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

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