A new notion of subharmonicity on locally smoothing spaces, and a conjecture by Braverman, Milatovic, Shubin

Given a strongly local Dirichlet space and $\lambda \ge 0$, we introduce a new notion of $\lambda$–subharmonicity for $L_{\text{loc}}^1$–functions, which we call \emph{local $\lambda$–shift defectivity}, and which turns out to be equivalent to distributional $\lambda$–subharmonicity in the Riemannian case. We study the regularity of these functions on a new class of strongly local Dirichlet, so called locally smoothing spaces, which includes Riemannian manifolds (without any curvature assumptions), finite dimensional RCD spaces, Carnot groups, and Sierpinski gaskets. As a byproduct of this regularity theory, we obtain in this general framework a proof of a conjecture by Braverman, Milatovic, Shubin on the positivity of distributional $L^q$-solutions of $\mathrm{\Delta }f\le f$ for complete Riemannian manifolds.