Edge Operators in Geometry

Abstract

Differential operators with degeneracies of various sorts have been studied by a large number of mathematicians, with results ranging from those concerning very special operators to those concerning analytic and algebraic properties of general classes of operators. Much of the analysis has focused on elliptic operators, and the main concern has been to show that many properties usually associated with elliptic operators or elliptic boundary problems have analogues in a more general context. In this note we describe how a certain class of operators with uniformly controlled degeneracies may be analyzed. These are called edge operators because they arise when nondegenerate elliptic operators are written in polar coordinates around an edge of a domain, or indeed around any distinguished submanifold; however as we shall indicate, they also arise in many other geometrically natural situations. Their analysis is undertaken by constructing an algebra of pseudodifferential operators sharing the same type of degeneracy which is large enough to contain parametrices for many of the elliptic elements of the original ring of edge differential operators. These parametrices are constructed explicitly, and this allows us to conclude a great deal of detailed information about solutions of the original equations. Operators in a somewhat restricted subclass of the class of elliptic operators have very good analytic properties on a scale of weighted Sobolev or Hölder spaces for all but a discrete set of values of the weight parameter — for many values of the weight parameter the operators are semi-Fredholm, with infinite dimensional kernel or cokernel, although occasionally it will have neither hence be Fredholm. On the other hand for certain values of the weight parameter it may have both, hence the most we can conclude is that the range is closed. Simultaneously we obtain a structure theory for the generalized inverses for these operators as well as the ‘Bergman’ projectors onto the kernel or cokernel.

The author was partially supported by an NSF Postdoctoral Fellowship