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On the global Gaussian Lipschitz space. (English) Zbl 1375.26007

In their previous contribution [Rev. Mat. Iberoam. 32, No. 4, 1189–1210 (2016; Zbl 1372.46028)], the authors have characterized by means of a Lipschitz-type continuity condition the Lipschitz space introduced in [A. E. Gatto and W. O. Urbina R., Quaest. Math. 38, No. 1, 1–25 (2015; Zbl 1404.42053)] by means of bounded functions. They investigate the issue further in this paper where they present a similar Lipschitz space in the same setting, but now without the boundedness condition. After deriving some properties of the Gaussian Poisson integral, the authors show the main result of this article which states that this newly introduced space can be described by a continuity condition, too. Moreover, the functions in this space turn out to have at most logarithmic growth at infinity, an example of such a function is provided.

MSC:

26A16 Lipschitz (Hölder) classes
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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