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<p>We give a simple proof of Fefferman-Stein type characterization of the space CMO(ℝ<sup><i>n</i></sup>), that is, <i>f</i> ∈ CMO(ℝ<sup><i>n</i></sup>) if and only if <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10587_2025_3124_Article_Equ1.gif" Format="GIF" Height="51" Rendition="HTML" Resolution="72" Type="Linedraw" Width="134" /> </MediaObject> <EquationSource Format="TEX">\(f=\phi+\sum_{j=1}^{n}R_{j}\varphi_{j},\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>f</mi> <mo>=</mo> <mi>ϕ</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>n</mi> </mrow> </munderover> <msub> <mi>R</mi> <mrow> <mi>j</mi> </mrow> </msub> <msub> <mi>φ</mi> <mrow> <mi>j</mi> </mrow> </msub> <mo>,</mo> </math></EquationSource> </Equation> where <i>ϕ</i>, <i>φ</i><sub><i>j</i></sub> ∈ <i>C</i><sub>0</sub>(ℝ<sup><i>n</i></sup>) and <i>R</i><sub><i>j</i></sub>, <i>j</i> = 1, 2,…, <i>n</i>, are the Riesz transforms. Notice that this result was established by G. Bourdaud (2002), but his proof depends on the Fefferman-Stein type decomposition of the space VMO(∝<sup><i>n</i></sup>) obtained by D. Sarason (1975). We will provide a direct method to prove this conclusion.</p>

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A simple proof of Fefferman-Stein type characterization of CMO(ℝn) space

  • 期刊论文

  • 发表日期：

    2025年2月6日

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  • Qingdong Guo,
  • Zeqiang Linli,
  • Kang Hu

摘要

We give a simple proof of Fefferman-Stein type characterization of the space CMO(ℝn), that is, f ∈ CMO(ℝn) if and only if \(f=\phi+\sum_{j=1}^{n}R_{j}\varphi_{j},\) f = ϕ + ∑ j = 1 n R j φ j , where ϕ, φj ∈ C0(ℝn) and Rj, j = 1, 2,…, n, are the Riesz transforms. Notice that this result was established by G. Bourdaud (2002), but his proof depends on the Fefferman-Stein type decomposition of the space VMO(∝n) obtained by D. Sarason (1975). We will provide a direct method to prove this conclusion.