The ${{Q}_{p}}$ spaces of holomorphic functions on the disk, hyperbolic Riemann surfaces or complex unit ball have been studied deeply. Meanwhile, there are a lot of papers devoted to the $Q_{p}^{\#}$ classes of meromorphic functions on the disk or hyperbolic Riemann surfaces. In this paper, we prove the nesting property (inclusion relations) of $Q_{p}^{\#}$ classes on hyperbolic Riemann surfaces. The same property for ${{Q}_{p}}$ spaces was also established systematically and precisely in earlier work by the authors of this paper.

A borderline case function $f$ for ${{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$ spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter $\beta \,>\,0$. On its support ${{I}_{0}}\,=\,{{\left[ 0,\,1 \right]}^{n}},\,f\left( x \right)$ can be expressed by the binary expansions of the coordinates of $x$. In particular, $f\,=\,{{f}_{\beta }}\,\in \,{{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$ if and only if $\alpha \,<\,\beta \,<\frac{n}{2}$ , while for $\beta \,=\,\alpha $, it was shown by Yue and Dafni that $f$ satisfies a John–Nirenberg inequality for ${{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$. When $\beta \,\ne \,1$, $f$ is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside ${{I}_{0}}$. In addition, it is not monotone along any coordinate direction in any small cube. When the parameter $\beta \,\in \,\left( 0,\,1 \right)$, $f$ is onto from ${{I}_{0}}$ to $\left[ -\frac{1}{1-{{2}^{-\beta }}},\,\frac{1}{1-{{2}^{-\beta }}} \right]$ , and the graph of $f$ has a non-integer fractal dimension $n\,+\,1\,-\beta$.

It is shown that given any sequence of automorphisms ${{\left( {{\phi }_{k}} \right)}_{k}}$ of the unit ball ${{\mathbb{B}}_{N}}$ of ${{\mathbb{C}}^{N}}$ such that $\left\| {{\phi }_{k}}\left( 0 \right) \right\|$ tends to 1, there exists an inner function $I$ such that the family of “non-Euclidean translates” ${{\left( I\,\text{o}\,{{\phi }_{k}} \right)}_{k}}$ is locally uniformly dense in the unit ball of ${{H}^{\infty }}\left( {{\mathbb{B}}_{N}} \right)$ .

Sufficient conditions for the analytic coefficients of the linear differential equation

are found such that all solutions belong to a given -space, or to the Dirichlet type subspace Dp of the classical Hardy space Hp, where 0 < p ≤ 2. For 0 < q < ∞, the space consists of those functions f, analytic in the unit disc D, for which |f(z)|(1 – |z|2)q is uniformly bounded in D, and f ∈ Dp if the integral ∫D |f′(z)|p(1 – |z|2)p–1dσz converges.

It is known that the derivative of a Blaschke product whose zero sequence lies in a Stolz angle belongs to all the Bergman spaces ${{A}^{P}}$ with $0<p<3/2$. The question of whether this result is best possible remained open. In this paper, for a large class of Blaschke products $B$ with zeros in a Stolz angle, we obtain a number of conditions which are equivalent to the membership of ${B}'$ in the space ${{A}^{p}}\left( p>1 \right)$. As a consequence, we prove that there exists a Blaschke product $B$ with zeros on a radius such that ${B}'\,\notin \,{{A}^{3/2}}$.

It is shown that there exists an inner function $I$ defined on the unit ball ${{\text{B}}^{n}}$ of ${{\mathbb{C}}^{n}}$ such that each function holomorphic on ${{\text{B}}^{n}}$ and bounded by 1 can be approximated by “non-Euclidean translates” of $I$.

It is shown that the invariant subspace of the Bergman space $A^p$ of the unit disk, generated by either an $A^p$-interpolating sequence or a singular inner function with a single point mass on the unit circle, is complemented in $A^p$.

We show that there exists a singular inner function $S$ which is universal for noneuclidean translates; that is one for which the set $\{S(\frac{z\,+\,{{z}_{n}}}{1\,+\,{{{\bar{z}}}_{n}}z})\,:\,n\,\in \,\mathbb{N}\}$ is locally uniformly dense in the set of all zero-free holomorphic functions in $\mathbb{D}$ bounded by one.

Following the investigations of $\text{B}$. Abrahamse $[1]$, F. Forelli $[11]$, M. Heins $[14]$ and others, we continue the study of the Pick-Nevanlinna interpolation problem in multiply-connected planar domains. One major focus is on the problem of characterizing the extreme points of the convex set of interpolants of a fixed data set. Several other related problems are discussed.

This paper studies conditions on an analytic function that imply it belongs to ${{M}_{\alpha }}$, the set of multipliers of the family of functions given by $f(z)\,=\,{{\int }_{\left| \zeta \right|=1}}\,\frac{1}{{{(1-\overline{\zeta }z)}^{\alpha }}}d\mu (\zeta )\,(\left| z \right|\,<\,1)$ where $\mu $ is a complex Borel measure on the unit circle and $\alpha \,>\,0$. There are two main theorems. The first asserts that if $0\,<\,\alpha \,<\,1$ and ${{\sup }_{\left| \zeta \right|=1}}\,\int_{0}^{1}{}\left| {f}'(r\zeta ) \right|{{(1-r)}^{\alpha -1}}\,dr<\infty \text{then}f\in {{M}_{\alpha }}$. The second asserts that if $0\,<\,\alpha \,\le \,1,f\,\in \,{{H}^{\infty }}$ and $\sup {{}_{t\int_{0}^{\pi }{{}}}}\frac{\left| f({{e}^{i(t+s)}})-2f({{e}^{it}})+f({{e}^{i(t-s)}}) \right|}{{{s}^{2-\alpha }}}\,ds\,<\,\infty$ then $f\in {{M}_{\alpha }}$. The conditions in these theorems are shown to relate to a number of smoothness conditions on the unit circle for a function analytic in the open unit disk and continuous in its closure.

We study the function spaces ${{Q}_{p}}(R)$ defined on a Riemann surface $R$, which were earlier introduced in the unit disk of the complex plane. The nesting property ${{Q}_{p}}(R)\,\subseteq \,{{Q}_{_{q}}}(R)$ for $0\,<\,p\,<\,q\,<\,\infty $ is shown in case of arbitrary hyperbolic Riemann surfaces. Further, it is proved that the classical Dirichlet space $\text{AD(}R\text{)}\,\subseteq \,{{Q}_{p}}(R)$ for any $p$, $0\,<\,p\,<\,\infty $, thus sharpening T. Metzger's well-known result $\text{AD(}R\text{)}\,\subseteq \,\text{BMOA}(R)$. Also the first author's result $\text{AD}(R)\,\subseteq \,\text{VMOA}(R)$ for a regular Riemann surface $R$ is sharpened by showing that, in fact, $\text{AD(}R\text{)}\,\subseteq \,{{Q}_{p,0}}(R)$ for all $p$, $0\,<\,p\,<\,\infty $. The relationships between ${{Q}_{p}}(R)$ and various generalizations of the Bloch space on $R$ are considered. Finally we show that ${{Q}_{p}}(R)$ is a Banach space for $0\,<\,p\,<\,\infty $.

In this paper we consider questions of recapturing an analytic function in a Banach space from its values on a uniqueness set. The principal method is to use reproducing kernels to construct a sequence in the Banach space which converges in norm to the given functions. The method works for several classical Banach spaces of analytic functions including some Hardy and Bergman spaces.

Let G denote a plane domain bounded by finitely many closed, non-intersecting Jordan curves. We show the following refinement of the stable rank one property of : Suppose that for there exists δ > 0 such that . Then there exist such that

f + hg = exp (k).

Also we obtain a partial result for the algebra H∞(G).

We show that if B is α Blaschke product with nontangential zero set {zk} and 0 < p < 1, 1/2 < αp < 1, then the condition sup0<r<1(l — r) Mp(r, D1+αB) < ∞ is equivalent to the condition {(1 - |zk|(1/p)-αKα} ∊ l∞.