On sufficient conditions for the convergence of double series over rectangles

We prove convergence almost everywhere on $[0,2\pi]\times[0,2\pi]$ of the double Fourier series of functions $f(x,y)$ with modulus of continuity
$$ \omega(f,\delta)=O\biggl(\frac1{\bigl(\ln\frac1\delta\bigr)^{1+\varepsilon}}\biggr) $$
for $\varepsilon>0$.