Diameters of certain sets in function spaces

The precise values of the diameters (according to A. N. Kolmogorov) of the class $H_C^\omega$ in the space $C[a,b]$ and lower bounds for the diameters of the class $H_p^\omega$ in the spaces $\widetilde {L_p}(o,2\pi)$ ($1\le p\le\infty$), for any modulus of continuity $\omega(\delta)$, are obtained. The latter bounds give the exact values of the odd-numbered diameters of the class $H_2^{1/2}=H_2^{\delta^{1/2}}$ and the exact order of decay of the diameters of the class $H_1^\omega$.