Hilbert transform and exponential integral estimates of rectangular sums of double Fourier series

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.