Littlewood-Paley theory on Gaussian spaces

Abstract

In this article we prove a number of inequalities of Littlewood-Paley-Stein (LPS) type for functions on general Gaussian spaces (s. below). In finite dimensional Euclidean spaces (with Lebesgue measure) the power of such inequalities has been demonstrated in Stein’s book [12]. In his second book [13], Stein treats other spaces too: also the situation of a general measure space ( X, μ ). However the latter case is too general to allow for a rich class of inequalities (cf. Theorem 10 in [13]).