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It is well known that Luzin’s conjecture has a positive solution in one dimensional case and it is still open in multidimensional case for the spherical partial sums:

Historically progress with solving the Luzin’s conjecture has been made by considering easier problems. One of such easier problems for   was suggested by V. A. Il’in in 1968 and this problem is called the generalized localization principle for the spherical partial sums.

Recently, the problem of generalized localization was completely solved in papers [1] and [2].

In this talk we first give a short survey on convergence a.e. of one dimensional and multidi-mensional Fourier series, then indicate a sketch of the above mentioned result’s proof.

References:

[1] R. R. Ashurov, Generalized localization for spherical partial sums of multiple Fourier series, J. Fourier Anal. Appl., 25 (2019), 3174–3183.

[2] R. R. Ashurov, Generalized Localization for Spherical Partial Sums of Multiple Fourier Series, Dokl. Math., 100 (2019), 505-507.