One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ ($C$ probably depends on $s$) such that we have an estimate of the form
$$\|u\|_{s+l} \leq C(\|u\|_s + \|Lu\|_s)$$
My question is whether the coefficient $s$ on $\|u\|_s$ can be weakened over compact Riemannian manifolds. The question is motivated by a result that we can find a pseudo-differential operator $S$ of order $-l$ such that $SL - I = K$ continuously maps compactly supported distributions to smooth functions.
This would seem to imply we can prove a form of the above estimate by writing
$$u = SLu - Ku$$
Then using the continuity of inclusions $H^r \to \mathcal{E}' = \mathcal{D}'$ (compactness of $M$ key here) and $C^\infty \to H^{s+l}$ to bound the latter term, and the operator norm of $S$ to bound the former. This seems disturbing to me as it would imply we just need any information on a Sobolev norm of $u$, not the $s$ norm in particular.