<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_436_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu$</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> be a Radon measure on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_436_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {R}}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> which may be non-doubling and only satisfies <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_436_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="128" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu (Q(x,l))\le C_{0}l^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>C</mi> <mn>0</mn> </msub> <msup> <mi>l</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_436_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$x\in {\mathbb {R}}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_436_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$l(Q)>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>l</mi> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, with some fixed constants <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_436_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$C_{0}>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_436_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$n\in (0,d]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>d</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. We introduce a new type of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_436_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$bmo(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mi>m</mi> <mi>o</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> space which looks bigger than the <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_436_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$rbmo(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mi>b</mi> <mi>m</mi> <mi>o</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> space of Dachun Yang(JAMS, 2005). And its four equivalent norms are established by constructing some special types of auxiliary doubling cubes. Then we further obtain that this new <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_436_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$rbmo(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mi>b</mi> <mi>m</mi> <mi>o</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> space actually coincides with the <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_436_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$rbmo(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mi>b</mi> <mi>m</mi> <mi>o</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> space of Dachun Yang.</p>

A new type of bmo space for non-doubling measures

期刊论文
发表日期：

2025年5月16日

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Shining Li,
Haijing Zhao,
Baode Li

摘要

Let $\mu$ $μ$ be a Radon measure on ${\mathbb {R}}^{d}$ $R^{d}$ which may be non-doubling and only satisfies $\mu (Q(x,l))\le C_{0}l^{n}$ $μ (Q (x, l)) \leq C_{0} l^{n}$ for all $x\in {\mathbb {R}}^{d}$ $x \in R^{d}$ , $l(Q)>0$ $l (Q) > 0$ , with some fixed constants $C_{0}>0$ $C_{0} > 0$ and $n\in (0,d]$ $n \in (0, d]$ . We introduce a new type of $bmo(\mu )$ $b m o (μ)$ space which looks bigger than the $rbmo(\mu )$ $r b m o (μ)$ space of Dachun Yang(JAMS, 2005). And its four equivalent norms are established by constructing some special types of auxiliary doubling cubes. Then we further obtain that this new $rbmo(\mu )$ $r b m o (μ)$ space actually coincides with the $rbmo(\mu )$ $r b m o (μ)$ space of Dachun Yang.