Regularity points and Jensen measures for $R(X)$

Borel正则性等价于Lusin定理与Borel代表元的存在性

Borel regularity is equivalent to Lusin's theorem and the existence of Borel representatives

本文在一般拓扑测度空间中通过测度正则性刻画了Lusin定理与Borel代表元的存在性，并证明测度的Borel正则性是这些结果在度量测度空间中成立的充要条件。

关于测度空间对的不等式及其应用

Inequalities for Pairs of Measure Spaces and Applications

本文在一般乘积测度框架下建立了一种Jensen型不等式，扩展了Hölder和Minkowski等经典不等式，并应用于幂均、熵型和擦除鲁棒不等式以及卷积型算子和加权离散模型。

Sobolev embedding implies regularity of measure in metric measure spaces

We prove that if the Sobolev embedding $M^{1,p}(X)\hookrightarrow L^q(X)$ holds for some $q>p\geq 1$ in a metric measure space $(X,d,μ),$ then a constant $C$ exists such that $μ(B(x,r))\geq Cr^n$ for all $x\in X$ and all $0<r\leq 1,$ where $\frac{1}{p}-\frac{1}{q}=\frac{1}{n}.$ This was proved in \cite{Gor17} assuming a doubling condition on the measure $μ.$

图凸包界作为广义 Jensen 不等式

Graph Convex Hull Bounds as generalized Jensen Inequalities

本文通过研究函数图像的凸包，为任意域上的任意函数给出期望的精确上下界，将 Jensen 不等式推广到有限/无限维及条件期望、Markov 算子情形。

On regularity for measures in multiplicative free convolution semigroups

Given a probability measure $μ$ on the real line, there exists a semigroup $μ_t$ with real parameter $t>1$ which interpolates the discrete semigroup of measures $μ_n$ obtained by iterating its free convolution. It was shown in \cite{[BB2004]} that it is impossible that $μ_t$ has no mass in an interval whose endpoints are atoms. We extend this result to semigroups related to multiplicative free convolution. The proofs use subordination results.

Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras

We establish several deep existence criteria for conditional expectations on von Neumann algebras, and then apply this theory to develop a noncommutative theory of representing measures of characters of a function algebra. Our main cycle of results describes what may be understood as a `noncommutative Hoffman-Rossi theorem' giving the existence of weak* continuous `noncommutative representing measures' for so-called D-characters. These results may also be viewed as `module' Hahn-Banach extension theorems for weak* continuous `characters' into possibly noninjective von Neumann algebras. In clos…

一种测度论方法用于Lipschitz正则性及其Haar型小波分析

A measure theoretic approach to Lipschitz regularity and its Haar type wavelet analysis

本文通过广义Haar系统将函数的二进制Lipschitz正则性特征扩展到非原子概率空间，提出新的分析方法。

Regularity versus smoothness of measures

The Assouad and lower dimensions and dimension spectra quantify the regularity of a measure by considering the relative measure of concentric balls. On the other hand, one can quantify the smoothness of an absolutely continuous measure by considering the $L^p$ norms of its density. We establish sharp relationships between these two notions. Roughly speaking, we show that smooth measures must be regular, but that regular measures need not be smooth.

Sobolev regularity of occupation measures and paths, variability and compositions

We prove a result on the fractional Sobolev regularity of composition of paths of low fractional Sobolev regularity with functions of bounded variation. The result relies on the notion of variability, proposed by us in the previous article [43, arXiv:2003.11698]. Here we work under relaxed hypotheses, formulated in terms of Sobolev norms, and we can allow discontinuous paths, which is new. The result applies to typical realizations of certain Gaussian or Lévy processes, and we use it to show the existence of Stieltjes type integrals involving compositions.

Regularity of monotone transport maps between unbounded domains

The regularity of monotone transport maps plays an important role in several applications to PDE and geometry. Unfortunately, the classical statements on this subject are restricted to the case when the measures are compactly supported. In this note we show that, in several situations of interest, one can to ensure the regularity of monotone maps even if the measures may have unbounded supports.

On fractional regularity of distributions of functions in Gaussian random variables

We study fractional smoothness of measures on $\mathbb{R}^k$, that are images of a Gaussian measure under mappings from Gaussian Sobolev classes. As a consequence we obtain Nikolskii--Besov fractional regularity of these distributions under some weak nondegeneracy assumption.

矩序列的绝对连续表示测度：侧重于密度的正则性

Absolutely Continuous Representing Measures of Moment Sequences with an Emphasis on the Regularity of the Densities

本文刻画了具有绝对连续表示测度的线性泛函，重点研究了密度g的正则性及其与矩s_α=L(x^α)的关联。

Fredholm积分方程在上Ahlfors正则集上解的Hölder正则性

Hölder regularity of the solutions of Fredholm integral equations on upper Ahlfors regular sets

本文研究了在满足上Ahlfors增长条件的度量测度空间中，Fredholm二类积分方程解的Hölder正则性问题。

On dimension and regularity of bundle measures

In this paper we quantify the notion of antisymmetry of the Fourier transform of certain vector valued measures. The introduced scale is related to the condition appearing in Uchiyama's theorem and is used to give a lower bound for the rectifiable dimension of those measures. Moreover, we obtain an estimate of the lower Hausdorff dimension assuming certain more restrictive version of the $2$-wave cone condition. Results of our considerations can be viewed as an uncertainty-type principle in the following way: it is impossible to simultaneously localize a (bundle) measure and a direction of its…

随机动力系统不变测度的Lipschitz正则性

Lipschitz regularity of the invariant measure of random dynamical systems

本文研究随机动力系统不变测度分解的正则性，通过构造类Wasserstein-Kantorovich度量的各向异性赋范空间并分析转移算子动力学，利用泛函分析技术证明谱隙，进而证明Lipschitz可观测量的指数衰减相关性和统计性质。

Quasi-invariance of low regularity Gaussian measures under the gauge map of the periodic derivative NLS

The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasi-invariance of the Gaussian measure on $L^2(\T)$ with covariance $[1+(-\D)^{s}]^{-1}$ under these transformations for any $s>\frac12$. This extends previous achievements by Nahmod, Ray-Bellet, Sheffield and Staffilani (2011) and Genovese, Lucà and Valeri (2018), who proved the result for integer values of the regularity parameter $s$.