New classes of function spaces and singular operators

This article is dedicated to the memory of Garnik Al’bertovich Karapetyan and it contains a review of results obtained by G. A. Karapetyan and his colleagues within the joint Russian–Armenian project of RFBR. In the first section, we look at multi-anisotropic spaces which were intensively studied by Karapetyan and his students. The second section is devoted to a new class of singular Hausdorff and Hausdorff–Berezin operators. In the third section, we study the connection between real function spaces and operator algebras in a Hilbert space, established by means of a quantization procedure. UDK: 517.518.