1. CJM 2010 (vol 62 pp. 1182)
 Yue, Hong

A Fractal Function Related to the JohnNirenberg 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
JohnNirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When
$\beta\neq 1$, $f$ is a selfaffine 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 $[\frac{1}{12^{\beta}}, \frac{1}{12^{\beta}}]$, and the
graph of $f$ has a noninteger fractal dimension $n+1\beta$.
Keywords:Haar wavelets, Q spaces, JohnNirenberg inequality, Greedy expansion, selfaffine, fractal, Box dimension Categories:42B35, 42C10, 30D50, 28A80 

2. CJM 1999 (vol 51 pp. 977)
 Fisher, Stephen D.; Khavinson, Dmitry

Extreme PickNevanlinna Interpolants
Following the investigations of B.~Abrahamse [1], F.~Forelli [11],
M.~Heins [14] and others, we continue the study of the
PickNevanlinna interpolation problem in multiplyconnected 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)
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
$\sup_{\zeta=1} \int^1_0 f'(r\zeta) (1r)^{\alpha1} \,dr<\infty$
then $f \in {\cal M}_\alpha$. The second asserts that if $0<\alpha
\leq 1$, $f \in H^\infty$ and $\sup_t \int^\pi_0 {f(e^{i(t+s)}) 
2f(e^{it}) + f(e^{i(ts)}) \over s^{2\alpha}} \, ds < \infty$ 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 
