The diagonal mapping in mixed norm spaces

Guangbin Ren¹ ; Jihuai Shi²

¹Department of Mathematics University of Science and Technology of China Hefei, Anhui 230026, P.R. China and Department of Mathematics University of Aveiro 3810-193 Aveiro, Portugal E-mail: [e-mail]
²Department of Mathematics University of Science and Technology of China Hefei, Anhui 230026, P.R. China

Studia Mathematica, Tome 163 (2004) no. 2, pp. 103-117

Résumé

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 $0 q\le\infty$.

Détail

DOI : 10.4064/sm163-2-1

Keywords: holomorphic function unit polydisc mathbb consider its restriction diagonal function unit disc mathbb defined mathcal dots prove diagonal mapping mathcal maps mixed norm space alpha polydisc mixed norm space alpha n unit disc infty infty