New characterizations for Fock spaces

We show that the maximal Fock space $F^\infty_\alpha$ on $C^n$ is a Lipschitz space, that is, there exists a distance $d_\alpha$ on $C^n$ such that an entire function $f$ on $C^n$ belongs to $F^\infty_\alpha$ if and only if $$|f(z)-f(w)|\le Cd_\alpha(z,w)$$ 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 $|f(z)-f(w)|\le C\beta(z,w)$ for all $z,w\in B_n$, where $\beta(z,w)$ is the distance on $B_n$ in the Bergman metric. We also present a new approach to Hardy-Littlewood type characterizations for $F^p_\alpha$.