Almost Everywhere Convergence of Multiple Trigonometric Fourier Series of Functions from Sobolev Classes

In this paper, we study the almost everywhere convergence of spherical partial sums of multiple Fourier series of functions from Sobolev classes. It is proved that almost everywhere convergence will take place under the same conditions on the order of smoothness of the expanded function as for multiple Fourier integrals; these conditions were found in a well-known paper of Carbery and Soria (1988). Our reasoning is largely based on the methodology developed in the work of Kenig and Thomas (1980).