On a class of pseudo-differential operators in IRâ ¿

Abstract

The class of pseudo-differential operators with symbols from Sm (superscript) po̧̧ (subscipt)(Ωx IR⠿) has been extensively studied.The main assumption which characterises this class of symbols is that a(x,Ȩ) є Sm (superscript)po̧̧ (subscipt)(Ωx IR⠿) should have a polynomial growth in the Ȩ variable only. The x-variable is controlled on compact subsets of Ω. A polynomial growth in both the x and Ȩ variables on a C°°(lR²⠿) function a(x,Ȩ) gives rise to a different class of symbols and a corresponding class of operators. In this work, such symbols and the action of the operators on the functional spaces S(lR⠿) , S'(lR⠿) and the Sobolev spaces Qs (superscript) (lR⠿) (s є lR⠿) are studied. A study of the calculus (i.e. transposes, adjoints and compositions) and the functional analysis of these operators is done with special attention to L-boundedness and compactness. The class of hypoelliptic pseudo-differential operators in IR⠿ is introduced as a subclass of those considered earlier.These operators possess the property that they allow a pseudo- inverse or parametrix. In conclusion. the spectral theory of these operators is considered. Since a general spectral theory would be beyond the scope of this work, only some special cases of the pseudo-differential operators in IR⠿ are considered. A few applications of this spectral theory are discussed