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

TL;DR

This paper proves that complete Riemannian manifolds are $L^p$-positivity preserving for all $p$ in (1,∞), confirming a conjecture and introducing new regularity and Liouville theorems for subharmonic distributions.

Contribution

It establishes the $L^p$-positivity preserving property for all $p eq 1$, confirming a longstanding conjecture and developing new regularity and approximation techniques.

Findings

Complete Riemannian manifolds are $L^p$-positivity preserving for $p eq 1$

A new a-priori regularity result for positive subharmonic distributions

A Liouville type theorem for distributional solutions

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.