Composition operators on strictly pseudoconvex domains with smooth symbol (Q2510063)

Let \(D\) be a bounded strictly pseudoconvex domain in \(\mathbb{C}^n\) with smooth boundary. When \(D\) is the unit disk in \(\mathbb{C}\), every composition operator is bounded on the weighted Bergman spaces and the Hardy spaces by Littlewood's subordination principle. However, when \(D\) is the unit ball \(\mathbb{B}_n\) in \(\mathbb{C}^n\) with \(n\geq 2\), it is clear that not all the composition operators are bounded on the weighted Bergman spaces or the Hardy spaces. When \(\phi\) is smooth up to the boundary, \textit{W. R. Wogen} [Oper. Theory, Adv. Appl. 35, 249--263 (1988; Zbl 0685.46029)] found a necessary and sufficient condition for \(C_\phi\) to be bounded on \(H^p(B_n)\). The first author and \textit{W. Smith} [J. Math. Anal. Appl. 329, No. 1, 617--633 (2007; Zbl 1115.32004)] generalized this to \(A_\alpha^p(B_n)\), moreover, they showed the jump phenomenon. Now, the authors in this paper generalize the boundedness criteria and the jump phenomenon of composition operators with smooth symbols to bounded strictly pseudoconvex domains \(D\) in \(\mathbb{C}^n\). The authors also show that this condition is equivalent to the compactness of the composition operator from a Hardy of Bergman space into the Bergman space whose weight is \(\frac{1}{4}\) bigger.