On embedding $H_p^{\omega_1,\dots,\omega_\nu}$ classes

Necessary and sufficient conditions are obtained for embedding the function classes $H_p^{\omega_1,\dots,\omega_\nu}$, with given majorants of partial $L_p$-moduli of continuity, in the space $L_q([0,1]^\nu)$ ($1\leqslant p<q<\infty$). In particular, for Lipschitz classes $H_p^{\delta^{\alpha_1},\dots,\delta^{\alpha_\nu}}$ ($0<\alpha_i\leqslant1$) a criterion is obtained for embedding in $L_q$ with limit exponent $q=\frac p{1-\overline\alpha p}$, where $\overline\alpha=\bigl(\frac1{\alpha_1}+\dots+\frac1{\alpha_\nu}\bigr)^{-1}$.
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