Riesz Transforms of Schrödinger Operators on Manifolds

Abstract

We consider Schrödinger operators A=−Δ+V on L ^p(M) where M is a complete Riemannian manifold of homogeneous type and V=V ⁺−V ⁻ is a signed potential. We study boundedness of Riesz transform type operators $\nabla A^{-\frac{1}{2}}$ and $|V|^{\frac{1}{2}}A^{-\frac{1}{2}}$ on L ^p(M). When V ⁻ is strongly subcritical with constant α∈(0,1) we prove that such operators are bounded on L ^p(M) for $p\in(p_{0}', 2]$ where $p_{0}'=1$ if N≤2, and $p_{0}'=(\frac{2N}{(N-2)(1-\sqrt{1-\alpha })})' \in (1, 2)$ if N>2. We also study the case p>2. With additional conditions on V and M we obtain boundedness of ∇A ^−1/2 and |V|^1/2 A ^−1/2 on L ^p(M) for p∈(1,inf (q ₁,N)) where q ₁ is such that $\nabla(-\Delta)^{-\frac{1}{2}}$ is bounded on L ^r(M) for r∈[2,q ₁).