Shota Rustaveli National Science Foundation of GeorgiaStart Date: 2012-12-20 End Date: 2015-12-19
The project deals with the study of mapping properties of linear and non-linear harmonic analysis integral operators, approximation of functions in so called new function spaces, problems of Fourier analysis and wavelet theory, matrix-function spectral factorization. Applications to boundary value problems of analytic (generalized analytic) and harmonic functions and PDEs will be also investigated in the frame of the above-mentioned spaces. Under the new function spaces we understand the following spaces: variable exponent Lebesgue and amalgam spaces, variable exponent Morrey spaces, grand Lebesgue spaces, generalized Morrey spaces and their weighted analogs. Research novelties are: • the project aims to study the mapping properties of a wide range of integral and Fourier operators in new function spaces. For instance, it is proposed to establish boundedness criteria in generalized Morrey spaces for commutators of various type singular integrals and potentials defined on quasimetric measure spaces; applications to the norm estimates of solutions of elliptic type differential equations; • the project anticipates for the first time to explore necessary and sufficient condition ensuring the boundedness/compactness of weighted kernel operator in variable Lebesgue and amalgam spaces; to prove two-weighted inequality for maximal function in variable exponent amalgam spaces; • we intend for the first time to explore the approximation problems in new function spaces on the base of extensions of Bernstein and Nikolsky inequalities for trigonometric polynomials in the above-mentioned spaces; • we intend to study the representability of multivariate continuous real-valued functions as the series of single, generally speaking different variables; • one of our goals is to prove a general theorem on essentially divergence in measure of Fourier series with respect to a given double orthonormal system; • one of our goals is to study the integrability properties of Paley functions, the majorants of partial sums for the double Fourier-Haar series; • we propose to extend well-known Menshov-Rademacher and Menshov theorems for the summability by “variable” triangle matrix; • to study the approximation of measurable functions in the sense of almost everywhere convergence by the subsequences of trigonometric series having lacunas as large as possible; • to examine the class of functions φ ensuring almost everywhere convergence of multiple separable wavelet of fºφ for f with the same property and some smoothness; • we will elaborate: method of construction of wavelet matrices with rational coefficients, a new effective method which by a given first row of a wavelet matrix enables us to construct the remaining rows; • in the frame of variable exponent analysis we will solve the linear conjugation boundary value problem we oscillating conjugacy coefficients taken from more general class that I. Simonenko’s class; • we intend to introduce variable exponent Hardy and Smirnov classes for generalized analytic functions and apply PDE; • in domains with non-smooth boundaries we expect to solve the Riemann and Riemann-Hilbert boundary value problems for generalized functions: to prove the solvability conditions, the influence of angles sizes of non-smooth boundary, variable exponent and boundary functions on the number of linear independent solutions of homogeneous problem will be illuminated