Geodesic
The Hardy--Littlewood Theorem for Multiple Fourier Series with Monotone Coefficients
Matematičeskie zametki, Tome 99 (2016) no. 4, pp. 502-510.

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It was proved earlier that, for multiple Fourier series whose coefficients are monotone in each index, the classical Hardy–Littlewood theorem is not valid for $p\le 2m/(m+1)$, where $m$ is the dimension of the space. We establish how the theorem must be modified in this case.
Keywords: Hardy–Littlewood theorem, multiple Fourier series, trigonometric polynomial. 
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M. I. Dyachenko; E. D. Nursultanov; M. E. Nursultanov. The Hardy--Littlewood Theorem for Multiple Fourier Series with Monotone Coefficients. Matematičeskie zametki, Tome 99 (2016) no. 4, pp. 502-510. http://geodesic.mathdoc.fr/item/MZM_2016_99_4_a2/

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