On the Global Gaussian Lipschitz Space

Journal article, 2017

© 2017 Edinburgh Mathematical Society. It is well known that the standard Lipschitz space in Euclidean space, with exponent α ϵ (0, 1), can be characterized by means of the inequality |δPtf/δt|≤tα-1, where is the Poisson integral of the function f. There are two cases: One can either assume that the functions in the space are bounded, or one can not make such an assumption. In the setting of the Ornstein-Uhlenbeck semigroup in Rn Gatto and Urbina defined a Lipschitz space by means of a similar inequality for the Ornstein-Uhlenbeck Poisson integral, considering bounded functions. In a preceding paper, the authors characterized that space by means of a Lipschitz-type continuity condition. The present paper defines a Lipschitz space in the same setting in a similar way, but now without the boundedness condition. Our main result says that this space can also be described by a continuity condition. The functions in this space turn out to have at most logarithmic growth at infinity.