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I am looking for a reference for the following $\epsilon$-regularity statement: let

  • $(M,g)$ be a Riemannian manifold of dimension $n$, $\Delta=dd^*+d^*d$,
  • $B_r$ denotes a ball of radius $r$ around a fixed point in $M.$

Then $\exists\epsilon>0,C\leq\infty$ such that if $f$ is a non-negative function satisfying the inequality $$ \Delta f\leq f^2, $$ then we have $$ \|f\|_{L^{\frac{n}{2}}(B_{\frac{r}{2}})}\leq\epsilon\implies\|f\|_{L^\infty(B_\frac{r}{2})}\leq C\|f\|_{L^{\frac{n}{2}}(B_r)} $$ Note, the $n=4$ case is given here.

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  • $\begingroup$ It seems that the set of functions satisfying $\Delta f\leq f^2$ can not be very big. Because your question remind me of a theorem of Grothendieck which says : if $|.|_{\infty}$ is dominated by $|.|_2 $ on a a subspace of $L^{\infty}\cap \ell^2$ then the subspace is necessarily a finite dimensional space $\endgroup$
    – Ali Taghavi
    Commented Sep 10 at 16:50
  • $\begingroup$ on the other hand the condition $\Delta f\leq f^2 $ is preserved by "addition" $\endgroup$
    – Ali Taghavi
    Commented Sep 10 at 16:52
  • $\begingroup$ So for a compact Riemannian manifold a natural question is to assign an upper bound for the maximum number of independent positive functions $f_1, f_2,\ldots,f_n$ with $\Delta f_i \leq f_i^2$ $\endgroup$
    – Ali Taghavi
    Commented Sep 10 at 16:56
  • $\begingroup$ @AliTaghavi I do not see the connection between $\Delta f \leq f^2$ and Grothendieck's result, other than the presence of an inequality and something being squared. How is Grothendieck's result relevant to what the OP is actually asking about? $\endgroup$
    – Yemon Choi
    Commented Sep 10 at 18:50
  • $\begingroup$ @YemonChoi the domination of norm infinity by norm-2. A result attributed to Grothendieck. I found it in Royden real analysis. $\endgroup$
    – Ali Taghavi
    Commented Sep 10 at 19:21

1 Answer 1

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First, it is straightforward to adapt Haslhofer's proof to higher dimensions, using the Sobolev inequality: For any compactly supported $u$ such that $\|\nabla u\|_2 < \infty$, $$ \|u\|_{\frac{2n}{n-2}} \le C\|\nabla u\|_2. $$ I encourage you to try.

If you google epsilon regularity, you’ll find several explanations of it, including anther Math Overflow questionn.

The beauty of this simple and elegant proof is that it does not require coordinates and the dependence of the constant in the inequality on the geometry of the manifold is purely through the Sobolev constant.

There must be other good references for this, but it is proved in Appendix B of my paper:

Convergence of riemannian manifolds with integral bounds on curvature. II Annales scientifiques de l’É.N.S. 4e série, tome 25, no 2 (1992), p. 179-199

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