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

D. V. Gorbachev

Matematičeskie zametki, Tome 62 (1997) no. 2, pp. 192-205

Résumé

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