<p>In this paper we prove that for non-negative measurable functions <i>f</i>, <Equation ID="Equ45"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2069_Article_Equ45.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="492" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} I_\alpha f \in BMO({\mathbb {R}}^n) \text { if and only if } I_\alpha f \in BMO^\beta ({\mathbb {R}}^n) \text { for } \beta \in (n-\alpha ,n]. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>I</mi> <mi>α</mi> </msub> <mi>f</mi> <mo>∈</mo> <mi>B</mi> <mi>M</mi> <mi>O</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mspace width="0.333333em" /> <mtext>if and only if</mtext> <mspace width="0.333333em" /> <msub> <mi>I</mi> <mi>α</mi> </msub> <mi>f</mi> <mo>∈</mo> <mi>B</mi> <mi>M</mi> <msup> <mi>O</mi> <mi>β</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mspace width="0.333333em" /> <mtext>for</mtext> <mspace width="0.333333em" /> <mi>β</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mi>α</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Here <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2069_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$I_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> denotes the Riesz potential of order <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2069_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2069_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$BMO^\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mi>M</mi> <msup> <mi>O</mi> <mi>β</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> represents the space of functions of bounded <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2069_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-dimensional mean oscillation introduced in [<CitationRef CitationID="CR12">12</CitationRef>].</p>

A Self-Improving Property of Riesz Potentials in BMO

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2025年6月24日

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You-Wei Benson Chen

摘要

In this paper we prove that for non-negative measurable functions f, $\begin{aligned} I_\alpha f \in BMO({\mathbb {R}}^n) \text { if and only if } I_\alpha f \in BMO^\beta ({\mathbb {R}}^n) \text { for } \beta \in (n-\alpha ,n]. \end{aligned}$ $\begin{matrix} I_{α} f \in B M O (R^{n}) if and only if I_{α} f \in B M O^{β} (R^{n}) for β \in (n - α, n] . \end{matrix}$ Here $I_\alpha $ $I_{α}$ denotes the Riesz potential of order $\alpha $ $α$ and $BMO^\beta $ $B M O^{β}$ represents the space of functions of bounded $\beta $ $β$ -dimensional mean oscillation introduced in [12].