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 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.• In general, Mellin convolution operators are not bounded in neither Besov nor Bessel potential spaces. Therefore, in order to study equations (2) in the spaces of interest, one has to find a subclass of multipliers with the boundedness property.

Mixed boundary value problems for the Laplace–Beltrami equation

“…are locally compact due to the Sobolev's embedding theorem and the compact perturbation does not influences the local invertibility. The second part of the assertion, concerning the solvability in the Bessel potential space settings (0.10b) and (0.16), follows from the first part and Proposition 4.3, exposed below and proved in [19,8], which states that these solvability properties are equivalent. To formulate the next theorem we need to introduce Fourier convolution and Bessel potential operators.…”

“…This accomplishes the proof of the first part of the assertion, concerning the solvability in the Sobolev-Slobodečkii space settings (0.10a) and (0.15). The second part of the assertion, concerning the solvability in the Bessel potential space settings (0.10b) and (0.16), follows from the first part and Proposition 4.3, exposed below and proved in [19,8], which states that these solvability properties are equivalent. 2…”

“…1 Inner points of C. The investigation of the boundary integral equation system (0.13) is based on recent results on Mellin convolution equations with meromorphic kernels in Bessel potential spaces (see R. Duduchava [19], R. Duduchava and V. Didenko [8]).…”

“…In this section we expose auxiliary results from [19] (also see [11,16,8]), which are essential for the investigation of boundary integral equations from the foregoing section. Let a(ξ) be a N × N matrix function a ∈ CM 0 p (R), continuous on the real axis R with the only possible jump at infinity.…”

“…In this section we expose auxiliary results from [19] (also see [11,16,8]), which are essential for the investigation of boundary integral equations from the foregoing section.…”