Commutator estimates regarding pseudo-differential operators

MathOverflow Asked by Shaoyang Zhou on December 15, 2020

Forgive me if what I’m asking is too naive for Mathoverflow.

Given a compact Riemannian manifold $(M,g)$ with Hodge Laplacian $Delta$. Recall that the Riesz potential $(-Delta)^{-frac{1}{2}}$ defined by
(-Delta)^{-frac{1}{2}} = int_{0}^{infty}e^{-tDelta}frac{dt}{sqrt{t}}

is a pseudo-differential operator on $M$ of order one. I’m wondering whether there exists a commutator identity/estimates regarding the Riesz potential and the Riemannian gradient $nabla$. In other words, let $r$ be an $(s,t)$-tensor, then what can we say about $[nabla, (-Delta)^{-frac{1}{2}}]r$ in terms of the norm of an $(s+1,t)$-tensor? When commuting two full covariant derivatives, the commutator is usually controlled by the curvature term. But I’m not familiar with the pseudo-differential version of the story. Any insight or help would be appreciated.

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