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 λ ⩾ 0 , we introduce a new notion of λ λ -subharmonicity for L¹ₗoc L loc 1 -functions, which we call local λ λ -shift defectivity , and which turns out to be equivalent to distributional λ 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 L q -solutions of Δ f f Δ f ⩽ f for complete Riemannian manifolds.