Two-parameter Hardy-Littlewood inequalities

Ferenc Weisz¹

¹

Studia Mathematica, Tome 118 (1996) no. 2, pp. 175-184

Résumé

The inequality (*) $(∑_{|n|=1}^{∞} ∑_{|m|=1}^{∞} |nm|^{p-2} |f̂(n,m)|^p)^{1/p} ≤ C_p ∥ƒ∥_{H_p}$ (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 of an arbitrary function from $H_1$ converges a.e. and also in $L_1$ norm to that function.

Détail

DOI : 10.4064/sm-118-2-175-184

Keywords: Hardy spaces, rectangle p-atom, atomic decomposition, Hardy-Littlewood inequalities