Geodesic
Fourier coefficients of piecewise-monotone functions of several variables
Izvestiya. Mathematics , Tome 62 (1998) no. 2, pp. 247-259.

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As the main result of this paper we obtain the best possible estimates for the Hardy–Littlewood sums of the Fourier coefficients of piecewise-monotone functions in the spaces $L_p([-\pi,\pi)^m)$, where $2$ . 
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M. I. Dyachenko. Fourier coefficients of piecewise-monotone functions of several variables. Izvestiya. Mathematics , Tome 62 (1998) no. 2, pp. 247-259. http://geodesic.mathdoc.fr/item/IM2_1998_62_2_a1/

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