Abstract:
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
$$
M_r(\omega)
=\sup_{F\in W_0^rH_\omega^{(n)}}\|F\|_C.
$$
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