Том 51, № 3 (2016)

Let G be a homogeneous group, and let X₁, X₂, · · ·, X_p0 be left-invariant real vector fields on G that are homogeneous of degree one with respect to the dilation group of G and satisfy Hörmander’s condition. We establish a regularity result in the Orlicz spaces for the following equation:

, where a_ij(x) are real valued, bounded measurable functions defined on G, satisfying the uniform ellipticity condition, and belonging to the space VMO(G) with respect to the subelliptic metric induced by the vector fields X₁, X₂ · · ·, X_p0.

The notion of a Mazur-Ulam space, introduced by C. P. Niculescu in [6] by using the midpoints, is extended here for an arbitrary weight λ ∈ (0, 1). A similar characterization in terms of a class of isometries and their unique fixed point is obtained for the rational case λ = m/n and under more complicated conditions than that of in [6] or [7, p. 166].

The paper is devoted to the construction in the half-plane, for delta-subharmonic functions, of an analog of the part of the theory of M. M. Djrbashian-V. S. Zakarian, which relates to the factorization of the ω-weighted subclasses of meromorphic functions of bounded type in the unit disc. Some ω-weighted classes of delta-subharmonic functions with bounded Tsuji type characteristics are introduced in the upper half-plane and the descriptive representations of these classes are found.

The paper is devoted to the normal families of meromorphic functions and shared functions. Generalizing a result of Chang (2013), we prove the following theorem. Let h (≠≡ 0,∞) be a meromorphic function on a domain D and let k be a positive integer. Let F be a family of meromorphic functions on D, all of whose zeros have multiplicity at least k + 2, such that for each pair of functions f and g from F, f and g share the value 0, and f^(k) and g^(k) share the function h. If for every f ∈ F, at each common zero of f and h the multiplicities m_f for f and m_h for h satisfy m_f ≥ m_h + k + 1 for k > 1 and m_f ≥ 2m_h + 3 for k = 1, and at each common pole of f and h, the multiplicities nf for f and nh for h satisfy n_f ≥ n_h + 1, then the family F is normal on D.