Extension of a theorem of P. Wolfe on singular integral equations

Abstract

The integral operator which sends f (ζ) into

$$\int_{|\xi |< 1} {\tfrac{{f(\xi )}}{{|x - \xi |^{p - 1} }}} d\xi = g(x)(|x|< 1)$$

x and ζ ∈ R^P, is shown to map the space of functions f (ζ) on |ζ|<1, square integrable with respect to the weight (1-|ξ|²)^1/2 one to one onto the space of functions g (x) on |x|<1 which possess gradients δg(x) in the sense of distribution theory, with both g(x) and δg(x) being square integrable with respect to the weight (1-|x|²)^1/2. This extends to p≥3 a result of P. Wolfe's in the case p=2.