Let G be a locally compact group. For a compact generating neighborhood \(\Omega\) of the identity and \(g\in G\), put \(| g| =\inf \{n\); \(g\in \Omega^ n\}\) and \(\gamma (t)= Haar\) measure of \(\{| g| \leq t\}\). Write \(F^{*p}\) for the pth convolution power F*F*...*F (p times).
I. If F is a sufficiently positive, symmetric, compactly supported function, \(F\in L^ 1\cap L^ 2\), \(n\geq 0\), and if \(\| F^{*p}\|_{\infty}=O(p^{-n/2})\), then this inequlity also holds for every other \(F_ 1\) which satisfies the stated conditions on F.
II. If there exists \(C>0\) and \(a\geq 0\) such that \(\gamma\) (t)\(\geq C t^ a\), then for every F as above, \(\| F^{*n}\|_{\infty}=O(n^{- b/2})\), \(\forall b<a\). The author says he can prove this with \(b=a.\)
III. A converse of II.
The proofs presented here are an intricate combination of inequalities developed elsewhere mainly by the author, involving infinitely decomposable functions, a convergent iteration process due to J. Moser, a certain convolution semigroup, Dirichlet forms,....