Composition operators on the bidisc

For 0<p<∞ and α>−1, let A_α^p(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 C_Φ acting from A_α^p(D²) to A_β^p(D²). A theorem of Stessin and Zhu ([18]) establishes that C_Φ:A_α^p(D²)→A_2α+2^p(D²) is always bounded. Additionally, it is known ([9]) that C_Φ:A_α^p(D²)↛A_α+1/4−ϵ^p(D²) for any ϵ>0 if C_Φ:A_α^p(D²)↛A_α^p(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., C_Φ:A_α^p(D²)↛A_2α+2−ϵ^p(D²) for any ϵ>0 if C_Φ:A_α^p(D²)↛A_α+1/2^p(D²) and Φ∈C¹(D²‾). Furthermore, we provide a complete characterization of when C_Φ:A_α^p(D²)→A_α+1/4^p(D²) is bounded for Φ∈C²(D²‾) and when C_Φ:A_α^p(D²)→A_α+1/2^p(D²) is bounded for Φ∈C¹(D²‾).