Two-parameter Hardy-Littlewood inequalities

The inequality (*) ${\displaystyle {\left({\sum }_{\left|n\right|=1}^{\infty }{\sum }_{\left|m\right|=1}^{\infty }{\left|nm\right|}^{p-2}{|f̂(n,m)|}^p\right)}^{\frac{1}{p}}\le C_p\parallel ƒ{\parallel }_{H_p}}$ (0 < p ≤ 2) is proved for two-parameter trigonometric-Fourier coefficients and for the two-dimensional classical Hardy space ${\displaystyle H_p}$ on the bidisc. The inequality (*) is extended to each p if the Fourier coefficients are monotone. For monotone coefficients and for every p, the supremum of the partial sums of the Fourier series is in ${\displaystyle L_p}$ whenever the left hand side of (*) is finite. From this it follows that under the same condition the two-dimensional trigonometric-Fourier series of an arbitrary function from ${\displaystyle H_1}$ converges a.e. and also in ${\displaystyle L_1}$ norm to that function.