Pseudo-differential operators with nonregular symbols

Abstract

where a power of 27~ is ignored and G(tr ,..., 5,) is the Fourier transform of U(Xi ,...) x,). We shall call xi the space variables and fi the dual variables. The notion of pseudo-differential operators has grown in recent years out of attempt to obtain sharp a priori estimates for the solutions of the partial differential equations. Since its discovery it has been found to be one of the most powerful tools in attacking various problems in partial differential equations such as the existence and uniqueness of the boundary value problems [I], regularity of the solutions of the partial differential equations [2], solvability of a general partial differential operator [3], etc. The basic calculus formulas for the pseudo-differential operators are due to J. Kohn and L. Nirenberg [4] as a development of the theory of Singular Integral Operators of Calderon and Zygmund. They constructed an algebra of pseudo-differential operators with symbols having bounded derivatives in the space variables and proved the boundedness of such operators. However their algebra is too restrictive for many purposes. One of the