Zi Jian Xu | Searcharxiv

arXiv2025

The Littlewood-Paley-Stein inequality for Dirichlet space tamed by signed measured curvature lower bounds

The notion of tamed Dirichlet space by distributional lower Ricci curvature bounds was proposed by Erbar--Rigoni--Sturm--Tamanini as the Dirichlet space having a weak form of Bakry--Émery curvature lower bounds in distribution sense. In this framework, we establish the Littlewood--Paley--Stein inequality for $L^p$-functions which partially generalizes the result by Kawabi--Miyokawa.

math.PR↗

arXiv2023

Riesz transforms for Dirichlet spaces tamed by distributional curvature lower bounds

The notion of tamed Dirichlet space was proposed by Erbar, Rigoni, Sturm and Tamanini as a Dirichlet space having a weak form of Bakry-Émery curvature lower bounds in distribution sense. After their work, Braun established a vector calculus for it, in particular, the space of $L^2$-normed $L^{\infty}$-module describing vector fields, $1$-forms, Hessian in $L^2$-sense. In this framework, we establish the Littlewood-Paley-Stein inequality for $1$-forms as an element of $L^p$-cotangent bundles and boundedness of Riesz transforms, which partially solves the problem raised by Kawabi-Miyokawa.

math.PR↗