberg inequalities (when the parameter is optimal). Moreover, we study these results with the exact remainders to provide direct understandings to the sharp constants, as well as the existence and non-existence of the optimizers of the Hardy inequalities and Caffarelli-Kohn-Nirenberg inequalities. As an application of our identities, we establish some sharp versions with optimal constants and theirs attainability of the stability of the Heisenberg Unc…

分数阶Gagliardo–Nirenberg不等式：逐点估计、尖锐渐近行为及最优目标空间

Fractional Gagliardo--Nirenberg Inequalities: Pointwise Estimates,Sharp Asymptotics, and Optimal Target Spaces

该研究针对分数阶差分算子建立逐点估计，推导含BMO端点的分数阶Gagliardo–Nirenberg不等式，确定其渐近因子最优阶，完全刻画重排不变目标空间并解答了开放问题，结果在非对角和BMO情形下为全新成果。

整数上分数 $p$-拉普拉斯算子的最优 Hardy 不等式

Optimal Hardy Inequalities for Fractional $p$-Laplacians on the Integers

本文针对整数上分数 $p$-拉普拉斯算子建立了最优 Hardy 不等式，并推广到 Riesz 分数情形，同时揭示了 $p$-零临界性与无穷远处最优性的联系。