Morrey type function spaces and applications to partial differential equations
Morrey type function spaces and applications to partial differential equations
Year 2019-2019
The more we understand the action of a given operator in as many functional spaces as possible, the more we understand the nature of the problem we are considering. In the last 20 years, in many countries around the world, intensive studies have been carried out on the properties of various operators of the theory of functions in Morrey-type spaces and their applications to the theory of partial differential equations. In particular, a significant amount of research in this direction has been carried out by participants of the project.
This project aims to make a significant contribution to the development of the theory of operators in general Morrey-type spaces. The boundedness of the Hausdorff operator, maximal operator, Riesz potential, operators of convolution type, integral operators in general local and global Morrey-type spaces, as well as in discrete Morrey-type spaces will be investigated, and embedding theorems will be proved for general Morrey-type spaces and for Nikolskii-Morrey spaces. As an application to the theory of partial differential equations, the regularity of solutions of elliptic and parabolic equations in Morrey type spaces will be investigated.
List of key publications on the project:
- O. G. Avsyankin. Compactness of Some Operators of Convolution Type in Generalized Morrey Spaces. Mathematical Notes, 2018, 104 - 3, 331-338,
- E. I. Berezhnoi. Exact Calculation of Sums of the Lorentz Spaces Λα and Applications.Math Notes (2018) 104: 628
- Goldman M. L., Bakhtigareeva E G. Some general properties of operators in Morrey-type spaces. Springer Proceedings in Mathematics & Statistics, 2019
- GOLDMAN M.L., BAKHTIGAREEVA E.G.. MODULAR INEQUALITIES FOR POSITIVELY HOMOGENEOUS OPERATORS ON THE CONE OF DECREASING FUNCTIONS IN A WEIGHTED ORLICZ SPACE. 2018, 17-18
- BAKHTIGAREEVA ELZA, GOLDMAN MIKHAIL. INEQUALITIES FOR OPERATORS ON THE CONE OF DECREASING FUNCTIONS IN WEIGHTED ORLICZ SPACE. 2018, 44-47
- Berezhnoi E.I.. Calderon-Lozanovskii construction on local Morrey spaces. 2018, 21-22.
- Burenkov V. I., Тararykova T. V., Denny I. Hakim, Eiichi Nakai, Yoshihiro Sawano, Takuya Sobukawa. Complex interpolation of the predual of Morrey spaces over measure spaces. Georgian Mathematical Journal, 2019.
Project goals
- The research of the boundedness and compactness of the Hausdorff operator, the maximal operator, Riesz potential, convolution type operators, integral operators in general local and global Morrey – type spaces, as well as in discrete Morrey type spaces.
- Embedding theorems for general Morrey type spaces and for Nikol'skii-Morrey spaces.
- Investigation of the regularity of solutions of elliptic and parabolic equations in Morrey type spaces.
Project leader
All participants
Burenkov Viktor
Doctor of Sciences, Professor, Professor of RUDN University
Project results
Sufficient conditions for compactness in generalized Morrey spaces are obtained for the composition of a convolution operator and a multiplication operator by a essentially bounded function.
The invertibility in Morrey spaces is studied for the sum of the identity operator and an integral convolution operator.
The norm of the sum of a collection of Lorentz spaces is exactly calculated. An extrapolation theorem is obtained for collections of Lorentz, Lebesgue, and Martsinkevich spaces with exact constant. For a pair of local Morrey spaces, a formula is obtained for calculating the Calderon – Lozanovskii construction for this pair.
The conditions for boundedness of Hausdorff operators in Morrey-type spaces (local and global) are investigated. Necessary and sufficient conditions are obtained for the precompactness of sets in general local Morrey - type spaces.
For operators acting from rearrangement-invariant spaces into generalized Morrey spaces, the equivalence is established for the norms in local and global Morrey-type spaces and the norms that use decreasing rearrangements of functions. The criteria are established for the boundedness of the embedding operator and the maximal Hardy-Littlewood operator from a rearrangement-invariant space to local and global generalized Morrey-type spaces. The criterion is established for embedding of a weighted Lorentz space with general weights into a generalized Morrey space.
For entire functions of exponential type an inequality of Bernstein type is proved, as well as an inequality of different metrics for Morrey spaces. Closedness with respect to the complex interpolation is proved for the scale of spaces that are pre-dual to Morrey spaces.
Partners