On one singular integral equation arising from the radiative transfer theory (Q2813438)

The authors study the following two-dimensional integral equation with a singularity with respect to one variable: NEWLINE\[NEWLINE (Lu)(x,t)= \int_a^b\int_{-1}^1\frac{s k(x,y)}{s-t}u(y,s)\,ds\,dy+u(x,t)+ \int_a^b\int_{-1}^1\frac{tk(x,y)}{s-t}u(y,t)\,ds\,dy=f(x,t)NEWLINE\]NEWLINE with \(x\in[a,b]\), \(t\in(-1,1)\) and a continuous kernel \(k\). Let \(D^*\) denote the class of functions of two variables that satisfy the Hölder type condition \(H^*\) in the second variable. The main result of the paper is a necessary and sufficient condition for solvability of the above equation in the class \(D^*\) under the assumption that \(f\in D^*\).