Composition operators on weighted Fock spaces induced by \(A_{\infty }\)-type weights

In this paper, we study the composition operators \(C_{\varphi }\) $C_{\phi }$ acting on the weighted Fock spaces \(F^p_{\alpha ,w}\) $F_{\alpha ,w}^p$ , where w is a weight satisfying some restricted \(A_{\infty }\) $A_{\infty }$ -conditions. We first characterize the boundedness and compactness of the composition operators \(C_{\varphi }:F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}\) $C_{\phi }:F_{\alpha ,w}^p\to F_{\beta ,v}^q$ for all \(0<p,q<\infty\) $0<p,q<\infty$ in terms of certain Berezin type integral transforms. A new condition for the bounded embedding \(I_d:F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu )\) $I_d:F_{\alpha ,w}^p\to L^q(\mathbb{C},\mu )$ in the case \(p>q\) $p>q$ is also obtained. Then, in the case that \(w(z)=(1+|z|)^{mp}\) $w(z)={(1+|z|)}^{\mathit{mp}}$ for \(m\in \mathbb {R}\) $m\in \mathbb{R}$ , using some Taylor coefficient estimates, we establish an upper bound for the approximation numbers of composition operators acting on \(F^p_{\alpha ,w}\) $F_{\alpha ,w}^p$ .