Littlewood-Paley Formulas and Carleson Measures for Weighted Fock Spaces Induced by A ∞ $A_{\infty }$ -Type Weights

Abstract

We obtain Littlewood-Paley formulas for Fock spaces ${\mathcal {F}}_{\beta ,\omega }^{q}$ induced by weights $\omega \in {A}_{\infty }^{restricted} = \cup _{1 \le p < \infty } {A}_{p}^{restricted}$ , where $ {A}_{p}^{restricted} $ is the class of weights such that the Bergman projection Pα, on the classical Fock space ${\mathcal {F}}_{\alpha }^{2}$ , is bounded on $${\mathcal{L}}_{\alpha,\omega}^{p} := \left\{f:\, {\int}_{\mathbb{C}}|f(z)|^{p} e^{-p\frac{\alpha}{2}|z|^{2}}\,\omega(z)dA(z)<\infty \right\}. $$ Using these equivalent norms for ${\mathcal {F}}_{\beta ,\omega }^{q}$ we characterize the Carleson measures for weighted Fock-Sobolev spaces ${\mathcal {F}}_{\beta ,\omega }^{q,n}$ .