Operator theory and functional spaces

Subject/research area

Research on the theory of operators and functional spaces cover problems of evaluations of norms of operators of classical analysis and their generalizations in various functional spaces: shared ideal and Banach function spaces, generalized Morrey, Sobolev-Morrie spaces, Orlicz-Lorentz spaces and others.

Goals and tasks

Obtaining accurate estimates of operator norms and their constrictions on various cones of nonnegative functions with monotonicity properties, applying this evaluation to construct the theory of optimal embedding of generalized Bessel and Riesz potentials, obtaining new results on the properties of Petre's K - functional for pairs of spaces including Morrie spaces, and constructing an interpolation theory.
The problem of constructing an optimal shell for cones of measurable functions is very relevant. It is an important part of the general problem of optimal embedding of function spaces.

Scope of application of results

The obtained results will be applied in the general theory of optimal embedding, in the theory of boundary value problems for differential operators, in the theory of approximation and the theory of optimal reconstruction. Evaluation of the norms of hardy-type operators, the maximum operator, play an important role in the theory of series and Fourier integrals, the spectral theory of differential operators, in particular, in the problems of convergence of spectral expansions.