Computational Methods and Function Theory Volume 00 (0000), No. 0, 1–? CMFT-MS XXYYYZZ Functions of ω-Bounded Type in the Half-Plane Armen Jerbashian and Vahagn Jerbashian Abstract. The paper is devoted to introduction and investigation of func- tions of ω-bounded type in the half-plane and also to investigation of some properties of the Banach spaces A p ω,γ , which are natural subsets of functions of ω-bounded type, such as Hardy classes are in Nevanlinna’s class N . The classes of δ-subharmonic functions of ω-bounded type are defined by a weighted integrability condition of Tsuji’s characteristics. The canonical representa- tions of these classes by some Green type potentials and an analog of Poisson integral are obtained. Particularly, these representations become canonical factorizations for the corresponding meromorphic classes of ω-bounded type. A theorem on the orthogonal projection from L 2 ω,0 to A 2 ω,0 , a Paley–Wiener type theorem and a theorem on an explicitely written isometry between A 2 ω,0 and the Hardy space H 2 are proved. Then a theorem on projection from the Lebesgue spaces L p ω,0 to A p ω,0 is proved. Keywords. Weighted spaces of regular functions. 2000 MSC. Primary 32A35; Secondary 31A05. 1. Introduction Classes of meromorphic functions in |z | < 1, defined by a weighted integrabil- ity condition of characteristics, initially where considered by R. Nevanlinna [1, Section 216]. He proved that if a function, meromorphic in |z | < 1, satisfies (1)  1 0 (1 - r) α T (r, f )dr < +∞, -1 <α< +∞, then the sets {a μ } and {b ν } of its zeros and poles are of the density  k (1 -|a k |) α+2 < +∞,  n (1 -|b n |) α+2 < +∞. Later, M. M. Djrbashian’s works [2, 3] of 1945 and 1948 revealed the canonical representations of the weighted classes of meromorphic functions satisfying (1) by some Blaschke type product and a Schwarz type integral in the exponent. For further developments in the field, see particularly [4] – [9]. We call Nevanlinna’s classes (1), as well as the classes investigated in [5, Chapter IX] classes of func- tions of α-bounded type in |z | < 1, and the classes N {ω} of [6] and other classes ISSN 1617-9447/$ 2.50 c  20XX Heldermann Verlag