Mathematics

This paper continues the investigation of the relation between the geometry of a circle embedding and the values of its Fourier coefficients. First, we answer a question of Kovalev and Yang concerning the support of the Fourier transform of a starlike embedding. An important special case of circle embeddings are homeomorphisms of the circle onto itself. Under a one-sided bound on the Fourier support, such homeomorphisms are rational functions related to Blaschke products. We study the structure of rational circle homeomorphisms and show that they form a connected set in the uniform topology.

Generic spherical quadrilaterals are classified up to isometry. Condition of genericity consists in the requirement that the images of the sides under the developing map belong to four distinct circles which have no triple intersections. Under this condition, it is shown that the space of quadrilaterals with prescribed angles consists of finitely many open curves. Degeneration at the endpoints of these curves is also determined.

We consider a family of analytic and normalized functions that are related to the domains H(s) , with a right branch of a hyperbolas H(s) as a boundary. The hyperbola H(s) is given by the relation 1 ρ = (2cos φ s ) s (0<s≤1, |φ|<(πs)/2 ). We mainly study a coefficient problem of the families of functions for which z f ′ /f or 1+z f ′′ / f ′ map the unit disk onto a subset of H(s) . We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.

Bounded and compact differences of two composition operators acting from the weighted Bergman space A p ω to the Lebesgue space L q ν , where 0<q<p<∞ and ω belongs to the class D of radial weights satisfying a two-sided doubling condition, are characterized. On the way to the proofs a new description of q -Carleson measures for A p ω , with p>q and ω∈D , involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q -Carleson measures for the classical weighted Bergman space A p α with −1<α<∞ to the setting of doubling weights. The case ω∈ D ˆ is also briefly discussed and an open problem concerning this case is posed.

Let Ω be a bounded convex domain in C n , n≥2 , 1≤q≤(n−1) , and ϕ∈C( Ω ¯ ) . If the Hankel operator H q−1 ϕ on (0,q−1) --forms with symbol ϕ is compact, then ϕ is holomorphic along q --dimensional analytic (actually, affine) varieties in the boundary. We also prove a partial converse: if the boundary contains only `finitely many' varieties, 1≤q≤n , and ϕ∈C( Ω ¯ ) is analytic along the ones of dimension q (or higher), then H q−1 ϕ is compact.

We study the compactness of composition operators on the Bergman spaces of certain bounded pseudoconvex domains in C n with non-trivial analytic disks contained in the boundary. As a consequence we characterize that compactness of the composition operator with a holomorphic, continuous symbol (up to the closure) on the Bergman space of the polydisk.

By considering intrinsic geometric conditions, we introduce a new class of domains in complex Euclidean space. This class is invariant under biholomorphism and includes strongly pseudoconvex domains, finite type domains in dimension two, convex domains, C -convex domains, and homogeneous domains. For this class of domains, we show that compactness of the ∂ ¯ -Neumann operator on (0,q) -forms is equivalent to the boundary not containing any q -dimensional analytic varieties (assuming only that the boundary is a topological submanifold). We also prove, for this class of domains, that the Bergman metric is equivalent to the Kobayashi metric and that the pluricomplex Green function satisfies certain local estimates in terms of the Bergman metric.

Let D denote the unit disc of C and let Ω denote the unit ball B n of C n or the unit polydisc D n , n≥2 . Given a non-constant holomorphic function b:Ω→D , we study the corresponding family σ α [b] , α∈∂D , of Clark measures on ∂Ω . For Ω= B n and an inner function I: B n →D , we show that the property σ 1 [I]≪ σ 1 [b] is related to the membership of an appropriate function in the de Branges--Rovnyak space H(b) .

Let (X,ω) be a compact Kähler manifold. We prove that all Monge-Ampère capacities are comparable. Using this we give an alternative direct proof of the integration by parts formula for non-pluripolar products recently proved by M. Xia.

In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.