Given a strongly local Dirichlet space and \(\lambda \geqslant 0\) , we introduce a new notion of \(\lambda \) -subharmonicity for \(L^1_\textrm{loc}\) -functions, which we call 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 \(\Delta f\leqslant f\) for complete Riemannian manifolds.
