Geodesic
On the convergence of double Fourier series of functions from $L_p$, $p>1$
Matematičeskie zametki, Tome 21 (1977) no. 6, pp. 777-788

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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. 
@article{MZM_1977_21_6_a4,
     author = {I. L. Bloshanskii},
     title = {On the convergence of double {Fourier} series of functions from $L_p$, $p>1$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {777--788},
     publisher = {mathdoc},
     volume = {21},
     number = {6},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_21_6_a4/}
}
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I. L. Bloshanskii. On the convergence of double Fourier series of functions from $L_p$, $p>1$. Matematičeskie zametki, Tome 21 (1977) no. 6, pp. 777-788. http://geodesic.mathdoc.fr/item/MZM_1977_21_6_a4/