Geodesic
Hilbert transform and exponential integral estimates of rectangular sums of double Fourier series
Sbornik. Mathematics, Tome 187 (1996) no. 3, pp. 365-384 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@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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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/

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