BLO spaces associated with the Ornstein–Uhlenbeck operator

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Abstract

Let (Rn,||,dγ) be the Gauss measure metric space, where Rn denotes the n-dimensional Euclidean space, || the Euclidean norm and dγ(x)πn/2e|x|2dx for all xRn the Gauss measure. In this paper, for any a(0,), the authors introduce some BLOa(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 BLOa(γ). Moreover, a characterization of the space BLOa(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 BLOa(γ).

MSC

primary
42B35
secondary
42B25
42B30

Keywords

BLO(γ)
BMO(γ)
Local (natural) Hardy–Littlewood maximal function
Gauss measure
Maximal singular integral
Imaginary power
Riesz transform
Ornstein–Uhlenbeck operator

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1

The author is supported by the National Science Foundation for Distinguished Young Scholars (Grant No. 10425106) of China.