John C. Polking

Removable singularities of solutions of linear partial differential equations

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 Riemanns 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). Bochners 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 Bochners 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 Bochners 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(~)

A Survey of Removable Singularities

Suppose \( P\left( {x,D} \right) = \sum\limits_{{1\alpha 1m}} {{{a}_{\alpha }}\left( x \right){{D}^{\alpha }}} \) is a linear partial differential operator defined on an open set ⋂ ⊂ ℝ n, and that A ⊂ ⋂ is closed.

Bounded point evaluations and approximation in Lp by solutions of elliptic partial differential equations

Let P(x, D) be an elliptic operator of order m with infinitely differentiable coefficients defined in an open subset Ω of Rn. Let E ⊂ Ω be a compact set and denote by N(E) the set of distributions defined in a neighborhood of E which satisfy the homogeneous equation P(x, D)f = 0 in this neighborhood. A point x0 ϵ E is called a bounded point evaluation for N(E) ⊂ Lp(E) if evaluation at x0 is continuous on N(E) in the Lp norm. In this paper we make a complete analysis of the relationship between the existence of bounded point evaluation and the density of N(E) in Lp(E), 1 ⩽ p < ∞. One of the two main tools is a characterization of bounded point evaluations in terms of Bessel capacities. Comparison with the known analysis in terms of Bessel capacities of when N(E) is dense in Lp(E) enables us to relate the problems. The final information comes from the second main tool, the construction of a set E which has no bounded point evaluations but for which N(E) is dense in Lp(E).

The Cauchy-Riemann Equations in Convex Domains

The use of explicit kernels to solve the Cauchy-Riemann equations has by now a long history. The injection of the Cauchy-Leray-Fantappie-Koppelman formalism into this area by Ramirez [RE], Grauert and Lieb [GL], Lieb [L], and Henkin [H] began the push to find solutions which satisfied optimal estimates. The first results in this direction on weakly pseudo convex domains were due to Range [RR]. Recently there has appeared the work of Bonneau and Diederich [BD], which seems to be the first result where no finite type condition is assumed.