Sequence space representations of \({\mathcal D}((M_k),L^p)\) and \({\mathcal D}(\{M_k\}, L^p)\) (Q2725635)

For \(r\in [1,\infty[\) and a sequence \((M_p)_{p\in\mathbb{N}_0}\) of positive members denote by \({\mathcal D}((M_p), L^r)\) (resp. \({\mathcal D}(\{M_p\}, L^r)\) the space of all \(C^\infty\)-functions \(f\) on \(\mathbb{R}^n\) for which for each \(h> 0\) and \(c>0\)) such that NEWLINE\[NEWLINE\|f^{(\alpha)}\|_{L^r}\leq Ch^{|\alpha|} M_{|\alpha|} \forall\alpha\in \mathbb{N}^0_0.NEWLINE\]NEWLINE These spaces are endowed with their natural Fréchet (resp. (DF)) topology. Using arguments that were introduced by Valdivia and Vogt, the authors show: If \((M_p)_{p\in \mathbb{N}_0}\) satisfies the conditions (M1), (M2\('\)) and (M3\('\)) of Komatsu then NEWLINE\[NEWLINE{\mathcal D}((M_p), L^r)\cong L_M(k^{1/n}, \infty)\widehat{\otimes} \ell^r,\;{\mathcal D}(\{M_p\}, L^r)\cong L_M(k^{1/n}, 0)'\widehat{\otimes} \ell^r,NEWLINE\]NEWLINE where \(M\) denotes the associated function of \((M_p)\) (extended to \(\mathbb{R}\) as an odd function) and where NEWLINE\[NEWLINEL_f(\alpha, R):= \{x\in \mathbb{C}^{\mathbb{N}}:\|x\|_\rho:= \sum|x_j|\exp(f(\rho\alpha_j))< \infty\forall\rho< R\}.NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0960.00019].