On Mellin convolution operators in Bessel potential spaces

Abstract: Mellin convolution equations acting in Bessel potential spaces are considered. The study is based upon two results. The first one concerns the interaction of Mellin convolutions and Bessel potential operators (BPOs). In contrast to the Fourier convolutions, BPOs and Mellin convolutions do not commute and we derive an explicit formula for the corresponding commutator in the case of Mellin convolutions with meromorphic symbols. These results are used in the lifting of the Mellin convolution operators acting on Bessel potential spaces up to operators on Lebesgue spaces. The operators arising belong to an algebra generated by Mellin and Fourier convolutions acting on $\mathbb{L}_p$-spaces. Fredholm conditions and index formulae for such operators have been obtained earlier by R. Duduchava and are employed here. Note that the results of the present work find numerous applications in boundary value problems for partial differential equations, in particular, for equations in domains with angular points.