Multidimensional analogue of a theorem of Privalov

A criterion is established for the continuity of functions that are conjugate in the sense of Cesari to a given function in the class
$$ H\bigl(\omega_j(\delta),j\in B,T^N\bigr)=\bigl\{f\in C(T^N):\omega_j(f,\delta) =O[\omega_j(\delta)],\ j\in B\bigr\}, $$
where $B\subseteq M=\{1,\dots,N\}$, $T^N=(-\pi,\pi )^N$, $\omega_j(f,\delta)$ ($1\leqslant j\leqslant N$) are the partial moduli of continuity of $f(\bar x)$ and $\omega_j(\delta)$ ($j\in B$) are moduli of continuity. Best possible estimates of the partial modulus of continuity of a function conjugate to $f\in H(\omega _j,j\in M,T^N)$ are obtained in the case when the $\omega_j(\delta)$ ($j\in M$) satisfy two specific conditions. These conditions on the modulus of continuity $\omega(\delta)$ are shown to be necessary and sufficient in order that the conjugation operator violate the invariance of the class $H$ $(\omega_j=\omega,j\in M,T^N)$ in the same way as it violates that of $\operatorname{Lip}\bigl(\alpha,C(T^N)\bigr)$ ($0<\alpha<1$).