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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.)

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  • $\begingroup$ Shouldn't the bound depend on how you choose your sequence of mollifiers? (Just re-index $K_n = J_{2^{n}}$ and you get a different growth rate.) Maybe I am misunderstanding your question: what do you mean by "upper bound for the growth"? $\endgroup$
    – Willie Wong
    Mar 3, 2017 at 3:47
  • $\begingroup$ Yes, certainly the growth would depend on how the mollifiers are scaled. My question is not that precisely stated indeed, sorry about that. What I mean is: given a specific sequence of mollifiers, how does one obtain an upper bound (hopefully sharp) of these norms which should have the form: number depending on $n$ times a Sobolev norm of $\phi$. $\endgroup$
    – S.Z.
    Mar 3, 2017 at 22:25

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