Oscillatory integrals of symbols of pseudo-differential operators on $R^n $ and operators of Fredholm type

Abstract

Introduction. In this paper we shall introduce the oscillatory integral o the orm 0--[[e-'p(, x)dxd or C-unction p(, x)o class (defined in Section 1), and by using this integral study the algebra o pseudo-differential operators o2 class S?.,., 0=<3=<p=<1, 31, whose basic weight function 2 2(x, ) varies even in x and may increase in polynomial order. *) The Friedrichs part P o the operator P o class S,,. will be defined as in . Then, the Lboundedness or the operator P o class S.,. for /p, can be proved by using Pr and the Calderon-Vaillancourt theorem in have to note that all the results obtained there hold even for operator-valued symbols as in Grushin [3].