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 $L^{\mathrm{\infty }}$-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.