On the Maz'ya-Shaposhnikova formula in Orlicz-Sobolev spaces on Carnot groups

On the limit as $s\to 0^+$ of fractional Orlicz-Sobolev spaces

An extended version of the Maz'ya-Shaposhnikova theorem on the limit as $s\to 0^+$ of the Gagliardo-Slobodeckij fractional seminorm is established in the Orlicz space setting. Our result holds in fractional Orlicz-Sobolev spaces associated with Young functions satisfying the $Δ_2$-condition, and, as shown by counterexamples, it may fail if this condition is dropped.

Maz'ya-Shaposhnikova表示法用于具有弱反倍增性质的同质类型空间上的球准Banach函数空间的准范数

Maz'ya--Shaposhnikova Representation of Quasi-Norms of Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type with Weak Reverse Doubling Property

本文提出弱反倍增条件和弱测度密度条件，建立了球准Banach函数空间的准范数表示公式，并发现当空间有限时该极限趋于零。

一个区分两个极大算子的巴拿赫空间

A Banach space that distinguishes two maximal operators

本文构造了一个局部可积函数的翻译不变的巴拿赫空间，在该空间上极大卷积算子$M^\diamond$有界，但尖锐极大算子$M^\sharp$无界，回答了Maz'ya提出的75个分析问题中的一个问题。

Dunkl 框架下 Besov 半范数与非局部周长的极限公式

On Limit Formulas for Besov Seminorms and Nonlocal Perimeters in the Dunkl Setting

本文在 Dunkl 框架下推广了 Besov 半范数极限公式与非局部周长极限公式，提出了无维数依赖的 Maz'ya--Shaposhnikova 公式，并弱化了边界正则性条件，结果在经典拉普拉斯情形下亦为新发现。

Anisotropic Fractional Sobolev Norms

Bourgain, Brezis & Mironescu showed that (with suitable scaling) the fractional Sobolev $s$-seminorm of a function $f\in W^{1,p}(\rn)$ converges to the Sobolev seminorm of $f$ as $s\to 1^-$. The anisotropic $s$-seminorms of $f$ defined by a norm on $\rn$ with unit ball $K$ are shown to converge to the anisotropic Sobolev seminorm of $f$ defined by the norm with unit ball $\,\ompd K$, the polar $L_p$ moment body of $K$. The limiting behavior for $s\to 0^+$ is also determined (extending results by Maz$'$ya & Shaposhnikova).

关于度量测度空间中分数阶Sobolev半范数的渐近行为：渐近体积比、体积熵与刚性

On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: asymptotic volume ratio, volume entropy and rigidity

本文研究一般度量测度空间中分数阶Sobolev半范数的渐近行为，给出Maz'ya-Shaposhnikova定理和Ludwig结果的新简短证明，建立渐近公式与渐近体积比之间的联系，并证明两个新的刚性结果。

Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori

This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative $d$-torus $\mathbb{T}^d_θ$ (with $θ$ a skew symmetric real $d\times d$-matrix). These spaces share many properties with their classical counterparts. We prove, among other basic properties, the lifting theorem for all these spaces and a Poincaré type inequality for Sobolev spaces. We also show that the Sobolev space $W^k_\infty(\mathbb{T}^d_θ)$ coincides with the Lipschitz space of order $k$, already studied by Weaver in the case $k=1$. We establish the embedding inequalities of all these spac…