Consider the Dirichlet Laplacian $\Delta$ on a compact Riemannian manifold (with boundary). Consider the operator $T = \sqrt{-\Delta}$. My question is: is there any Leibniz/product rule? Can we say, for $u, v \in C^\infty_0(M)$, we have $$T(uv) = uT(v) + vT(u) ? $$ Otherwise, is there a ``nice'' estimate on the commutator $[u, T]$?

Also, if $T$ is a first order differential operator, then we have $$[u, [v, T]] = 0$$. Here $T$ is like a first order differential operator in a sense, though not quite. Would we still have the above result though?