Reverse Hölder Inequalities for log-Lipschitz Functions

延拓、分离与同构反向等周

Extension, separation and isomorphic reverse isoperimetry

本文研究了度量空间的Lipschitz延拓模与分离模，通过有限维赋范空间的随机聚类几何结构获得了两者的上下界，并首次精确评估了特定赋范空间Lipschitz延拓模的阶。

BMO-estimates for non-commutative vector valued Lipschitz functions

We construct Markov semi-groups $\mathcal{T}$ and associated BMO-spaces on a finite von Neumann algebra $(\mathcal{M}, τ)$ and obtain results for perturbations of commutators and non-commutative Lipschitz estimates. In particular, we prove that for any $A \in \mathcal{M}$ self-adjoint and $f: \mathbb{R} \rightarrow \mathbb{R}$ Lipschitz there is a Markov semi-group $\mathcal{T}$ such that for $x \in \mathcal{M}$, \[ \Vert [f(A), x] \Vert_{{\rm BMO}(\mathcal{M}, \mathcal{T})} \leq c_{abs} \Vert f' \Vert_\infty \Vert [A, x] \Vert_\infty. \] We obtain an analogue of this result for more general v…

Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order

Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequality for non-necessarily Lipschitz functions $f\colon \R^n \to \R$ with bounded derivatives of higher orders, which hold when the underlying measure satisfies a family of Sobolev type inequalities $\|g- \E g\|_p \le C(p)\|\nabla g\|_p.$ Such Sobolev type inequalities hold, e.g., if the underlying measure satisfies the log-Sobolev inequality (in which case $C(p) \le C\sqrt{p}$) or the Poincaré inequality (then $C(p) \le Cp$). Our concentration …

Lipschitz spaces adapted to Schrödinger operators and regularity properties

Consider the Schrödinger operator $\mathcal{L}=-Δ+V$ in $\mathbb{R}^n, n\ge 3,$ where $V$ is a nonnegative potential satisfying a reverse Hölder condition of the type \begin{equation*} \left( \frac{1}{|B|}\int_B V(y)^qdy\right)^{1/q}\le \frac{C}{|B|}\int_B V(y)dy, \, \text{ for some }q>n/2. \end{equation*} We define $Λ^α_{\mathcal{L}},\, 0<α<2,$ the class of measurable functions such that $$ \|ρ(\cdot)^{-α}f(\cdot)\|_\infty<\infty \quad \, \, \text{and}\:\: \quad \sup_{|z|>0}\frac{\|f(\cdot+z)+f(\cdot-z)-2f(\cdot)\|_\infty}{|z|^α}<\infty, $$ where $ρ$ is the critical radius function associated…

薛定谔框架下BMO与Lipschitz空间的若干刻画

Some characterizations of BMO and Lipschitz spaces in the Schrödinger setting

本文针对薛定谔算子相关的BMO和Lipschitz空间，给出若干新刻画，将已有研究结果推广到权类范围大于经典Muckenhoupt类的加权情形。

调和函数在度量测度空间上的Hölder正则性

Hölder regularity of harmonic functions on metric measure spaces

本文研究了度量测度空间上调和函数的Hölder正则性，证明了多个条件等价，并应用这些结果解决了反Hölder不等式和热核估计的问题。