For a nondecreasing function $K: [0, \infty)\rightarrow [0, \infty)$ and $0<s<\infty $, we introduce a Morrey type space of functions analytic in the unit disk $\mathbb {D}$, denoted by $\mathcal {D}^s_K$. Some characterizations of $\mathcal {D}^s_K$ are obtained in terms of K-Carleson measures. A relationship between two spaces $\mathcal {D}^{s_1}_K$ and $\mathcal {D}^{s_2}_K$ is given by fractional order derivatives. As an extension of some known results, for a positive Borel measure $\mu $ on $\mathbb {D}$, we find sufficient or necessary condition for the embedding map $I: \mathcal {D}^{s}_{K}\mapsto \mathcal {T}^s_{K}(\mu)$ to be bounded.

Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$. We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$.

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

Classical results about peaking from complex interpolation theory are extended to polynomials on a closed disk, and on the complement of its interior. New results are obtained concerning interpolation by univalent polynomials on a Jordan domain whose boundary satisfies certain smoothness conditions.

Let $(z_k)$ be a sequence of distinct points in the unit disc $\mathbb {D}$ without limit points there. We are looking for a function $a(z)$ analytic in $\mathbb {D}$ and such that possesses a solution having zeros precisely at the points $z_k$, and the resulting function $a(z)$ has ‘minimal’ growth. We focus on the case of non-separated sequences $(z_k)$ in terms of the pseudohyperbolic distance when the coefficient $a(z)$ is of zero order, but $\sup _{z\in {\mathbb D}}(1-|z|)^p|a(z)| = + \infty$ for any $p > 0$. We established a new estimate for the maximum modulus of $a(z)$ in terms of the functions $n_z(t)=\sum \nolimits _{|z_k-z|\le t} 1$ and $N_z(r) = \int_0^r {{(n_z(t)-1)}^ + } /t{\rm d}t.$ The estimate is sharp in some sense. The main result relies on a new interpolation theorem.

We prove a number of results concerning the Hausdorff and packing dimension of sets of points which escape (at least in average) to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we generalize the results proved by Sixsmith in 2016 and answer his question on annular itineraries for exponential maps.

This paper presents an approach, based on interpolation theory of operators, to the study of interpolating sequences for interpolation Banach spaces between Hardy spaces. It is shown that the famous Carleson result for H∞ can be lifted to a large class of abstract Hardy spaces. A description is provided of the range of the Carleson operator defined on interpolation spaces between the classical Hardy spaces in terms of uniformly separated sequences. A key role in this description is played by some general interpolation results proved in the paper. As by-products, novel results are obtained which extend the Shapiro–Shields result on the characterisation of interpolation sequences for the classical Hardy spaces Hp. Applications to Hardy–Lorentz, Hardy–Marcinkiewicz and Hardy–Orlicz spaces are presented.

Let f be analytic in the unit disk $\mathbb {D}=\{z\in \mathbb {C}:|z|<1 \}$ and let ${\mathcal S}$ be the subclass of normalised univalent functions with $f(0)=0$ and $f'(0)=1$, given by $f(z)=z+\sum _{n=2}^{\infty }a_n z^n$. Let F be the inverse function of f, given by $F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$ for $|\omega |\le r_0(f)$. Denote by $ \mathcal {S}_p^{* }(\alpha )$ the subset of $ \mathcal {S}$ consisting of the spirallike functions of order $\alpha $ in $\mathbb {D}$, that is, functions satisfying

$$\begin{align*}{\mathrm{Re}} \ \bigg\{e^{-i\gamma}\dfrac{zf'(z)}{f(z)}\bigg\}>\alpha\cos \gamma, \end{align*}$$

for $z\in \mathbb {D}$, $0\le \alpha <1$ and $\gamma \in (-\pi /2,\pi /2)$. We give sharp upper and lower bounds for both $ |a_3|-|a_2| $ and $ |A_3|-|A_2| $ when $f\in \mathcal {S}_p^{* }(\alpha )$, thus solving an open problem and presenting some new inequalities for coefficient differences.

In this paper, we will show that the unitary equivalence of two multiplication operators on the Bergman spaces on polygons depends on the geometry of the polygon.

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

We establish the global regularity of multilinear Fourier integral operators that are associated to nonlinear wave equations on products of $L^p$ spaces by proving endpoint boundedness on suitable product spaces containing combinations of the local Hardy space, the local BMO and the $L^2$ spaces.

This note characterizes, in terms of interpolating Blaschke products, the symbols of Hankel operators essentially commuting with all quasicontinuous Toeplitz operators on the Hardy space of the unit circle. It also shows that such symbols do not contain the complex conjugate of any nonconstant singular inner function.

In this paper we investigate the range of validity of Ruelle’s property. First, we show that every finitely generated Fuchsian group has Ruelle’s property. We also prove the existence of an infinitely generated Fuchsian group satisfying Ruelle’s property. Concerning the negative results, we first generalize Astala and Zinsmeister’s results [Mostow rigidity and Fuchsian groups. C. R. Math. Acad. Sci. Paris 311 (1990), 301–306; Teichmüller spaces and BMOA. Math. Ann. 289 (1991), 613–625] by proving that all convergence-type Fuchsian groups of the first kind fail to have Ruelle’s property. Finally, we give some results about second-kind Fuchsian groups. [-3.2pc]

We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$.

The standard version of Cauchy’s argument principle, applied to a holomorphic function f, requires that f has no zeros on the curve of integration. In this note, we give a generalisation of such a principle which covers the case when f has zeros on the curve, as well as an application.

We characterise bounded and compact generalised weighted composition operators acting from the weighted Bergman space $A^p_\omega $, where $0<p<\infty $ and $\omega $ belongs to the class $\mathcal {D}$ of radial weights satisfying a two-sided doubling condition, to a Lebesgue space $L^q_\nu $. On the way, we establish a new embedding theorem on weighted Bergman spaces $A^p_\omega $ which generalises the well-known characterisation of the boundedness of the differentiation operator $D^n(f)=f^{(n)}$ from the classical weighted Bergman space $A^p_\alpha $ to the Lebesgue space $L^q_\mu $, induced by a positive Borel measure $\mu $, to the setting of doubling weights.

We establish Bohr inequalities for operator-valued functions, which can be viewed as analogues of a couple of interesting results from scalar-valued settings. Some results of this paper are motivated by the classical flavour of Bohr inequality, while others are based on a generalized concept of the Bohr radius problem.

For arbitrary closed countable subsets Z of the unit circle examples of topologically mixing operators on Hilbert spaces are given which have a densely spanning set of eigenvectors with unimodular eigenvalues restricted to Z. In particular, these operators cannot be ergodic in the Gaussian sense.

We prove that if $f:\mathbb{B}^n \to \mathbb{B}^n$, for n ≥ 2, is a homeomorphism with bounded skew over all equilateral hyperbolic triangles, then f is in fact quasiconformal. Conversely, we show that if $f:\mathbb{B}^n \to \mathbb{B}^n$ is quasiconformal then f is η-quasisymmetric in the hyperbolic metric, where η depends only on n and K. We obtain the same result for hyperbolic n-manifolds. Analogous results in $\mathbb{R}^n$, and metric spaces that behave like $\mathbb{R}^n$, are known, but as far as we are aware, these are the first such results in the hyperbolic setting, which is the natural metric to use on $\mathbb{B}^n$.