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Zbl 0857.42009
Lu, Shanzhen; Yang, Dachun
The central BMO spaces and Littlewood-Paley operators.
(English)
[J] Approximation Theory Appl. 11, No.3, 72-94 (1995). ISSN 1000-9221

Let $1<p<\infty$. A function $f\in L^p_{\text{loc}}(\bbfR^n)$ is said to belong to $\text{CBMO}_p(\bbfR^n)$ (central bounded mean oscillation space), if $$\sup_{r>0} \Biggl(|B(r)|^{-1} \int_{B(r)}|f(x)-f_{B(r)}|^pdx\Biggr)^{1/p}<\infty,$$ where $f_{B(r)}$ is the integral mean of $f$ over the ball $B(r)$ with center at the origin and radius $r$. This space is a local version of the usual $\text{BMO}(\bbfR^n)$, and a dual space of a kind of Hardy space associated with the Herz space. The authors give a characterization of $\text{CBMO}_2(\bbfR^n)$ in terms of the central Carleson measure. Using this, they give some results on $\text{CBMO}_2(\bbfR^n)$ boundedness of several classes of general Littlewood-Paley operators.
[K.Yabuta (Nara)]
MSC 2000:
*42B25 Maximal functions
42B30 Hp-spaces (Fourier analysis)

Keywords: BMO; CBMO; central bounded mean oscillation; Hardy space; Herz space; central Carleson measure; Littlewood-Paley operators

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