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In this paper we prove that a complete Riemannian manifold is$L^p$-positivity preserving for any $p\in(1,\infty)$. This means that any $L^p$function which solves $(-\Delta + 1)u\ge 0$ in the sense of distributions isnecessarily non-negative. In particular, the case $p=2$ of our result answersin the affermative a conjecture formulated by M. Braverman, O. Milatovic and M.Shubin in 2002. The two main ingredients are a new a-priori regularity resultfor positive subharmonic distributions, which in turn permits to prove aLiouville type theorem, and a Brezis-Kato inequality on Riemannian manifolds.Both these results rely on a smooth monotonic approximation of distributionalsolutions of $\Delta u \ge \lambda(x) u$ of independent interest.

$L^{p}$ Positivity Preserving and a conjecture by M. Braverman, O. Milatovic and M. Shubin