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Project Name : Modern problems of harmonic analysis, approximation theory and the theory of integral operators in new function spaces; applications in boundary value problems
Donor Organization : Shota Rustaveli National Science Foundation
Budget: 150000
Grant number : FR/25/5-100/12
Direction : 5 Mathematics, Mechanics
Subdirection : 5-100 Mathematical Analysis
Key word : Integral operators in new function spaces
Description : The goal of the proposed research is to study various problems in operator theory, linear and non-linear harmonic analysis, approximation theory, Fourier analysis in new function spaces such as variable exponent weighted Lebesgue and Morrey spaces, generalized Grand Lebesgue weighted spaces, with applications to BVPs of analytic and harmonic functions appearing in different contexts but strongly interrelated in many respects: motivation, techniques and methodology. A considerable part of the proposed project is related to the study of boundedness problems of singular and various type fractional integral operators, and Fourier operators. It was realized the necessity for the study of above-mentioned new function spaces because of their rather essential role in various fields. In the theory of nonlinear PDE, it turns out that Grand Lebesgue spaces and Morrey spaces are right spaces in which some nonlinear equations have been considered. The project intends to treat the boundedness problems for both classical integral operators and integral transforms defined, generally speaking, on quasi-metric measure spaces. We intend to solve in Grand Lebesgue spaces one of the challenging problem of operator and function spaces theory, namely, the trace problem for potentials and fractional maximal functions. One of our purposes of the presented project is to give a complete characterization of weights governing the validity of Sobolev type theorem for potentials in generalized Grand Lebesgue space with weight. We intend to study mapping properties in generalized grand Morrey spaces of maximal functions, potentials and singular integrals defined on quasi-metric measure spaces; to explore boundedness problems for modified maximal functions and potentials defined on non-homogeneous spaces; to study the mapping properties of the Riemann-Liouville and Weyl transforms with product kernels in weighted grand Lebesgue spaces. The project envisages the research of some problems of multiple series. Namely, to study the so-called Fubini type phenomenon for the summability means of general function series; to establish an extensions of Kolmogorov, Shipp and Bočkarev’s well-known theorems on divergence of Fourier trigonometric and orthonormal series; Important characteristic feature of the project is a tight connection between the theory of nonlinear harmonic analysis operators in nonstandard Banach function spaces and the boundary value problems of analytic and harmonic functions. The project aims to study new aspects of BVPs within the frame of nonstandard Banach function spaces; to introduce and study weighted Hardy and Smirnov classes with function exponents in multiply connected domains; to explore the Dirichlet problem for harmonic functions of variable exponent Smirnov classes in multiply connected domains with piecewise smooth boundaries; to reveal an influence of boundary’s geometry on a solvability picture; to construct solutions in explicit form; we intend to solve the Riemann problem of linear conjugation for analytic functions in the class of Cauchy type integrals with densities of grand Lebesgue spaces. The project envisages to study spectral factorization problems of matrix functions. Recently, it was constructed an efficient spectral factorization algorithm by the team leader. One of the goals of the project is a further simplification and refinement of the above-mentioned algorithm in the polynomial case.
Duration : 12/04/2013 - 12/04/2016
Leader : No leader
Project manager :
Project participant(s) : Alexander Meskhi 12/04/2013 - 12/04/2016 Vakhtang Kokilashvili 15/04/2013 - up to date
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