Recall the Kato-Ponce estimate for fractional powers of the operator $J = (1-\Delta)$, $$ \| J^s(fg) \|_{L^r} \lesssim \| J^s f \|_{L^{p_1}} \| g \|_{L^{q_1}} + \| J^s g \|_{L^{p_2}} \| f \|_{L^{q_2}}, $$ where $\frac{1}{r} = \frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2}$, $1<r<\infty$, $1<p_1, p_2, q_1, q_2 <\infty$, and $s>0$ need not be an integer. On $\mathbb{R}^n$ this is well-established, see e.g. https://arxiv.org/pdf/1609.01780 .
Is it known that this estimate also holds on smooth compact manifolds?
In this paper (Lemma 3.1) Sogge states the above estimate on smooth compact manifolds, but references the original paper of Kato-Ponce which is on $\mathbb{R}^n$. Is it somehow obvious that it extends easily to compact manifolds?
