The commutators of Bochner-Riesz operators for elliptic operators

Abstract

We study $L^p$-boundedness of commutators of Bochner-Riesz operators for elliptic self-adjoint operators which satisfy the finite speed of propagation property for the corresponding wave equation. Our results can be applied to Schrödinger operators with inverse square potentials on $\mathbb{R}^n$, elliptic operators on compact manifolds, and Schrödinger operators on asymptotically conic manifolds

Our proof is new even for the commutator of the classical Bochner-Riesz operator when $L$ is the Laplace operator $\Delta=\sum_{i=1}^n\partial_{x_i}^2$ on the Euclidean space $\mathbb{R}^n$.