We show that the transference method of Coifman and Weiss can be extended to Hardy and Sobolev spaces. As an application we obtain the de Leeuw restriction theorems for multipliers.

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...