Functions of bounded $q$-integral $p$-variation and imbedding theorems

For a function of one real variable there is defined a notion of $q$-integral $p$-variation generalizing Wiener $p$-variation. In terms of this notion there is given a necessary and sufficient condition that a function in $L_q$ have a higher derivative in $L_p$ ($p\leqslant q$), and also that the derivative have a definite smoothness in $L_p$. In addition, embedding theorems with inversion are proved in the periodic case for generalized Lipschitz classes in $L_p$.
Bibliography: 9 titles.