Local Sobolev regularity conjecture for Schrödinger operators under Assumption A

Let $M$ be a Riemannian manifold, let $E$ be a Hermitian vector bundle over $M$, and let $V$ be a potential satisfying Assumption A: $V=V_{+}+V_{-}$, with $V_{+}(x)8e0$ and $V_{-}(x)8e0$ as linear operators on each fiber, and for every compact $K\subset M$ there are constants $a_K<1$ and $C_K$ such that

$$\left(\int_K |V_-|^2|u|^2\,d\mu\right)^{1/2}\leq a_K\|\Delta_Mu\|+C_K\|u\|,$$

for all $u\in C_c^\infty(M)$. Let $H_{V,\max}$ denote the maximal domain of the associated Schrödinger-type operator. Local regularity conjecture. If $V$ satisfies Assumption A, then

$$\operatorname{Dom}(H_{V,\max})\subset W^{1,2}_{\operatorname{loc}}(E).$$

The conjecture is motivated by the established local regularity theorem under the additional assumption that $D^*D$ has a scalar principal symbol; the authors note that the assertion is nontrivial even when $M=\mathbb{R}^n$.