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

Given a strongly local Dirichlet space and λ⩾0, we introduce a new notion of λ-subharmonicity for Lloc1-functions, which we call localλ-shift defectivity, and which turns out to be equivalent to distributional λ-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 Lq-solutions of Δf⩽f for complete Riemannian manifolds.