Pseudo-localization of singular integrals and noncommutative Littlewood-Paley inequalities

Noncommutative Khintchine and Paley inequalities via generic factorization

We reprove an inequality for Rademacher series with coefficients in the Schatten class $S_1$. Our method yields the same estimate for coefficients after suitable gaps in $S_1$-valued trigonometric series; this was known for scalar-valued functions. A very similar method gives a new proof of the extension to $S_1$-valued $H^1$ functions of Paley's theorem about lacunary coefficients.

关于非交换Littlewood-Paley-Stein不等式的最佳常数

On the best constants of the noncommutative Littlewood-Paley-Stein inequalities

本文研究了半有限von Neumann代数上非交换Littlewood-Paley-Stein不等式的最佳常数，证明了当$p\to\infty$时上界常数$β_p$的最优阶为$p$，并给出了其他情形下常数的上下界。

Vector-valued Littewood-Paley-Stein theory for semigroups II

Inspired by a recent work of Hytönen and Naor, we solve a problem left open in our previous work joint with Mart\'ınez and Torrea on the vector-valued Littlewood-Paley-Stein theory for symmetric diffusion semigroups. We prove a similar result in the discrete case, namely, for any $T$ which is the square of a symmetric Markovian operator on a measure space $(Ω, μ)$. Moreover, we show that $T\otimes{\rm Id}_X$ extends to an analytic contraction on $L_p(Ω; X)$ for any $1<p<\infty$ and any uniformly convex Banach space $X$.

半群的全纯泛函演算与向量值Littlewood-Paley-Stein理论

Holomorphic functional calculus and vector-valued Littlewood-Paley-Stein theory for semigroups

基于全纯泛函演算新方法，研究正则压缩半群的向量值Littlewood-Paley-Stein理论，导出了相关常数关于p的最优增长阶，并解决了Naor和Young关于经典Poisson和热半群最佳常数最优阶的猜想。