Theory of pseudo-differential operators of ultradifferentiable class
Waichirô Matsumoto · Kyoto journal of mathematics · 1987
compact in R', 3 C, R > 0 , V a Z ' , , sup 1(-) 6 .fI u e K ax W e call this the ultradifferentiable space of class {M f i } ( = t h e ul.d.space o f class { M } ) .A typical ul.d.class is a Gevrey class, i.e., M = n ! ( v .:1 ) .When v=1, it is the real analytic class.In the study of the spaces of the admissible data in the Cauchy problems, we were led to introduce some ul.d.classes wider than any Gevrey c la s s .(See W. Matsumoto [29], [30] and [32].)In a systematic treatment of the problems in a ul.d.class, a theory of pseudo-differential operators (=ps.d.op's) of ul.d.class is required.However, it is not yet well investigated except the case of the Gevrey classes.The theory of ps.d.op's of C -class has been well studied.(See J.J. Kohn and L. Nirenberg [20], L. Heirmander [14], H. Kumano-go [26] and [27], etc.) Since their formulations are slightly different each other, we mean, in this paper, Kumano-go's theory [27] by the theory of ps.s.op's of C -class.Hereafter, we shall try to construct a theory of ps.d.op's of general ul.d.class corresponding to that of C -class.We are mainly interested in the ul.d.classes wider than any Gevrey class because that of Gevrey classes has been well investigated.(SeeL.