Potentials and Removability of Singularities

We shall deal with results concerning removability of singularities of certain solutions of partial differential equations. Let us first briefly recal l what one usually understands by a set IT of removable singularities. Let U ⊂ R N be an open s et and consider a differential operator of the form 1 $$ {\rm P(D)} = \sum\limits_{a \in {\rm M}} {{\rm a}_a \,{\rm D}^a } $$ acting on distributions in U; here M is a finite set of multi-indices $$ a = [a_1,...,a_{\rm N} ] $$ and we write, as usual $$ D = D_{l}^{{{{a}_{l}}}} \cdots D_{N}^{{{{a}_{N}}}} $$ where Dk=-i∂k and ∂k denotes differentiation with respect to the k - th variable. For the sake of simplicity we shall always suppose that a a are infinitely differentiable functions in U or complex constants. Given a class K(U) of distributions in U, then a relatively closed subset F⊂U is termed removable for K(U) with respect to P(D) provided P(D)u = 0 in the whole U whenever u∈K(U) satisfies P(D)u = 0 in U—F. Results concerning removability of singularities usually introduce an adequate measure of massiveness of a set and show that sets which are not very massive are re-movable. As an example we shall mention here a theorem of R.HARVEY and J. POLKING [27] dealing with the class $$ \rm K(U) = K\delta (U) $$ consisting of functions satisfying locally in U the Hölder condition with exponent $$ \delta,0