Author

Luqi Wang

Date of Award

7-1997

Degree Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Supervisor

Eric Sawyer

Abstract

The theory of pseudo-differential operators is one of the most important tools
in modern mathematics. It has found important applications in many mathematical
developments. It was used in a crucial way in the proof of the Atiyah-Singer
Index theorem in [AtSi] and in the regularity of elliptic differential equations.
In the theory of several complex variables, pseudo-differential operators
are indispensable in studying the [symbol removed]-Neumann problem. The theory of subelliptic
and hypoelliptic differential operators achieved its current satisfactory
state largely because of pseudo-differential operators. In the solution to the
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
properties and the compositions of the associated special pseudo-differential
operators.

In this thesis, we consider the mapping properties and symbolic calculus
of an important class of pseudo-differential operators, the symbolic class of
Hörmander type with rough coefficients. We will prove some new results for
these operators. These operators arise naturally from problems in nonlinear partial differential equations. After the introduction of the classical symbol
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
[BeFe]). It is observed by Guan-Sawyer in [GuSa1] that the oblique derivative
problem can be reduced to the problem of pseudodifferential equations on the
boundary with a parametrix in the class [symbol removed]. That discovery led them to establish complete optimal regularity for the oblique derivative problem with
smooth data. Later, they used the class [symbol removed] to study some nonlinear oblique
derivative problems in [GuSa2]. While observing that the symbols arising here
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].

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 having
1 degree less smoothness but no loss of order. Moreover this property persists
for each of the symbols τ₁ and τ₂, etc., resulting in such symbols enjoying the
mapping properties of the better behaved class [symbol removed].

There have been many developments regarding the mapping properties
and 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 for
Hʳp sobolev spaces (Theorem 2.3.2) and finally we consider pseudo-differential
operators of symbol class [symbol removed], and prove that they have better mapping
properties (Theorem 2.3.3).

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