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The notion of tamed Dirichlet space was proposed by Erbar, Rigoni, Sturm andTamanini as a Dirichlet space having a weak form of Bakry-\'Emery curvaturelower bounds in distribution sense. After their work, Braun established avector 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-Steininequality for $1$-forms as an element of $L^p$-cotangent bundles andboundedness of Riesz transforms, which partially solves the problem raised byKawabi-Miyokawa.

Riesz transforms for Dirichlet spaces tamed by distributional curvature lower bounds