<p>Fix <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2801_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Consider the Bessel operator <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2801_Article_IEq2.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">$\triangle _\lambda :=-\frac{d^2}{dx^2}-\frac{2\lambda }{x}\frac{d}{dx}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>▵</mi> <mi>λ</mi> </msub> <mo>:</mo> <mo>=</mo> <mo>-</mo> <mfrac> <msup> <mi>d</mi> <mn>2</mn> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <mi>λ</mi> </mrow> <mi>x</mi> </mfrac> <mfrac> <mi>d</mi> <mrow> <mi mathvariant="italic">dx</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2801_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R_+}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2801_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R_+}:=(0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2801_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$dm_\lambda :=x^{2\lambda }dx$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <msub> <mi>m</mi> <mi>λ</mi> </msub> <mo>:</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow> <mn>2</mn> <mi>λ</mi> </mrow> </msup> <mi>d</mi> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> with <i>dx</i> the Lebesgue measure. We provide a deeper study of the Bessel Riesz transform and fractional integral operator via the related Besov and Triebel–Lizorkin spaces associated with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2801_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$\triangle _\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>▵</mi> <mi>λ</mi> </msub> </math></EquationSource> </InlineEquation>. Moreover, we investigate some possible characterization of the commutator of fractional integral operator, which was missing in the literature of the Bessel setting.</p>

The Riesz Transform and Fractional Integral Operators in the Bessel Setting

期刊论文
发表日期：

2025年6月5日

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Jorge J. Betancor,
Xuan Thinh Duong,
Ming-Yi Lee,
Ji Li,
Brett D. Wick

摘要

Fix $\lambda >0$ $λ > 0$ . Consider the Bessel operator $\triangle _\lambda :=-\frac{d^2}{dx^2}-\frac{2\lambda }{x}\frac{d}{dx}$ $▵_{λ} : = - \frac{d^{2}}{d x^{2}} - \frac{2 λ}{x} \frac{d}{dx}$ on $\mathbb {R_+}$ $R_{+}$ , where $\mathbb {R_+}:=(0,\infty )$ $R_{+} : = (0, \infty)$ and $dm_\lambda :=x^{2\lambda }dx$ $d m_{λ} : = x^{2 λ} d x$ with dx the Lebesgue measure. We provide a deeper study of the Bessel Riesz transform and fractional integral operator via the related Besov and Triebel–Lizorkin spaces associated with $\triangle _\lambda $ $▵_{λ}$ . Moreover, we investigate some possible characterization of the commutator of fractional integral operator, which was missing in the literature of the Bessel setting.