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<p>In this paper, we present a self-contained proof of the fact that the (modified) Riesz transforms in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {R}}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> induce well-defined linear and bounded operators on Sarason’s space VMO, of functions of vanishing mean oscillations. Among other things, we use this result to provide a quantitative characterization of VMO in terms of the action of the (modified) Riesz transforms. In the final section, we indicate how a larger class of singular integral operators of convolution type in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathbb {R}}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> are bounded when acting from VMO into itself.</p>

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RIESZ TRANSFORMS ON SPACES OF VANISHING MEAN OSCILLATIONS

  • 期刊论文

  • 发表日期：

    2026年2月18日

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  • Dorina Mitrea,
  • Marius Mitrea

摘要

In this paper, we present a self-contained proof of the fact that the (modified) Riesz transforms in \({\mathbb {R}}^n\) R n induce well-defined linear and bounded operators on Sarason’s space VMO, of functions of vanishing mean oscillations. Among other things, we use this result to provide a quantitative characterization of VMO in terms of the action of the (modified) Riesz transforms. In the final section, we indicate how a larger class of singular integral operators of convolution type in \({\mathbb {R}}^n\) R n are bounded when acting from VMO into itself.