Characterization of the separativity of ultradifferentiable classes
Let \(\{M_ n\}^{\infty}_{n=0}\) be a logarithmicly convex and nondecreasing sequence of positive numbers. We assume that \(\lim_{n\to \infty}(M_ n)^{1/n}=\infty\). We set \({\mathcal B}_ R\{M_ n\}=\{f(x)\in C^{\infty}({\mathbb{R}}^{\ell});\| f\|_{\{M_ n\},R}\equiv \sup | f^{(\alpha)}(x)| <\infty\), where x runs over \({\mathbb{R}}^{\ell}\) and \(\alpha\) does over \({\mathbb{Z}}_+^{\ell}\}\), and \({\mathcal B}\{M_ n\}= \lim_{R\to \infty}{\mathcal B}_ R\{M_ n\}\). We call \({\mathcal B}\{M_ n\}\) the ultradifferentiable class with weight \(\{M_ n\}.\) When we consider problems on \({\mathcal B}\{M_ n\}\), the following condition sometimes plays an essential role: (S.1) \(\exists R\geq 1\), \(\forall n\), \(m\gg 1\), \(M_{n+m}\leq R^{n+m}M_ nM_ m.\) We call it the separativity condition. In this paper, we gave three equivalent conditions to (S.1). However, in order to consider a theory of pseudo-differential operators acting on \({\mathcal B}\{M_ n\}\), more various equivalent representations are required. We set \(T(r)=\sup_{n\in {\mathbb{Z}}_+}r^ n/M_ n\) \((r>0)\) and call it the associated function of \(\{M_ n\}.\) Theorem 1. The following conditions are equivalent to (S.1). (S.2) \(\exists R\geq 1\), \(\forall n\gg 1\), \((M_{2n})^{1/2n}\leq R(M_ n)^{1/n}.\) (S.3) \(\exists R\geq 1\), \(\forall n\gg 1\), \(M_{2n+1}/M_{2n}\leq RM_{n+1}/M_ n.\) (S.4) \(\exists R\geq 1\), \(\forall n\gg 1\), \(M_{n+1}/M_ n\leq R(M_ n)^{1/n}.\) (S.5) \(\exists R\geq 1\), \(\forall r\gg 1\), \(T(r)\geq T(r/R)^ 2.\) In Appendix B in [\textit{W. Matsumoto}, Theory of pseudo-differential operators of ultradifferentiable class, to appear in J. Math. Kyoto Univ.], we have proved the part which did not appear in this paper. Further, in this paper, we obtained some necessary conditions for (S.1). Theorem 2. (S.1) implies the following equivalent conditions. The converse is not always true. (S.6) \(\exists \nu >0\), \(\forall n\gg 1\), \(M_ n\leq n!^{\nu}.\) (S.7) \(\exists \nu >0\), \(\forall n\gg 1\), \((M_ n)^{1/n}\leq n^{\nu}.\) (S.8) \(\exists \nu >0\), \(\forall n\gg 1\), \(M_{n+1}/M_ n\leq n^{\nu}.\) (S.9) \(\exists \kappa >0\), \(\forall r\gg 1\), \(T(r)\geq \exp (r^{\kappa}).\) We have applied Theorems 1 and 2 to consider a theory of pseudo- differential operators acting on \({\mathcal B}\{M_ n\}\) in [loc. cit.].
scientific article; zbMATH DE number 3973605