The almost everywhere convergence of eigenfunction expansions of elliptic differential operators in the Torus

Many of the equations of physical sciences and engineering involve operators of elliptic type. Most important among these is non-relativistic quantum theory, which is based upon the spectral analysis of second order elliptic differential operators. Spectral theory of the elliptic differential operators is an extremely rich field which has been studied by many qualitative and quantitative techniques like Sturm-Liouville theory, separation of variables, Fourier and Laplace transforms, perturbation theory, eigenfunction expansions, variational methods, microlocal analysis, stochastic analysis and numerical methods including finite elements. We note here that the applications of second order elliptic operators to geometry and stochastic analysis are also now of great importance. In the present research we investigated the problems concerning the almost everywhere convergence of multiple Fourier series summed over the elliptic levels in the classes of Liouville functions on Tours. The sufficient conditions for the almost everywhere convergence problems, which are most difficult problems in Harmonic analysis, are obtained in the classes of Liouville. The difficulty is on the obtaining the suitable estimations for the maximal operator of the partial sums of the Fourier series, which guarantees the almost everywhere convergence of Fourier series. The process of estimating the maximal operator involves very complicated calculations which depends on the functional structure of the classes of functions. The main idea on the proving the almost everywhere convergence of the eigenfunction expansions in the interpolation spaces is estimation of the maximal operator of the partial sums in the boundary classes and application of the interpolation Theorem of the family of linear operators. It is well known that the theory of the eigenfunction expansions of the differential operators closely connected with the convergence problems of Fourier series and integrals. The one of the most important summation method which is called spherical summation method connected with the eigenfunction expansions of the Laplace operator, while the questions on convergence of the multiple Fourier series summed over the elliptic levels can be investigated by using the spectral theory of the elliptic differential operators. In chapter III and IV of the present thesis maximal operator of spherical and elliptic partial sums are estimated in the interpolation classes of Liouville and the almost everywhere convergence of the multiple Fourier series by spherical and elliptic summation methods are established. The considering multiple Fourier series as an eigenfunction expansions of the differential operators helps to translate the functional properties (for example smoothness) of the Liouville classes into Fourier coefficients of the functions which being expanded into such expansions. The sufficient conditions for convergence of the multiple Fourier series of functions from Liouville classes are obtained in terms of the smoothness and dimensions. Such results are highly effective in solving the boundary problems with periodic boundary conditions occurring in the spectral theory of differential operators. The investigations of multiple Fourier series in modern methods of harmonic analysis incorporates the wide use of methods from functional analysis, mathematical physics, modern operator theory and spectral decomposition. New method for the best approximation of the square-integrable function by multiple Fourier series summed over the elliptic levels are established in chapter V. Using the best approximation, the Lebesgue constant corresponding to the elliptic partial sums is estimated. The latter is applied to obtain an estimation for the maximal operator in the classes of Liouville.

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