Generalization of the Hardy--Littlewood theorem on functions with derivatives in the space~$H_1$

A. A. Pekarskii

Matematičeskie zametki, Tome 52 (1992) no. 1, pp. 87-93.

Résumé

Suppose $f$ is a function that is analytic in the disk $D=\{z:|z|1\}$ and belongs to the Hardy space $H_1$. Then, by the Hardy–Littlewood theorem, the following conditions are equivalent: (a) $f'\in H_1$; (b) $f$ coincides with some function of bounded variation almost everywhere on $\partial D$; (c) almost everywhere on $\partial D$, the function $f$ coincides with some absolutely continuous function; (d) for an integral modulus of continuity $f-\omega(f,\delta)$ for the function $f$, we have $\omega(f,\delta)=O(\delta)$. This article presents a generalization of this theorem to higher derivatives in the space $H_p$. The notions of generalized absolute continuity, generalized variation, and higher-order moduli of smoothness are used for this purpose.