Roland Duduchava

Mellin Convolution Operators in Bessel Potential Spaces with Admissible Meromorphic Kernels

abstract:
The paper is devoted to Mellin convolution operators with meromorphic kernels in Bessel potential spaces. We encounter such operators while investigating boundary value problems for elliptic equations in planar 2D domains with angular points on the boundary.
Our study is based upon two results. The first concerns commutants of Mellin convolution and Bessel potential operators: Bessel potentials alter essentially after commutation with Mellin convolutions depending on the poles of the kernel (in contrast to commutants with Fourier convolution operatiors.) The second basic ingredient is the results on the Banach algebra $\mathfrak{A}_p$ generated by Mellin convolution and Fourier convolution operators in weighted $\mathbb{L}_p$-spaces obtained by the author in 1970's and 1980's. These results are modified by adding Hankel operators. Examples of Mellin convolution operators are considered.

Mathematics Subject Classification: 47G30, 45B35, 45E10

Key words and phrases: Fourier convolution, Mellin convolution, Bessel potentials, meromorphic kernel, Banach algebra, symbol, fixed singularity, Fredholm property, index