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Weighted Classes of Regular Functions Area Integrable Over the Unit Disc

This preprint contains some generalizations of the main theorems of M.M.Djrbashian of 1945–1948 which laid ground for the theory of Aα (or initially H (α)) spaces and his factorization theory of classes N{ω} exhausting all functions meromorphic in the unit disc. Also some later results on Aα spaces and Nevanlinna’s weighted class are improved. The preprint contains the main analytic apparatus for generalizing almost all known results on Aα spaces within a new theory, where instead (1 − r)dr (−1 < α < +∞, 0 < r < 1) some weights of the form dω(r) are used. The obtained results make evident that the theory of Aω spaces and the factorization theory of M.M.Djrbashian are inseparable parts of a general theory of classes of regular functions associated with M.M.Djrbashian general integrodifferentiation. The author hopes that the publication of this preprint can lead to clarification of some priority misunderstandings in the field. Introduction. The earliest work of M.M.Djrbashian [1] published in 1945 (see also [2] containing the proofs of the representations of [1] and some additional theorems) was mainly aimed to improve Nevanlinna’s result (see [3], Sec. 216) on the density of zeros and poles of functions f(z) meromorphic in the unit disc, for which the Riemann–Liouville fractional integral of the growth characteristic T (r, f) is bounded, i.e. ∫ 1 0 (1− r)T (r, f)dr < +∞ (1) for a given α > −1. The same work [1] contains investigation of the similar Hardy type classesH(α) of holomorphic functions in |z| < 1, for which the Riemann–Liouville fractional integral of M p (r, f) = ∫ 2π 0 |f(re)|dθ is bounded, i.e. H(α) ≡ Aα : ∫∫ |ζ|<1 (1− |ζ|)|f(ζ)|dσ(ζ) < +∞, (2) where α ∈ (−1,+∞) and p ∈ [1,+∞) are any fixed numbers and dσ(ζ) is the Lebesgue’s area measure. Particularly, in [1] the representation formula of these classes was first obtained. It has to be mentioned, that H(α) ≡ Aα is a generalization of the non-weighted Hilbert space H(0) ≡ A0 : ∫∫ |ζ|<1 |f(ζ)|dσ(ζ) < +∞