Piecewise monotonic functions of several variables and a theorem of Hardy and Littlewood

The author discusses classes of periodic functions of $m$ variables that are either piecewise monotonic or piecewise monotonic in the sense of Hardy, and clarifies the connections, for such functions, between the property of belonging to $L_p$ space, $1<p<\infty$, and the convergence of series of their trigonometric Fourier coefficients,
$$ \sum_{n_1,\dots ,\ n_m=-\infty}^{+\infty}\big|a_{n_1\dots n_m}\big|^\alpha \left(\prod_{j=1}^m(|n_j|+1)\right)^{\alpha-2}. $$
We establish the existence, when $m>1$, of certain results that differ from the one-dimensional case.