Mathematics

In this paper we derive a Cauchy integral formula for holomorphic and hyperholomorphic functions in domains bounded by a Koch snowflake in two and three dimensional setting.

We give a proof of the Bourgain-Milman theorem using complex methods. The proof is inspired by Kuperberg's, but considerably shorter.

In \cite{CO-Tp-spaces}, the present authors initiated the study of composition operators on discrete analogue of generalized Hardy space T p defined on a homogeneous rooted tree. In this article, we give equivalent conditions for the composition operator C ϕ to be bounded on T p and on T p,0 spaces and compute their operator norm. We also characterize invertible composition operators as well as isometric composition operators on T p and on T p,0 spaces. Also, we discuss the compactness of C ϕ on T p and finally prove there are no compact composition operators on T p,0 spaces.

For any real β let H 2 β be the Hardy-Sobolev space on the unit disk $\D$. H 2 β is a reproducing kernel Hilbert space and its reproducing kernel is bounded when β>1/2 . In this paper, we study composition operators C ? on H 2 β for 1/2<β<1 . Our main result is that, for a non-constant analytic function $\varphi:\D\to\D$, the operator C ? has dense range in H 2 β if and only if the polynomials are dense in a certain Dirichlet space of the domain $\varphi(\D)$. It follows that if the range of C ? is dense in H 2 β , then ? is a weak-star generator of H ??. Note that this conclusion is false for the classical Dirichlet space D . We also characterize Fredholm composition operators on H 2 β .

Given any ϵ>0 and any planar region Ω bounded by a simple n-gon P we construct a ( 1+ϵ) -quasiconformal map between Ω and the unit disk in time C(ϵ)n . One can take C(ϵ)=C+Clog(1/ϵ)loglog(1/ϵ) .

We show that the manifold ( S 2 × S 2 )#( S 2 × S 2 ) does not admit a non-constant non-injective uniformly quasiregular self-map. This answers a question of Martin, Mayer, and Peltonen, and provides the first example of a quasiregularly elliptic manifold which is not uniformly quasiregularly elliptic. To obtain the result, we introduce conformally formal manifolds, which are closed smooth n -manifolds M admitting a measurable conformal structure [g] for which the (n/k) -harmonic k -forms of the structure [g] form an algebra. This is a conformal counterpart to the existing study of geometrically formal manifolds. We show that, similarly as in the geometrically formal theory, the real cohomology ring H ∗ (M;R) of a conformally formal n -manifold M admits an embedding of algebras Φ: H ∗ (M;R)↪ ∧ ∗ R n . We also show that uniformly quasiregularly elliptic manifolds M are conformally formal in a stronger sense, in which the wedge product is replaced with a conformally scaled Clifford product. For this stronger version of conformal formality, the image of Φ is closed under the Euclidean Clifford product of ∧ ∗ R n , which in turn is impossible for M=( S 2 × S 2 )#( S 2 × S 2 ) .

Important geometric or analytic properties of domains in the Euclidean space R n or its one-point compactification (the Möbius space) R ¯ ¯ ¯ ¯ n (n≥2) are often characterized by comparison inequalities between two intrinsic metrics on a domain. For instance, a proper subdomain G of R n is {\it uniform} if and only if the quasihyperbolic metric k G is bounded by a constant multiple of the distance-ratio metric j G . Motivated by this idea we first characterize the completeness of the modulus metric of a proper subdomain G of R ¯ ¯ ¯ ¯ n in terms of Martio's M -condition. Next, we prove that if the boundary is uniformly perfect, then the modulus metric is minorized by a constant multiple of a Möbius invariant metric which yields a new characterization of uniform perfectness of the boundary of a domain. Further, in the planar case, we obtain a new characterization of uniform domains. In contrast to the above cases, where the boundary has no isolated points, we study planar domains whose complements are finite sets and establish new upper bounds for the hyperbolic distance between two points in terms of a logarithmic Möbius metric. We apply our results to prove Hölder continuity with respect to the Ferrand metric for quasiregular mappings of a domain in the Möbius space into a domain with uniformly perfect boundary.

In the present paper, we derive several conditions of linear combinations and convolutions of harmonic mappings to be univalent and convex in one direction, one of them gives a partial answer to an open problem proposed by Dorff. The results presented here provide extensions and improvements of those given in some earlier works. Several examples of univalent harmonic mappings convex in one direction are also constructed to demonstrate the main results.

Let Ω⊂ C n be a strictly pseudoconvex Runge domain with C 2 -smooth defining function, l∈N, p∈(1,∞). We prove that the holomorphic function f has derivatives of order l in H p (Ω) if and only if there exists a sequence on polynomials P n of degree n such that ∑ k=1 ∞ 2 2lk |f(z)− P 2 k (z)| 2 ∈ L p (∂Ω).

We study the behavior of various set-functions under holomorphic motions. We show that, under such deformations, logarithmic capacity varies continuously, while analytic capacity may not.