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On the Global Gaussian Lipschitz Space
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 60 / Issue 3 / August 2017
- Published online by Cambridge University Press:
- 03 January 2017, pp. 707-720
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- Article
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It is well known that the standard Lipschitz space in Euclidean space, with exponent α ∈ (0, 1), can be characterized by means of the inequality
, 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.
, where 
is the Poisson integral of the function 