A new notion of subharmonicity on locally smoothing spaces, and a conjecture by Braverman, Milatovic, Shubin

TL;DR

This paper introduces a new concept of subharmonicity on locally smoothing spaces, establishing its equivalence to distributional subharmonicity in Riemannian cases, and proves a conjecture about positivity of solutions to a differential inequality.

Contribution

It defines local $ ext{ extlambda}$--shift defectivity, studies regularity on locally smoothing spaces, and proves a conjecture on positivity of solutions to $ riangle f \,\leq\,f$ on Riemannian manifolds.

Findings

New notion of $ ext{ extlambda}$--subharmonicity introduced

Regularity results on a broad class of spaces including Riemannian manifolds and fractals

Proof of Braverman-Milatovic-Shubin conjecture on solution positivity

Abstract

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.