Mellin Convolution Equations

In the present paper we collect results on Mellin convolution equations (MCEs), obtained recently. We start with the motivation and in the first section are exposed MCEs which encounter in applications. Further we expose the boundedness results of corresponding operators in the Lebesgue space with weight. Fourier convolution (Wiener-Hopf) equations are defined and their connection to MCEs is described. Solvability and Fredholm properties and the index formulae for MCEs are formulated in terms of the symbol functions assigned to them. Results on the Banach algebra generated by Mellin and Wiener-Hopf operators in the Lebesgue space are exposed: The symbol function is defined and the Fredholm criteria is formulated, the index formula is written. In conclusion we expose relatively new results on Fredholm property and index of MCEs in the Bessel potential spaces. These results are applied to the mixed boundary value problem (Mixed BVP) for the Laplace equation in an angle, which reduces to an equivalent MCE. Results on the Fredholm property, solvability and index of such equations are formulated and, in conclusion, applied to the above formulated mixed BVP for the Laplace equation.