AbstractGiven a strongly local Dirichlet space and $$\lambda \geqslant 0$$ λ ⩾ 0 , we introduce a new notion of $$\lambda $$ λ -subharmonicity for $$L^1_\textrm{loc}$$ L loc 1 -functions, which we call local$$\lambda $$ λ -shift defectivity, and which turns out to be equivalent to distributional $$\lambda $$ 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 $$L^q$$ L q -solutions of $$\Delta f\leqslant f$$ Δ f ⩽ f for complete Riemannian manifolds.