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 $2$ . 
@article{IM2_1998_62_2_a1,
     author = {M. I. Dyachenko},
     title = {Fourier coefficients of piecewise-monotone functions of several variables},
     journal = {Izvestiya. Mathematics },
     pages = {247--259},
     publisher = {mathdoc},
     volume = {62},
     number = {2},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1998_62_2_a1/}
}
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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/