Removable singularities of solutions of linear partial differential equations

Abstract

Suppose P(x, D) is a linear partial differential operator on an open set ~ contained in R ~ and that A is a closed subset of ~. Given a class ~(~) of distributions on ~, the set A is said to be removable for ~(~) if e ach /E ~(~), which satisfies P(x, D ) / = 0 in ~ A , also satisfies P(x, D)/= 0 in ~. The problem considered in this paper is the following. Given a class ~(~) of distributions on ~, what restriction on the size of A will ensure that A is removable for ~(~). We obtain results for Lroc (~) (p ~< ~ ) , C(~), and Lipa (~). The first result of this kind was the Riemann removable singularity theorem: if a function / is holomorphic in the punctured unit disk a n d / ( z ) = o ( H -1) as z approaches zero, then / is holomorphic in the whole disk. Bochner [1] generalized Riemann's result by considering the class ~(~) of functions f on ~ such that ](x)=o(d(x, A) -q) uniformly for x in compact subsets of ~, and giving a condition on the size of A which insures tha t A is removable for ~(~) (Theorem 2.5 below). Bochner's theorem is remarkable in that the condition on the size of A only depends on the order of the operator P(x, D). The theorem applies, therefore, to systems of differential operators, such as exterior differentiation in R n and ~ (the Cauchy-Riemann operator) in C n. The same can be said for the other results in this paper. The proof of Bochner's theorem provided the motivation for our results. I t is interesting to note tha t a very general result (Corollary 2.4) f o r / ~ (~) (due to Li t tman [7]) is an easy corollary of Bochner's work. Here the condition on the singular set A is expressed in terms of Minkowski content. In section 4 the case of Ll~oc(~) is studied again, and results in section 2 are improved by replacing Minkowski content with Hausdorff measure. In addition, the cases C(~)