Necessary & Sufficient Condition for first Order Differential Operator

This book consists of five chapters. The first chapter is devoted to the historical background and introductory concepts. Partial complex differentiations in the classical sense and in the sense of Sobolev generalized analytic functions associated differential operators associated spaces and interior estimates are introduced in Chapter II. The interior estimate for holomorphic functions in the supremum-norm is also obtained. In Chapter III first the concept of scales of Banach spaces are presented. In Chapter IV initial value problems defined by Son and Tutschke [25].In the last chapter an initial value problem for a first order evolution equation defined by N. Q. Hung [13] in the space of generalized regular functions in Quaternionic Analysis is discussed. Hung has proven only sufficient conditions for the related differential operators to be associated. We have obtained not only sufficient but also necessary conditions for the underlined differential operators to be associated [1].