Mixed norm spaces of analytic and harmonic functions. I.
We give new characterizations of the analytic Besov spaces Bp on the unit ball B of Cn in terms of oscillations and integral means over some Euclidian balls contained in B.
The Möbius metric is studied in the cases, where its domain is an open sector of the complex plane. We introduce upper and lower bounds for this metric in terms of the hyperbolic metric and the angle of the sector, and then use these results to find bounds for the distortion of the Möbius metric under quasiregular mappings defined in sector domains. Furthermore, we numerically study the Möbius metric and its connection to the hyperbolic metric in polygon domains.
A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. A similar phenomenon is valid for the families of parallel closed geodesics. We give a complete description of...