  Given a strongly local Dirichlet space and $\lambda\geq 0$, we introduce a
new notion of $\lambda$--subharmonicity for $L^1_\loc$--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 $\Delta f\leq f$ for complete Riemannian manifolds.
