The solution of mildly singular integral equation of the first kind on a disk

The integral equation $$\int_{\left| y \right| \leqslant 1} {\frac{{F(y)}}{{\left| {x - y} \right|^\lambda }}dy = G(x)} $$ x,y ∈ E², with 0 < λ < 2 is studied. Uniqueness for integrable solutions F is established under the assumption that G is integrable. Existence of an integrable solution F is then obtained under the further assumption that G ∈ C², with an explicit solution formula being given for F in terms of integral operators acting on derivatives of G.