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A Generalized Poisson Transform of an Lp -Function over the Shilov Boundary of the n-Dimensional Lie Ball
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- Journal:
- Canadian Journal of Mathematics / Volume 62 / Issue 6 / 14 December 2010
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1276-1292
- Print publication:
- 14 December 2010
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- Article
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Let
$D$ be the 
$n$ -dimensional Lie ball and let 
$B\text{(S)}$ be the space of hyperfunctions on the Shilov boundary 
$S$ of $D$ . The aim of this paper is to give a necessary and sufficient condition on the generalized Poisson transform 
${{P}_{l,\text{ }\!\!\lambda\!\!\text{ }}}f$ of an element 
$f$ in the space $B\text{(S)}$ for $f$ to be in 
${{L}^{p}}\left( S \right)$ , 
$1\,<\,p\,<\,\infty $ . Namely, if 
$F$ is the Poisson transform of some 
$f\in \,B(S)$ 
$F\,=\,{{P}_{l,\lambda }}f$ ), then for any 
$l\,\in \,Z$ ) and 
$\lambda \,\in \,C$ such that 
$Re[\text{i}\lambda ] > \frac{n}{2}\,-\,1$ , we show that 
$f\,\in \,{{L}^{p}}\text{(}S\text{)}$ if and only if $f$ satisfies the growth condition
$${{\left\| F \right\|}_{\lambda ,p}}=\underset{0\le r<1}{\mathop{\sup }}\,{{\left( 1\,-\,{{r}^{2}} \right)}^{\operatorname{Re}\left[ \text{i }\lambda \text{ } \right]-\frac{n}{2}+l}}{{\left[ \,\int_{s}{{{\left| F\left( ru \right) \right|}^{p}}du} \right]}^{\frac{1}{p}}}<\,+\infty $$
