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1. CJM 2010 (vol 62 pp. 1182)

Yue, Hong
 A Fractal Function Related to the John-Nirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$ A borderline case function $f$ for $Q_{\alpha}({\mathbb R^n})$ spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter $\beta>0$. On its support $I_0=[0, 1]^n$, $f(x)$ can be expressed by the binary expansions of the coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb R^n})$ 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 $Q_{\alpha}({\mathbb R^n})$. When $\beta\neq 1$, $f$ 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 $\beta\in (0, 1)$, $f$ is onto from $I_0$ to $[-\frac{1}{1-2^{-\beta}}, \frac{1}{1-2^{-\beta}}]$, and the graph of $f$ has a non-integer fractal dimension $n+1-\beta$. Keywords:Haar wavelets, Q spaces, John-Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimensionCategories:42B35, 42C10, 30D50, 28A80

2. CJM 1999 (vol 51 pp. 977)

Fisher, Stephen D.; Khavinson, Dmitry
 Extreme Pick-Nevanlinna Interpolants Following the investigations of 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. Categories:30D50, 30D99

3. CJM 1998 (vol 50 pp. 449)

Aulaskari, Rauno; He, Yuzan; Ristioja, Juha; Zhao, Ruhan
 $Q_p$ spaces on Riemann surfaces 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 $0 Categories:30D45, 30D50, 30F35 4. CJM 1998 (vol 50 pp. 595) Luo, Donghan; MacGregor, Thomas  Multipliers of fractional Cauchy transforms and smoothness conditions This paper studies conditions on an analytic function that imply it belongs to${\cal M}_\alpha$, the set of multipliers of the family of functions given by$f(z) = \int_{|\zeta|=1} {1 \over (1-\overline\zeta z)^\alpha} \,d\mu (\zeta)(|z|<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$\sup_{|\zeta|=1} \int^1_0 |f'(r\zeta)| (1-r)^{\alpha-1} \,dr<\infty$then$f \in {\cal M}_\alpha$. The second asserts that if$0<\alpha \leq 1$,$f \in H^\infty$and$\sup_t \int^\pi_0 {|f(e^{i(t+s)}) - 2f(e^{it}) + f(e^{i(t-s)})| \over s^{2-\alpha}} \, ds < \infty$then$f \in {\cal 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. Categories:30E20, 30D50
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