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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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