Solution of integral equations with the Legendre function in the kernels in \(L_{\eta,r}\)-spaces (Q2850794)

Four integral equations of the first kind on the positive half axis involving Legendre function in the kernels are studied. Using the representations of the integral operators in the left-hand sides of considering equations as compositions of fractional integral operators with power weights, the conditions for their boundedness from one weight spaces of summable functions into another spaces are proved. These results are applied to deduce explicit solutions of the considered integral equations in \({\mathfrak L}_{\eta,r}\).