Treffer: Approximate identities and geometric means of smooth positive functions

The author generalizes several results of his own in [Acta Math. 178, No. 2, 143-167 (1997; Zbl 0898.30040)] on geometric means of Lipschitz functions in \(\mathbb{R}^n\). Let \(\Lambda_\omega\) be the set of all bounded, continuous functions in \(\mathbb{R}^n\) for which \[ \bigl|f(t_1)-f(t_2)\bigr |\leq M\omega \bigl(|t_1-t_2 |\bigr), \tag{*} \] where \(\omega\) is a certain weight on \(\mathbb{R}^+\) for which \(\omega(t)/t\) is decreasing. This space is endowed with the usual norm \(\|f\|_{\Lambda_\omega} =\|f\|_\infty +\inf\{M >0:(*)\) holds\}. Let \(\Lambda_{\omega, \log}\) be the space of all positive functions \(f\in\Lambda_\omega\) for which \(\int_{\mathbb{R}^n} \log{\|f \|_\infty \over f(t)}dt 0\) and where \(d\mu_{K,(x,y)} (t)={1\over y^n} K({t-x\over y})dt\) for some normalized kernel \(K\). One of the main results of the paper tells us that these geometric means are certain approximate identities in the sense that whenever the support of the kernel \(K\) is compact then every \(f\in \Lambda_{\omega,\log}\) satisfies \[ \sup_{x\in\mathbb{R}^n} \biggl|\bigl(G_y^{[K]} f\bigr)(x) -f(x) \biggr|\leq C\|f\|_{\Lambda_\omega}\omega(y),\;y>0 \] for some constant \(C=C(n,K, \omega)\). A certain converse to such an estimate is given. Kernels with noncompact support are considered, too.