The theory of pseudo-differential operators is one of the most important toolsin modern mathematics. It has found important applications in many mathematicaldevelopments. It was used in a crucial way in the proof of the Atiyah-SingerIndex theorem in [AtSi] and in the regularity of elliptic differential equations.In the theory of several complex variables, pseudo-differential operatorsare indispensable in studying the [symbol removed]-Neumann problem. The theory of subellipticand hypoelliptic differential operators achieved its current satisfactorystate largely because of pseudo-differential operators. In the solution to thelocal solvability problem for differential equations by Beals-Fefferman [BeFe], pseudo-differential operators played the key role. Many boundary value problems for differential equations can be reduced to pseudo-differential equations, see for example, Hörmander [Hor2]. Roughly speaking, almost everything involving pseudodifferential operators can be reduced to two parts: the mappingproperties and the compositions of the associated special pseudo-differentialoperators. In this thesis, we consider the mapping properties and symbolic calculusof an important class of pseudo-differential operators, the symbolic class ofHörmander type with rough coefficients. We will prove some new results forthese operators. These operators arise naturally from problems in nonlinear partial differential equations. After the introduction of the classical symbolclass [symbol removed] in [KohNi], Hormander considered symbolic class [symbol removed] in [Horl]. Eventually, such classes of pseudodifferential operators played a key role in the local solvability problem for differential operators (see Beals-Fefferman[BeFe]). It is observed by Guan-Sawyer in [GuSa1] that the oblique derivativeproblem can be reduced to the problem of pseudodifferential equations on theboundary with a parametrix in the class [symbol removed]. That discovery led them to establish complete optimal regularity for the oblique derivative problem withsmooth data. Later, they used the class [symbol removed] to study some nonlinear obliquederivative problems in [GuSa2]. While observing that the symbols arising herelie in the symbol class [symbol removed], P. Guan and E. Sawyer [GuSal] discovered that such symbols actually behave much better than where [symbol removed]. Indeed, [equation removed] where [symbol removed], and [symbol removed]. Thus τ decomposes into two pieces, one term having order worse by ½ but no loss in smoothness, another term having1 degree less smoothness but no loss of order. Moreover this property persistsfor each of the symbols τ₁ and τ₂, etc., resulting in such symbols enjoying themapping properties of the better behaved class [symbol removed]. There have been many developments regarding the mapping propertiesand compositions of symbols in the class [symbol removed]. Specifically, the works of C.Fefferman, C. Fefferman and E. Stein, A. Calderon and R. Vaillancourt, R.Coifman-Y. Meyer, A. Miyachi, we refer to [St2] for the complete references.Our results in this paper can be viewed as a further step in this direction. In this thesis mapping properties of pseudo-differential operator are studied in various symbol classes. In the first result (Theorem 2.3.1) we consider the symbol class [symbol removed] and obtain L2 results extending those of [CoMe]. For the symbols in the class [symbol removed], mapping properties are obtained forHʳp sobolev spaces (Theorem 2.3.2) and finally we consider pseudo-differentialoperators of symbol class [symbol removed], and prove that they have better mappingproperties (Theorem 2.3.3).

Doctor of Philosophy (PhD)