On deformation of certain functional classes in the spaces $C(T^m)$ and $L(T^m)$

The problem of violation of the invariance of the functional classes $H(\omega_1,\dots,\omega_m;C(T^m))$ and $H(\omega_1,\dots,\omega_m;L(T^m))$ $(m\geqslant 2)$ under a multidimensional conjugation operator $\widetilde f_B$ is studied in the case when the moduli of continuity $\omega_i$ $(i=1,\dots,m)$ satisfy Zygmund's condition. Direct estimates are obtained and sharpness of these estimates is established.