Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels (Q2927725)

The Mellin convolution equation together, under certain conditions on the kernel, is studied while investigating a boundary value problem of PDE in the planar domains with angular points of different systems such as Lame, Maxwell, Cauchy-Riemann and others. The Mellin convolution operator is endowed and with admissible meromorphic kernels, the boundedness for the Bessel potential space (known as fractional Sobolev space, subspace of tempered Schwartz space of distributions) is proved by observing the Mellin and Fourier convolution operators with discontinuous symbols acting on Lebesgue space. Moreover, the commutant of the Mellin convolution operator is defined with the help of a few examples. Further, the Banach algebra generated by Mellin and Fourier convolution operators is examined. Simultaneously, the results obtained are employed to describe Fredholm properties and the index of Mellin convolution operators with admissible meromorphic kernels in Bessel potential spaces.