On the validity of the generalized localization for double trigonometric Fourier series of functions from \(L\ln^+L\ln^+\ln^+L\) (Q1594002)

Localization of the Fourier series in dimension one differs a lot from that in several dimensions. The absence of the classical localization for multiple Fourier series in many spaces led to the idea of generalized localization [see \textit{V. A. Il'yin}, Differ. Uravn. 6, 1159-1169 (Russian) (1970; Zbl 0201.40101); English transl. in Differ. Equations 6(1970), 884-897 (1972), and \textit{I. L. Bloshanskij}, Dokl. Akad. Nauk SSSR 242, 11-14 (1978) (Russian); English transl. in Sov. Math. Dokl. 19, 1019-1023 (1978; Zbl 0513.42027)]. The case of double Fourier series is quite representative here. One says that the generalized localization holds for a sequence of operators \(L_N\) on the space \(X\), if, for an arbitrary set \(E\subset [-\pi,\pi)^2\), \(\lim_{N\to\infty}L_N\left(f,x\right)=0\) for each \(f\in X\) vanishing on \(E,\) and for almost all \(x\in E.\) In view of Bloshanskij's result [\textit{I. L. Bloshanskij}, Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 4, 675-707 (Russian) (1989; Zbl 0701.42008); English transl. in Math. USSR, Izv. 35, No. 1, 1-35 (1990)] the generalized localization does not hold for the square partial sums on \(L=L_1.\) To provide the generalized localization for double Fourier series a natural idea is to consider a class of functions a bit smaller than \(L.\) In the paper this is proved for the class \(L\ln^+L\ln^+\ln^+L.\) The discussion on the sharpness of the result obtained is also of interest. Note that in the paper by \textit{E. Liflyand} and \textit{M. Skopina} [Analysis, München 18, No. 4, 333-343 (1998; Zbl 0927.42005)], it is proved that the generalized localization in \(L\) holds for special linear means of double Fourier series, which, in a sense, are not far away from square partial sums.