Let $M$ be a closed Riemannian manifold and denote by $H^s(M), \, s\in \mathbb{R} $ the standard Sobolev spaces on $M$ defined using powers of $1+\triangle$. Let $J_n: \mathcal{D}'(M)\rightarrow \mathcal{D}(M),\,n \in \mathbb{R}$ be a sequence of mollifiers on $M$.

Fix $s<0$ and $p\geq 2$. What is the best known upper bound for the growth (as $n \rightarrow \infty$) of the norms $\|J_n\phi \|_{L^p(M)}$ for all distributions $\phi \in H^s(M)$, in terms of $s$, $p$ and the dimension of $M$?

(Actually I will be happy with an answer only in the special case when $M$ is the standard $d$-dimensional sphere.)