Generalization of the Hardy-Littlewood theorem on functions with derivatives in the space H1

Suppose f is a function that is analytic in the disk D = {z:¦z1¦ < 1} and belongs to the Hardy space H₁. Then, by the Hardy-Littlewood theorem, the following conditions are equivalent: (a) f H₁; (b) f coincides with some function of bounded variation almost everywhere on D; (c) almost everywhere on D, the function f coincides with some absolutely continuous function; (d) for an integral modulus of continuity (f, ) for the function f, we have (f, ) = O(). 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.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 87–93, July, 1992.