Integral equations with diagonal and boundary singularities of the kernel (Q873756)

The authors study Fredholm and Volterra integral equations with weakly singular kernels \(K(x,y)\), \(x.y\in \Omega:=[a,b]\times[c,d]\), which occur not only on the diagonal \(x=y\) but also on the boundary \(\partial\Omega\) and are given through estimates. An example of such kernel is: \[ \left| \left(\frac{\partial}{\partial x}\right)^k\left(\frac{\partial}{\partial x} +\frac{\partial}{\partial y}\right)^l K(x,y)\right| \leq C| x-y| ^{-\nu-k} (y-a)^{-\lambda-l}(b-y)^{-\mu-l} \] where \(0\leq\nu,\lambda,\mu<1\) and \(k,l,k+l\leq m\). The weighted spaces of smooth functions having singularities on the boundary \(\partial\Omega\) are introduced and it is proved that they are optimal for solving Fredholm integral equations with singular kernels of the indicated type. These obtained results enhance earlier investigations of K. E. Arkinson, H. Brunner and others who considered the one-dimensional case.