Embedding theorems and best approximations

We establish necessary and sufficient conditions, in terms of best approximations, for a function in $L^p(0,2\pi)$ ($0<p<1$) to belong to $L^q(0,2\pi)$ ($q<p$). The proofs depend on the properties of equimeasurable functions, which were applied by Ul'yanov in the theory of the embedding of certain classes $H_p^\omega$ for $p\geqslant1$ (RZhMat., 1969, 2B109). We also obtain some relationships among moduli of continuity in different metrics, which let us generalize results of Hardy and Littlewood (Math. Z., 28, № 4 (1928), 612–634) to the case $0<p<1$ and prove converses for nonincreasing functions.
Bibliography: 11 titles.