The diagonal mapping in mixed norm spaces
Volume 163 / 2004
Studia Mathematica 163 (2004), 103-117
MSC: 32A37, 47B38.
DOI: 10.4064/sm163-2-1
Abstract
For any holomorphic function $F$ in the unit polydisc $U^n$ of ${\mathbb C}^n$, we consider its restriction to the diagonal, i.e., the function in the unit disc $U$ of $\mathbb C$ defined by ${\mathcal D}F(z)=F(z,\dots,z)$, and prove that the diagonal mapping ${\mathcal D}$ maps the mixed norm space $H^{p, q, \alpha}(U^n)$ of the polydisc onto the mixed norm space $H^{p,q,|\alpha|+(p/q+1)(n-1)}(U)$ of the unit disc for any $0< p< \infty$ and $0< q\le\infty$.
