A characterization of the Gaussian Lipschitz space and sharp estimates for the Ornstein–Uhlenbeck Poisson kernel

Abstract

Abstract. The Gaussian Lipschitz space was defined by Gatto and Urbina, by means of the Ornstein-Uhlenbeck Poisson kernel. We give a characterization of this space in terms of a combination of ordinary Lipschitz continuity conditions. The main tools used in the proof are sharp estimates of the Ornstein-Uhlenbeck Poisson kernel and some of its derivatives.

On the Global Gaussian Lipschitz Space

“…for some A > 0. These spaces and also Gaussian Besov spaces were studied in a series of works; see [2,3,5] and also the authors' paper [4].…”

“…In [4], the authors characterized GLip α , 0 < α < 1, in terms of a Lipschitz-type continuity condition. Indeed, Theorem 1.1 of [4] says that f ∈ GLip α if and only if there exists a positive constant K such that (1.2) |f (x) − f (y)| ≤ K min |x − y| α , |x − y x | 1 + |x|…”

“…The paper is organized as follows. Section 2 contains a needed improvement of the estimate for P t (x, y) and its derivatives in [4]. Some properties of the Gaussian Poisson integral are obtained in Section 3.…”

“…The integral involving K 2 (t, x, y) can also be estimated as in [4], because (1.7) applies in the support of K 2 (t, x, y).…”

“…These functions turn out to have at most logarithmic growth at infinity. The analogous Lipschitz space containing only bounded functions was introduced by Gatto and Urbina and has been characterized by the authors in [4].…”