Project Name :
Some Problems of Harmonic and Nonlinear Analysis in Nonclassical Setting with Applications in Differential Equations
Donor Organization :
Georgian National Scientific Foundation
Budget:
150000 lari
Grant number :
GNSF/ST07/3-169
Research Direction :
5 Mathematics, Mechanics
Sub Direction :
5-100 Mathematical Analysis
Keyword(s) :
Fourier Operators, Measure extension
Description :
The subject of the Project is related to the investigation of problems of harmonic analysis, the function approximation theory and nonlinear analysis in the nonclassical setting. The investigation of problems in such a setting is motivated not only by the need in the solution of applied problems, but also by the intrinsic need of the theory of differential equations and modern analysis.
One of the objectives of the Project is the investigation of two-weighted problems for integral transforms defined on general structures such as: nilpotent Lie groups, fractals and the products of quasimetric spaces with measures. Solving two-weighted problems means finding criteria of the boundedness (compactness) of integral operators from the one weighted space into another. The necessity for the investigation of two-weighted problems arises, for example, in the spectral theory of differential operators which is related to quantum statistics, namely: to problems of estimating from below the eigenvalues of Schroedinger’s operator, in the theory of partial differential equations of mathematical physics when the coefficients of equations degenerate on the domain boundary, in the theory of semilinear differential equations under minimal assumptions on the coefficient regularity and so on. The existence of positive solutions of nonlinear differential equations is equivalent to the validity of two-weighted inequalities for potential type operators, where weights are expressed in terms of the equation coefficients. Based on the experience accumulated by the team members of the Project in the solution of two-weighted problems in classical spaces for potentials, fractional maximal functions, Riemann-Liouville operators, the Project intends to study two-weighted problems in Banach spaces with nonstandard growth condition. It should be noted that the solution of two-weighted problems in variable Lebesgue spaces had remained in obscurity for a long time. The first successful steps in this direction were done by the team members and this fact is confirmed by their original works published on this topic in mathematical journals of high rating.
The Project intends to investigate problems of the approximation theory in the nonclassical two-weighted setting. Classical smoothness moduli cannot be used in weighted spaces and also in Banach spaces with nonstandard growth since in these spaces the shift operator is not continuous. The Project intends to introduce a new structural characteristic of functions and, using it, to prove fundamental direct and inverse theorems of the constructive function theory, and also to determine how a metric influences the estimate of generalized smoothness moduli and best approximations.
Based on the results of the investigation of objects (Cauchy singular integrals and their commutators) of nonlinear harmonic analysis, the Project intends to study non-Fredholmian cases for boundary value problems of the theory of analytic and harmonic functions in the framework of spaces with nonstandard growth. The emergence of such pathological cases is caused by a complicated geometric configuration of the boundary and, in particular, by the occurrence of cusps. We aim to explore the solvability problems of shifted boundary value problems.
The Project intends to carry out research in ergodic theory. The results obtained in this direction are expected to clarify a whole number of obscure questions in Hardy ergodic classes. Furthermore, the Project proposes to investigate problems of spectral factorization of matrix-functions. These problems take their origin in Wiener’s works which laid the foundation for the statistical communication theory. The spectral factorization will be investigated in the difficult case of the occurrence of poles and zeros on the unit circle boundary, which will contribute to the development of a factorization algorithm for matrices of arbitrary dimension.
Also, a part of Project is devoted to some qualitative topics of measure theory: measurability problems of functions and non-linear operators, various aspects of the general measure extension problem, invariant (quasiinvariant) measures on abstract groups and their typical properties such as the metrical transitivity (ergodicity), the uniqueness property, the Steinhaus property of sets of positive measure, nonseparability, etc. In this direction, further researches will be carried out in order to establish the conditions under which it is possible to extend a given measure (or a measure type functional) with preserving its “nice” structural properties, new methods will be developed for extending a measure to a nonseparable one, a wide class of non-linear sup-measurable operators will be investigated in connection with those differential equations whose right-hand side is of “pathological” nature.
Duration :
01/01/2008 - 31/12/2010
Project Leader :
Vakhtang Kokilashvili
Project manager :
Project participant(s) :
Lasha Ephremidze
01/01/2008 - 31/12/2010
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