Geodesic
Hilbert transform and exponential integral estimates of rectangular sums of double Fourier series
Sbornik. Mathematics, Tome 187 (1996) no. 3, pp. 365-384 
G. A. Karagulian. Hilbert transform and exponential integral estimates of rectangular sums of double Fourier series. Sbornik. Mathematics, Tome 187 (1996) no. 3, pp. 365-384. http://geodesic.mathdoc.fr/item/SM_1996_187_3_a2/
@article{SM_1996_187_3_a2,
     author = {G. A. Karagulian},
     title = {Hilbert transform and exponential integral estimates of rectangular sums of double {Fourier} series},
     journal = {Sbornik. Mathematics},
     pages = {365--384},
     year = {1996},
     volume = {187},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1996_187_3_a2/}
}
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%A G. A. Karagulian
%T Hilbert transform and exponential integral estimates of rectangular sums of double Fourier series
%J Sbornik. Mathematics
%D 1996
%P 365-384
%V 187
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Voir la notice de l'article provenant de la source Math-Net.Ru

A new integral estimate for rectangular partial sums of double Fourier series is obtained. The main result of the paper is the following. Theorem. {\it For any $f\in L\log L(\mathbf T^2)$ and $\delta>0$ there exists a set $E_{\delta,f}\in\mathbf T^2$, $|E_{\delta,f}|>(2\pi)^2-\delta$ such that} \begin{align*} &1)\quad \int_{E_{\delta,f}}\exp\biggl[\frac{c_1\delta|S_{N,M}(x,y,f)|}{\|f\|_{L\log L(\mathbf T^2)}}\biggr]^{1/2}\,dx\,dy\leqslant C_2, \qquad N,M=1,2,\dots, \\ &2)\quad \lim_{N,M\to\infty}\int_{E_{\delta,f}}\bigl[\exp(|S_{N,M}(x,y,f)-f(x,y)|)^{1/2}-1\bigr]\,dx\,dy=0. \end{align*} This theorem yields estimates almost everywhere for rectangular sums of double Fourier series and convergence in $L^p$ on sets of large measure.

[1] Garnett Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR | Zbl

[2] Getsadze R. D., “O raskhodimosti kratnykh ryadov Fure”, Soobsch. AN Gruz.SSR, 122:2 (1986), 269–271 | MR

[3] Gusman M., Differentsirovanie integralov v $\mathbb R^n$, Mir, M., 1978 | MR

[4] Hant R. A., “An estimate of conjugate function”, Studia Math., 44:4 (1972), 371–376 | MR

[5] Konyagin S. V., “O perestanovkakh funktsii i raskhodimosti ryadov Fure po kubam”, Tr. MIAN, 189, Nauka, M., 1989, 98–109 | MR | Zbl

[6] Oskolkov K. I., “Podposledovatelnosti summ Fure integriruemykh funktsii”, Tr. MIAN, 167, Nauka, M., 1985, 239–260 | MR | Zbl

[7] Sjolin P., “Convergence almost everywhere of sertain singular integrals and multiple Fourier series”, Arkiv for Mat., 9 (1971), 65–90 | DOI | MR

[8] Zigmund A., Trigonometricheskie ryady, t. 1, Mir, M., 1965 | MR

[9] Zhizhiashvili L. V., Nekotorye voprosy mnogomernogo garmonicheskogo analiza, Izd-vo un-ta, Tbilisi, 1983 | MR | Zbl