Hess-Schrader-Uhlenbrock inequality for the heat semigroup on differential forms over Dirichlet spaces tamed by distributional curvature lower bounds

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 Hess-Schrader-Uhlenbrock inequality for $1$-forms as an element of $L^2$-cotangent module (an $L^2$-normed $L^{\infty}$-module), which extends the Hess-Schrader-Uhlenbrock inequality by Braun under an additional condition.