Littlewood-Paley formulas and Carleson measures for weighted Fock spaces induced by $A_{\mathrm{\infty }}$-type weights

We obtain Littlewood-Paley formulas for Fock spaces ${\mathcal{F}}_{\beta ,\omega }^q$ induced by weights $\omega\in A^{restricted}\infty=\cup{1\le p<\infty}A^{restricted}{p}$,where$A^{restricted}{p}$istheclassofweightssuchthattheBergmanprojection$P_\alpha$,ontheclassicalFockspace$$\mathcal{F}$^2_{\alpha}$,isboundedon$$$\mathcal{L}$^p_{\alpha,\omega}:=\left{f:, \int_{$\mathbb{C}$}|f(z)|^pe^{-p\frac{\alpha}{2}|z|^2},\omega(z)dA(z)<\infty \right}. $$Usingtheseequivalentnormsfor$$\mathcal{F}$^q_{\beta,\omega}$ we characterize the Carleson measures for weighted Fock-Sobolev spaces ${\mathcal{F}}_{\beta ,\omega }^{q,n}$.