Littlewood-Paley formulas and Carleson measures for weighted Fock spaces induced by $A_{\infty }$-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 $P_{\alpha }$, 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 ${\mathcal{F}}_{\beta ,\omega }^{q,n}$.