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

I. L. Bloshanskii

Matematičeskie zametki, Tome 21 (1977) no. 6, pp. 777-788.

Résumé

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.