A note on the property of infinitely differentiable functions (Q1862719)

Consider infinitely differentiable functions \(f\) on \(\mathbb{R}\) such that for some \(p\in[1,\infty]\) all derivatives \(f^{(n)}\) lie in \(L_p\). It is known [\textit{Ha Huy Bang}, Proc. Am. Math. Soc. 108, 73-76 (1990; Zbl 0707.26015)] that \(\lim_{n\to\infty} \|f^{(n)}\|_p^{1/n}\) exists and equals \(\sigma_f:= \sup\{|\xi|: \xi\) in the support of \(\widehat f\}\), the Fourier transform \(\widehat f\) being taken in the distributional sense. The author considers the amalgam \((L_p,\ell_q)\) for \(1\leq p\), \(q\leq\infty\). It consists of the locally integrable functions \(f\) on \(\mathbb{R}\) for which the expression \[ \|f\|_{p,q}: =\left[\sum^\infty_{n= -\infty}\left( \int_n^{n +1} |f|^p \right)^{q/p} \right]^{1/q} \] is finite. He proves that if all derivatives of the infinitely differentiable function \(f\) lie in \((L_p, \ell_q)\), then \(\lim_{n\to\infty} \|f^{(n)} \|^{1/n}_{p,q}\) exists and equals \(\sigma_f\).