We establish two new norm convergence theorems for Henstock-Kurzweil integrals. In particular, we provide a unified approach for extending several results of R. P. Boas and P. Heywood from one-dimensional to multidimensional trigonometric series.

The inequality (*) $({\sum }_{\left|n\right|=1}^{\infty }{\sum }_{\left|m\right|=1}^{\infty }{\left|nm\right|}^{p-2}{\left|f̂(n,m)\right|}^p{)}^{1/p}\le C_p\parallel ƒ{\parallel }_{H_p}$ (0 < p ≤ 2) is proved for two-parameter trigonometric-Fourier coefficients and for the two-dimensional classical Hardy space $H_p$ 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 $L_p$ whenever the left hand side of (*) is finite. From this it follows that under the same condition the two-dimensional trigonometric-Fourier series...