An integral operator connected with the Helmholtz equation

AbstractLet Lu be the integral operator defined by (Lkϑ)(x, y) = ∝ s ∝ ϑ(x′, y′)(eikϱϱ) dx′ dy′, (x, y) ϵ S where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ⩾ 0, Im k ⩾ 0, and ϱ2 = (x −x′)2 + (y − y′)2. We define q(x, y) = [dist((x, y), ∂S)]12, (x, y) ϵ S; L2(q, S) = {ƒ : ∝ s ∝ ¦ ƒ(x, y)¦2 q(x, y) dx dy < ∞}; W21(q, S) = {ƒ : ƒ ϵ L2(q, S), ∂ƒ∂x, ∂f∂y ϵ L2(q, S)}, where in the definition of W21(q, S) the derivatives are taken in the sense of distributions. We prove that Lk is a continuous 1-l mapping of L2(q, S) onto W21(q, S)