Riesz transforms for Dirichlet spaces tamed by distributional curvature lower bounds

TL;DR

This paper develops harmonic analysis tools like Riesz transforms and Littlewood-Paley-Stein inequalities for tamed Dirichlet spaces with distributional curvature bounds, extending vector calculus in this setting.

Contribution

It establishes boundedness of Riesz transforms and Littlewood-Paley-Stein inequalities for 1-forms in tamed Dirichlet spaces, advancing analysis on these spaces.

Findings

Proved Littlewood-Paley-Stein inequality for 1-forms.

Established boundedness of Riesz transforms.

Extended vector calculus to tamed Dirichlet spaces.

Abstract

The notion of tamed Dirichlet space was proposed by Erbar, Rigoni, Sturm and Tamanini as a Dirichlet space having a weak form of Bakry-\'Emery 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.