On the convergence of double Fourier series of functions from $L_p$, $p>1$

It is proved that if a function from $L_p$, $p>1$, satisfies the condition
$$ \frac1{t\cdot\tau}\int_0^t\int_0^\tau|f(x+u,y+v)-f(x,y)|\,du\,dv=O\Bigl(\Bigl[\ln\frac1{t^2+\tau^2}\Bigr]^{-2}\Bigr), $$
then the double Fourier series of function $f$, under summation over a rectangle, converges almost everywhere.