Given a local Dirichlet space, we introduce a new notion of subharmonicity for locally integrable functions in terms of the heat kernel, which we we call 'local defectivity'. This notion does need any a priori W^{1,2}_loc type or sign assumption, and turns out to be equivalent to subharmonicity in the distributional sense on Riemannian manifolds. For 'locally smoothing spaces', we prove that any power of a nonnegative locally defective function has, a posteriori, a W^{1,2}_loc type regularity. We also show that locally defective functions admit a Brezis-Kato type inequality, and a maximum principle. As a consequence of these results, we prove that any intrinsically complete, locally smoothing, irreducible space is L^p-positivity preserving, which solves a conjecture by Braverman, Milatovic, Shubin. These results apply to every complete connected Riemannian manifold and every finite dimensional RCD-space. This is joint work with Stefano Pigola, Peter Stollmann, and Giona Veronelli.