Composition operators on strictly pseudoconvex domains with smooth symbol

Hyungwoon Koo and Song-Ying Li

Vol. 268 (2014), No. 1, 135–153

DOI: 10.2140/pjm.2014.268.135

Abstract

It is well known that the composition operator $C_{\varphi }$ is unbounded on Hardy and Bergman spaces on the unit ball $B_n$ in ${ℂ}^n$ when $n>1$ for a linear holomorphic self-map $\varphi$ of $B_n$. We find a sufficient and necessary condition for a composition operator with smooth symbol to be bounded on Hardy or Bergman spaces over a bounded strictly pseudoconvex domain in ${ℂ}^n$. Moreover, we show that this condition is equivalent to the compactness of the composition operator from a Hardy or Bergman space into the Bergman space whose weight is $\frac{1}{4}$ bigger. We also prove that a certain jump phenomenon occurs when the composition operator is not bounded. Our results generalize known results on the unit ball to strictly pseudoconvex domains.

Keywords

composition operator, strictly pseudoconvex domain, boundedness, smooth symbol

Mathematical Subject Classification 2010

Primary: 47B33

Secondary: 32T15, 32A36

Milestones

Received: 26 September 2012

Revised: 19 May 2013

Accepted: 13 June 2013

Published: 21 May 2014