We show that the maximal Fock space F 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 α ∞ if and only if | f ( z ) - f ( w ) | ≤ C d α ( z , w ) for some constant C and all z , w ∈ 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 ) | ≤ C β ( z , w ) for all z , w ∈ 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 α p .
Bao et al. (Thu,) studied this question.