The project proposal deals with new approaches in modern analysis on metric spaces, multidimensional and Applied Harmonic Analysis, applications to PDEs. It envisages to build a new scale of nonstandard Banach function spaces defined on quasi-metric measure spaces (QMMSs)and to explore them from the viewpoints of mapping properties of basic differential and integral operators.We emphasize that modern Harmonic Analysis enlarged the scope to encompass analytical problems posed on metric measure spaces. We expect that the introduced new function spaces will finely fit to the applications in PDEs and BVPs for analytic functions.
The proposal anticipates the solution of challenging two-weight and trace problems for linear and multilinear operators, solution of sharp exponent bound problem for one-sided fractional operators, proving the Rellich and Sobolev type weighted inequalities in new function spaces;establishing extrapolation results in new grand spaces. We intend to apply the obtained results e. g. in solvability of multilinear wave equations, to the regularity of elliptic equations in non-divergence form with VMO coefficients, to determine the space of high order derivatives of solutions to the elliptic equations with discontinuous coefficients when the known term belongs to fully measurable grand spaces.
In multidimensional Harmonic Analysiswe intend: to solve the construction problem of universal series with respect to the arbitrary measurable functions and to establish criteria of universality of function series; to solve coefficient reconstruction problem for convergent double lacunary Walsh series; to establish Fubini’s type assertion for convergent multiple function series, to establish necessary and sufficient condition ensuring the convergence in measure of the sequences of measurable functions.
A considerable part of the project proposaldeals with Applied Harmonic Analysis, namely, with matrix spectral factorization problem. We plan: to elaborate an efficient multivariate matrix spectral factorization algorithm and to obtain the convergence rate of new algorithm. The practical applications is intended for analyzing information flow in brain networks with non-parametric Granger causality, including the epilepsy curing methods in biomedicine

Project members:

• Vakhtang Kokilashvili (Principal Investigator)

• Alexander Meskhi (Coordinator)

• Tengiz Tetunashvili (Key Personnel)

• Lasha Ephremidze (Key Personnel)

• Nika Salia (Key Personnel)

• Giorgi Imerlishvili (Key Personnel)

• Tsira Tsanava (Key Personnel)