Shota Rustaveli National Science Foundation of GeorgiaStart Date: 2006-01-01 End Date: 2008-01-01
The project is closely related to such an intensively developing direction of the modern analysis and its results implementation as the Banach function spaces with non-standard variable growth order and the theory of operators in these spaces. Special properties of the above-mentioned spaces were singled out from the Banach spaces theory at 50-s of the last century. The initial works in this field belong to W. Orlicz and J. Musielak. Actually, their coming out was firstly caused by theoretical meanings only, but later, on the edge between the last and the current centuries, it was realized the necessity for study of these spaces, because of their rather essential role in the mathematical models of nonlinear elasticity theory and incompressible fluid’s mechanics, as well of their importance in the research of different physical events, via variational methods. In recent decades these circumstances stimulated quite extensive studies of Lebesgue spaces with variable exponent (i.e., Lavrentiev’s phenomena research by V. Zhikov), particularly, studies of different models of incompressible fluid’s mechanics integral operators and related to them, also the Sobolev spaces with variable exponent (L. Diening, M. Ružička, S. Samko, D.E. Edmunds and J. Rάkosnik, X. Fan and D. Zhao), research of p(x)-Laplacian nonlinear differential equations and function spaces associated with them that enable to describe physical events by “point variable” characteristics, for example, in the elasticity theory of nonhomogeneous medium (E. Acerbi & G. Mingione, P. Marcellini, X. Fan, H. Zhang and etc.). The project anticipates the research of classical problems through the nonstandard approaches specified by their application demands. We plan to develop the idea of non-standard variable growth in three directions. The project envisages research of function spaces with variable exponent, potentials of variable order and variable order summability of Fourier trigonometric series, in the scope of their interrelation and applications. A part of project will be devoted to certain nonstandard functions (Sierpinski-Zygmund functions, nontrivial solutions of the Cauchy functional equation, Vitali type functions, etc.) and their study from the standpoint of measurability with respect to various classes of invariant (quasiinvariant) measures. The existence of such functions and their unusual properties are important for a number of questions in abstract harmonic analysis and general theory of dynamical (quasidynamical) systems.