POST'S PSEUDO-DIFFERENTIAL OPERATOR IN S-TYPE SPACES

Abstract

During the last few decades, the theory of fractional differentiation and pseudo-differential operators, which naturally generalize and extend the concepts of classical derivative and differential operations, has been rapidly developing. The reason for this development is primarily the close connection of pseudo-differential operators and fractional differentiation with important problems of analysis and modern mathematical physics. It turned out that such player operators play an important role in the theory of analytical boundary-value problems (in the study of the index of the problem, in reduction to the boundary of the region, etc.), in microlocal analysis, in the theory of random processes, with the help of fractal differentiation operators heat-diffusive processes in porous media, etc. There are different approaches to the generalization of the classical derivative, the implementation of which gave rise to a variety of fractional differentiation and pseudodifferentiation operations. In this connection, there is a natural need for a comparative characterization of these generalizations, which is convenient to conduct through the prism of the classical form of fractional differentiation on elements with "sufficiently good" properties. In addition, the representation of this or that pseudo-differentiation operation in such a classical form makes it possible to use a rather convenient Fourier transform apparatus for the analysis of problems with these operations. In this work, the question of the possibility of representation in S type spaces of I.M. Gelfand is investigated. and Shilova G.E. pseudo-differential operator E. Post a(Dx) in the classical form of fractional differentiation, provided that its symbol a(·) is a convolution in the original space.