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