Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$s,b\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. This article is devoted to establishing the Fourier-analytical characterization of the pointwise multiplier space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">$M(B^{s,b}_{p,q}(\mathbb {R}^n))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mi>s</mi> <mo>,</mo> <mi>b</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the logarithmic Besov space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq3.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$B^{s,b}_{p,q}(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mi>s</mi> <mo>,</mo> <mi>b</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in the endpoint cases, that is, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$p,q\in \{1,\infty \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. The authors first obtain such a characterization for the cases where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$p=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$q=\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and where <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$p=\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$q=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Applying this, the authors then establish the duality formula <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq9.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="224" /> </InlineMediaObject> <EquationSource Format="TEX">$M(B^{s,b}_{p,q}(\mathbb {R}^n))=M(B^{-s,-b}_{p',q'}(\mathbb {R}^n)),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mi>s</mi> <mo>,</mo> <mi>b</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>M</mi> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>B</mi> <mrow> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>q</mi> <mo>′</mo> </msup> </mrow> <mrow> <mo>-</mo> <mi>s</mi> <mo>,</mo> <mo>-</mo> <mi>b</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$s,b\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">$p,q\in [1,\infty ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq12.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$p'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq13.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$q'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> are respectively the conjugate indices of p and q. This duality principle is further applied to establish the Fourier-analytical characterization of <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">$M(B^{s,b}_{p,q}(\mathbb {R}^n))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mi>s</mi> <mo>,</mo> <mi>b</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in the cases where <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq15.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$p=\infty =q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>∞</mi> <mo>=</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation> and where <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1129_Article_IEq16.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$p=1=q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>=</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation>.

Pointwise Multiplier Spaces of Logarithmic Besov Spaces: Duality Principle and Fourier-Analytical Characterization in Endpoint Cases

期刊论文
发表日期：

2025年9月20日

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Ziwei Li,
Dachun Yang,
Wen Yuan

摘要

Let $s,b\in \mathbb {R}$ $s, b \in R$ . This article is devoted to establishing the Fourier-analytical characterization of the pointwise multiplier space $M(B^{s,b}_{p,q}(\mathbb {R}^n))$ $M (B_{p, q}^{s, b} (R^{n}))$ for the logarithmic Besov space $B^{s,b}_{p,q}(\mathbb {R}^n)$ $B_{p, q}^{s, b} (R^{n})$ in the endpoint cases, that is, $p,q\in \{1,\infty \}$ $p, q \in {1, \infty}$ . The authors first obtain such a characterization for the cases where $p=1$ $p = 1$ and $q=\infty $ $q = \infty$ and where $p=\infty $ $p = \infty$ and $q=1$ $q = 1$ . Applying this, the authors then establish the duality formula $M(B^{s,b}_{p,q}(\mathbb {R}^n))=M(B^{-s,-b}_{p',q'}(\mathbb {R}^n)),$ $M (B_{p, q}^{s, b} (R^{n})) = M (B_{p^{'}, q^{'}}^{- s, - b} (R^{n})),$ where $s,b\in \mathbb {R}$ $s, b \in R$ , $p,q\in [1,\infty ]$ $p, q \in [1, \infty]$ , and $p'$ $p^{'}$ and $q'$ $q^{'}$ are respectively the conjugate indices of p and q. This duality principle is further applied to establish the Fourier-analytical characterization of $M(B^{s,b}_{p,q}(\mathbb {R}^n))$ $M (B_{p, q}^{s, b} (R^{n}))$ in the cases where $p=\infty =q$ $p = \infty = q$ and where $p=1=q$ $p = 1 = q$ .