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

TL;DR

This paper develops Littlewood-Paley formulas for weighted Fock spaces with $A_ abla$-type weights and characterizes Carleson measures for associated weighted Fock-Sobolev spaces, advancing understanding of these function spaces.

Contribution

It introduces Littlewood-Paley formulas for weighted Fock spaces induced by $A_ abla$-type weights and characterizes Carleson measures for weighted Fock-Sobolev spaces.

Findings

Established Littlewood-Paley formulas for these weighted Fock spaces.

Provided a characterization of Carleson measures for weighted Fock-Sobolev spaces.

Extended the analysis to weights where the Bergman projection is bounded.

Abstract

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