Pseudo-Differential operators

Abstract

<p>The theory of pseudo-differential operators is one of the most important tools<br />in modern mathematics. It has found important applications in many mathematical<br />developments. It was used in a crucial way in the proof of the Atiyah-Singer<br />Index theorem in [AtSi] and in the regularity of elliptic differential equations.<br />In the theory of several complex variables, pseudo-differential operators<br />are indispensable in studying the [symbol removed]-Neumann problem. The theory of subelliptic<br />and hypoelliptic differential operators achieved its current satisfactory<br />state largely because of pseudo-differential operators. In the solution to the<br />local 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 mapping<br />properties and the compositions of the associated special pseudo-differential<br />operators.</p> <p>In this thesis, we consider the mapping properties and symbolic calculus<br />of an important class of pseudo-differential operators, the symbolic class of<br />Hörmander type with rough coefficients. We will prove some new results for<br />these operators. These operators arise naturally from problems in nonlinear partial differential equations. After the introduction of the classical symbol<br />class [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<br />[BeFe]). It is observed by Guan-Sawyer in [GuSa1] that the oblique derivative<br />problem can be reduced to the problem of pseudodifferential equations on the<br />boundary with a parametrix in the class [symbol removed]. That discovery led them to establish complete optimal regularity for the oblique derivative problem with<br />smooth data. Later, they used the class [symbol removed] to study some nonlinear oblique<br />derivative problems in [GuSa2]. While observing that the symbols arising here<br />lie in the symbol class [symbol removed], P. Guan and E. Sawyer [GuSal] discovered that such symbols actually behave much better than where [symbol removed].</p> <p>Indeed,</p> <p>[equation removed]</p> <p>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 having<br />1 degree less smoothness but no loss of order. Moreover this property persists<br />for each of the symbols τ₁ and τ₂, etc., resulting in such symbols enjoying the<br />mapping properties of the better behaved class [symbol removed].</p> <p>There have been many developments regarding the mapping properties<br />and compositions of symbols in the class [symbol removed]. Specifically, the works of C.<br />Fefferman, C. Fefferman and E. Stein, A. Calderon and R. Vaillancourt, R.<br />Coifman-Y. Meyer, A. Miyachi, we refer to [St2] for the complete references.<br />Our results in this paper can be viewed as a further step in this direction.</p> <p>In this thesis mapping properties of pseudo-differential operator are studied in various symbol classes.</p> <p>In the first result (Theorem 2.3.1) we consider the symbol class [symbol removed] and obtain L2 results extending those of [CoMe].</p> <p>For the symbols in the class [symbol removed], mapping properties are obtained for<br />Hʳp sobolev spaces (Theorem 2.3.2) and finally we consider pseudo-differential<br />operators of symbol class [symbol removed], and prove that they have better mapping<br />properties (Theorem 2.3.3).</p>