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Search: MSC category 30E20 ( Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx] )

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1. CJM 2010 (vol 62 pp. 1276)

El Wassouli, Fouzia
A Generalized Poisson Transform of an $L^{p}$-Function over the Shilov Boundary of the $n$-Dimensional Lie Ball
Let $\mathcal{D}$ be the $n$-dimensional Lie ball and let $\mathbf{B}(S)$ be the space of hyperfunctions on the Shilov boundary $S$ of $\mathcal{D}$. The aim of this paper is to give a necessary and sufficient condition on the generalized Poisson transform $P_{l,\lambda}f$ of an element $f$ in the space $\mathbf{B}(S)$ for $f$ to be in $ L^{p}(S)$, $1 > p > \infty.$ Namely, if $F$ is the Poisson transform of some $f\in \mathbf{B}(S)$ (i.e., $F=P_{l,\lambda}f$), then for any $l\in \mathbb{Z}$ and $\lambda\in \mathbb{C}$ such that $\mathcal{R}e[i\lambda] > \frac{n}{2}-1$, we show that $f\in L^{p}(S)$ if and only if $f$ satisfies the growth condition $$ \|F\|_{\lambda,p}=\sup_{0\leq r < 1}(1-r^{2})^{\mathcal{R}e[i\lambda]-\frac{n}{2}+l}\Big[\int_{S}|F(ru)|^{p}du \Big]^{\frac{1}{p}} < +\infty. $$

Keywords:Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundle
Categories:32A45, 30E20, 33C67, 33C60, 33C55, 32A25, 33C75, 11K70

2. 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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