Abstract We consider the structure of divergence-free vector measures on plane. show that such can be decomposed into induced by closed simple curves. More generally, we if divergence a planar vector-valued measure is signed measure, then curves (not necessarily closed). As an application generalize certain rigidity properties fields to measures. Namely, locally finite has zero divergence, vani...

‎In this paper,  the sufficient conditions for the linear combinations of slanted half-plane harmonic mappings to be univalent and convex in the direction of $(-gamma) $ are studied. Our result improves some recent works. Furthermore, a illustrative example and imagine domains of the linear combinations satisfying the desired conditions are enumerated.

We describe a direct and transparent connection between the polyBergman type spaces on the upper half-plane and certain two dimensional singular integral operators. Mathematics Subject Classification (2000). 30G30, 45P05, 47B38.

In this paper we study the hyperbolic and Euclidean area distortion of measurable sets under some classes of K-quasiconformal mappings from the upper half-plane and the unit disk onto themselves, respectively.

A factorization theorem is proved in the Hardy spaces Hp of the bi-upper half plane, 0 < p ≤ 1. The proof is based on some fundamental work of Chang-Fefferman on atomic decompositions of Hp.