Removable singularities of weak solutions to linear partial differential equations

Abstract

Suppose that P(x, D) is a linear differential operator of order m > 0 with smooth coefficients whose derivatives up to order m are continuous functions in the domain G ⊂ ℝn (n ≥ 1), 1 ∞, s > 0, and q=p/(p − 1). In this paper, we show that if n, m, p, and s satisfy the two-sided bound 0 ≥ n − q(m − s)< n, then for a weak solution of the equation P(x, D)u=0 from the Sharpley-DeVore class C p s (G)loc, any closed set in G is removable if its Hausdorff measure of order n − q(m − s) is finite. This result strengthens the well-known result of Harvey and Polking on removable singularities of weak solutions to the equation P(x, D)u=0 from the Sobolev classes and extends it to the case of noninteger orders of smoothness.