Para-Differential Operators in Spaces of Triebel-Lizorkin and Besov Type

In this paper we study the regularity of solutions of nonlinear partial differential equations. Here we shall extend results of J. M. BONY [1] to Besov and Triebel Lizorkin spaces. J. M. BONY introduced in [1] the method of para-differential operators in order to prove some theorems about nonlinear partial differential equations. He considered solutions in generalized Sobolev. space H2 8 (s > 0) and in Holder spaces C, where s > 0 is not: an integer. In recent years, Y. MEYER extended the regularity results obtained by J. M. BONY to solutions in other classes of'functions spaces, cf. [3-5]. -. In this paper we consider the Besov spaces B ,q and Triebel-Lizorkin spaces in the Euclidean n-space R5 In Chapter 1 we introduce these spaces, which contain many classical spaces as special cases. In the spaces F, q and B, q we study the regularity of solutions of nonlinear partial differential equations and extend the results of J. M. BONY and Y. MEYER. In order to prove our results we use the method of dyadic decomposition and maximal functions, multiplication properties of Besov and Triebel-Lizorkin spaces and results with respect to the boundedness of pseudodifferential operator of the "exotic" class L. Applying the theory of para-differential operators introduced by J. M. BONY [1] and Y. MEYER [3-5], we are able to prove our regularity results. All immaterial positive numbers are denoted by ' c or c' etc.