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Two-parameter Hardy-Littlewood inequalities. (English) Zbl 0864.42003
Summary: The inequality \[ \left(\sum^\infty_{|n|=1}\sum^\infty_{|m|=1} |nm |^{p-2} \bigl|\widehat f(n,m) \bigr|^p \right)^{1/p} \leq C_p|f|_{H_p} \quad (0<p\leq 2) \tag{*} \] 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.

MSC:
42B05 Fourier series and coefficients in several variables
42B30 \(H^p\)-spaces
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