<p>Let <Equation ID="Equ10"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2913_Article_Equ10.gif" Format="GIF" Height="40" Rendition="HTML" Resolution="72" Type="Linedraw" Width="242" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \Delta _\lambda :=-\frac{\textrm{d}^2}{\textrm{d} x^2}+\frac{\lambda (\lambda -1)}{x^2}, \quad \lambda \ge 1, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </msub> <mo>:</mo> <mo>=</mo> <mo>-</mo> <mfrac> <msup> <mtext>d</mtext> <mn>2</mn> </msup> <mrow> <mtext>d</mtext> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mfrac> <mo>,</mo> <mspace width="1em" /> <mi>λ</mi> <mo>≥</mo> <mn>1</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>be the Bessel operator on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2913_Article_IEq1.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 consider its positive power <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2913_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$(\Delta _\lambda )^s$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2913_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$0<s<1/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo><</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>), which is a nonlocal operator defined by the heat (or Poisson) semigroup. In this note, we investigate the asymptotic behavior of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2913_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$(\Delta _\lambda )^s$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> </math></EquationSource> </InlineEquation> at <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2913_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$s=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. A similar conclusion for the negative power is also considered.</p>

Fractional Bessel Operators of Order Near Zero

期刊论文
发表日期：

2025年9月8日

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Wanghao Fu,
Bo Li,
Chao Zhang

摘要

Let $\begin{aligned} \Delta _\lambda :=-\frac{\textrm{d}^2}{\textrm{d} x^2}+\frac{\lambda (\lambda -1)}{x^2}, \quad \lambda \ge 1, \end{aligned}$ $\begin{matrix} Δ_{λ} : = - \frac{d^{2}}{d x^{2}} + \frac{λ (λ - 1)}{x^{2}}, λ \geq 1, \end{matrix}$ be the Bessel operator on $\mathbb {R}_+:=(0,\infty )$ $R_{+} : = (0, \infty)$ , and consider its positive power $(\Delta _\lambda )^s$ ${(Δ_{λ})}^{s}$ ( $0<s<1/2$ $0 < s < 1 / 2$ ), which is a nonlocal operator defined by the heat (or Poisson) semigroup. In this note, we investigate the asymptotic behavior of $(\Delta _\lambda )^s$ ${(Δ_{λ})}^{s}$ at $s=0$ $s = 0$ . A similar conclusion for the negative power is also considered.