On a property of infinitely differentiable functions
Let \(f\in\mathcal C^\infty(\mathbb R)\) be such that \(f^{(n)}\in L^p(\mathbb R)\), \(n=0,1\dots\), where \(1\leq p\leq\infty\). \textit{Ha Huy Bang} [Proc. Am. Math. Soc. 108, No. 1, 73-76 (1990; Zbl 0707.26015)] showed that there always exists the limit \(d_f=\lim_{n\to\infty}\|f^{(n)}\|_p^{1/n}\) which is equal to \(\sigma_f:=\sup\{|\xi|:\;\xi\in\text{supp} \hat f(\xi)\}\), \(\hat f\) denoting the Fourier transform of \(f\). This result has been extended to the case of any Orlicz norm by \textit{Ha Huy Bang} and \textit{M. Morimoto} [Tokyo J. Math. 17, No. 1, 141-147 (1994; Zbl 0813.46019)]. Orlicz norms are generated by convex functions. In this paper, the author extends Ha Huy Bang's theorem to the case of a norm generated by a concave function \(\Phi\): \[ \|f\|_{N_\Phi}:=\int_0^\infty\Phi(\text{mes}\{x:\;|f(x)|>y\}) dy, \] which is essentially different from the Orlicz norm case and leads to overcoming technical difficulties. The author also proves a \(2\pi\)-periodic function version of the above result.
\(L^p\)-infinitely differentiable functions
scientific article; zbMATH DE number 1406561