New Characterizations for the Weighted Fock Spaces

It is known that the standard weighted Bergman spaces over the complex ball can be characterized by means of Lipschitz type conditions. It is also known that the same spaces can be characterized, except for a critical case, by means of integrability conditions of double integrals associated with difference quotients of Bergman functions. In this paper we obtain characterizations of similar type for the class of weighted Fock spaces whose weights grow or decay polynomially at ∞. In particular, our result for double-integrability characterization shows that there is no critical case for the Fock spaces under consideration. As applications we also obtain similar characterizations for the corresponding weighted Fock–Sobolev spaces of arbitrary real orders.