New characterizations for Fock spaces
Guanlong Bao, Pan Ma, Kehe Zhu
TL;DR
The paper studies characterizations of Fock spaces $F^p_α$ on $\mathbb C^n$ by introducing a Gaussian-derived distance $d_α$ and proving that the maximal space $F^∞_α$ is a Lipschitz (distance-controlled) space, analogous to Bloch-space characterizations in Bergman theory via a Bergman-type metric. It establishes Theorem A, linking Lipschitz control under $d_α$ to membership in $F^∞_α$, and shows derivative-based equivalents that connect growth and gradient bounds to geometric Lipschitz properties. A new, simpler one-dimensional Hardy-Littlewood approach is presented for $F^p_α$, leveraging a CZ-type result to relate $f∈F^p_α$ to the behavior of $f'(z)/z$, with an induction framework suggested for higher dimensions. The work also outlines open problems and conjectures about broader Lipschitz-type characterizations with $d_α$, providing a bridge between analytic structure (kernels, radial derivatives) and geometric distance notions in Fock spaces.
Abstract
We show that the maximal Fock space $F^\infty_α$ on $C^n$ is a Lipschitz space, that is, there exists a distance $d_α$ on $C^n$ such that an entire function $f$ on $C^n$ belongs to $F^\infty_α$ if and only if $$|f(z)-f(w)|\le Cd_α(z,w)$$ for some constant $C$ and all $z,w\in C^n$. This can be considered the Fock space version of the following classical result in complex analysis: a holomorphic function $f$ on the unit ball $B_n$ in $C^n$ belongs to the Bloch space if and only if there exists a positive constant $C$ such that $|f(z)-f(w)|\le Cβ(z,w)$ for all $z,w\in B_n$, where $β(z,w)$ is the distance on $B_n$ in the Bergman metric. We also present a new approach to Hardy-Littlewood type characterizations for $F^p_α$.