BLO spaces associated with the Ornstein–Uhlenbeck operator

Abstract

Let ( R n , | ⋅ | , d γ ) be the Gauss measure metric space, where R n denotes the n-dimensional Euclidean space, | ⋅ | the Euclidean norm and d γ ( x ) ≡ π − n / 2 e − | x | 2 d x for all x ∈ R n the Gauss measure. In this paper, for any a ∈ ( 0 , ∞ ) , the authors introduce some BLO a ( γ ) space, namely, the space of functions with bounded lower oscillation associated with a given class of admissible balls with parameter a. Then the authors prove that the noncentered local natural Hardy–Littlewood maximal operator is bounded from BMO ( γ ) of Mauceri and Meda to BLO a ( γ ) . Moreover, a characterization of the space BLO a ( γ ) , via the local natural maximal operator and BMO ( γ ) , is given. The authors further prove that a class of maximal singular integrals, including the corresponding maximal operators of both imaginary powers of the Ornstein–Uhlenbeck operator and Riesz transforms of any order associated with the Ornstein–Uhlenbeck operator, are bounded from L ∞ ( γ ) to BLO a ( γ ) .