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

Abstract

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 is necessarily non-negative. In particular, the case $p=2$ of our result answers in 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 result for positive subharmonic distributions, which in turn permits to prove a Liouville type theorem, and a Brezis-Kato inequality on Riemannian manifolds. Both these results rely on a smooth monotonic approximation of distributional solutions of $\Delta u\ge \lambda (x)u$ of independent interest.

14 pages. This unpublished preprint is superseded by the two companion articles arXiv:2301.05159 and [Guneysu,Pigola,Stollmann,Veronelli, Regularity of subharmonic distributions on local spaces, and a conjecture by Braverman, Milatovic, Shubin, Preprint 2022]

References in corpus (2)

• Essential self-adjointness of Schroedinger type operators on manifolds
• The BMS conjecture