Two-parameter Hardy-Littlewood inequalities

Weisz, Ferenc

Studia Mathematica, Tome 119 (1996), p. 175-184 / Harvested from The Polish Digital Mathematics Library

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

Publié le : 1996-01-01

Zbl 0864.42003

EUDML-ID : urn:eudml:doc:216272

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