On a class of pseudo-differential operators in IRâ ¿ - Matjila, D M (Rhodes University, Faculty of Science, Department of Mathematics, 1988)
The class of pseudo-differential operators with symbols from Sm (superscript) poÌ§Ì§ (subscipt)(â„¦x IRâ ¿) has been extensively studied.The main assumption which characterises this class of symbols is that a(x,EÌ§) Ñ” Sm (superscript)poÌ§Ì§ (subscipt)(â„¦x IRâ ¿) should have a polynomial growth in the EÌ§ variable only. The x-variable is controlled on compact subsets of â„¦. A polynomial growth in both the x and EÌ§ variables on a CÂ°Â°(lRÂ²â ¿) function a(x,EÌ§) 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(lRâ ¿) , S'(lRâ ¿) and the Sobolev spaces Qs (superscript) (lRâ ¿) (s Ñ” lRâ ¿) 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 IRâ ¿ 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 IRâ ¿ are considered. A few applications of this spectral theory are discussed
Keywords: Pseudodifferential operators; Operator theory