An extremum problem on a class of differentiable functions of several variables

On the multidimensional class $W_0^rH_\omega^{(n)}$ of continuous periodic functions $F$ with the $r$th derivative $D^rF$ from
$$ H_\omega^{(n)} =\biggl\{f\in C\bigm| |f(x)-f(y)|\le\sum_{i=1}^n\omega_i(|x_i-y_i|) \forall x,y\in\mathbb R^n\biggr\} $$
(where the $\omega_i(x_i)$ are the convex moduli of continuity) and zero mean with respect to each variable, we obtain the exact value of
$$ M_r(\omega) =\sup_{F\in W_0^rH_\omega^{(n)}}\|F\|_C. $$