New characterizations for Fock spaces

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


.