Composition operators on the bidisc

• Published in: Journal of mathematical analysis and applications Vol. 554; no. 1; p. 129902
• Main Authors: Koo, Hyungwoon, Li, Song-Ying
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.02.2026
• Subjects: Composition operator; Jump phenomenon; Polydisk; Weighted Bergman space; Polydisk; Weighted Bergman space; Composition operator; Jump phenomenon
• ISSN: 0022-247X
• DOI: 10.1016/j.jmaa.2025.129902

For 0<p<∞ and α>−1, let Aαp(D2) denote the weighted Bergman space over the unit bidisc D2 in C2. Given a holomorphic self-map Φ of D2, we investigate the jump phenomenon of the composition operator CΦ acting from Aαp(D2) to Aβp(D2). A theorem of Stessin and Zhu ([18]) establishes that CΦ:Aαp(D2)→A2α+2p(D2) is always bounded. Additionally, it is known ([9]) that CΦ:Aαp(D2)↛Aα+1/4−ϵp(D2) for any ϵ>0 if CΦ:Aαp(D2)↛Aαp(D2) and Φ∈C1(D2‾). We show that there is a jump phenomenon at the upper end as well, providing a complete analysis of both lower and upper jumps in the weight. We prove that there is a minimal jump of size α+3/2 at the upper end, i.e., CΦ:Aαp(D2)↛A2α+2−ϵp(D2) for any ϵ>0 if CΦ:Aαp(D2)↛Aα+1/2p(D2) and Φ∈C1(D2‾). Furthermore, we provide a complete characterization of when CΦ:Aαp(D2)→Aα+1/4p(D2) is bounded for Φ∈C2(D2‾) and when CΦ:Aαp(D2)→Aα+1/2p(D2) is bounded for Φ∈C1(D2‾).