L^p-Poisson integral representations of solutions of the Hua system on the Lie ball in ℂⁿ

Abstract

Let D={z∈ℂ ⁿ /1−2z¯z ^t+|zz ^t|²>0 and |zz ^t|<1} be the Lie ball in ℂ ⁿ and let ℋ _q be the associated Hua operator. Then, for a complex number λ such that ℛe(iλ)>n/2−1, we establish that every function F satisfying the following Hua system of differential equations:

is the Poisson transform of an L ^p -function (1<p<+∞) over the Shilov boundary S of D if and only if it satisfies the following growth condition of Hardy type:

Keywords:

• Lie ball
• Shilov boundary
• Hua operator
• Hyperfunctions

Acknowledgements

I would like to thank Professor A. Boussejra for useful discussions.

Additional information

Notes on contributors

Fouzia El Wassouli

Email: [email protected]