<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha \in (0,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\beta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Fix <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$-\pi <\varphi \le \pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>π</mi> <mo><</mo> <mi>φ</mi> <mo>≤</mo> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$|\varphi |>\alpha \pi /2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>φ</mi> <mo stretchy="false">|</mo> <mo>></mo> <mi>α</mi> <mi>π</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We obtain asymptotic upper bounds on the Fourier transform of the radially symmetric tempered distribution <Equation ID="Equ64"> <EquationSource Format="TEX">$\begin{aligned} \mathbb {R}^n\ni x\mapsto E_{\alpha ,\beta }(e^{\dot{\imath } \varphi } |x|^{\sigma }), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>∋</mo> <mi>x</mi> <mo>↦</mo> <msub> <mi>E</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> </mrow> <msup> <mi>e</mi> <mrow> <mover accent="true"> <mi>ı</mi> <mo>˙</mo> </mover> <mi>φ</mi> </mrow> </msup> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mi>σ</mi> </msup> <mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sigma >(n-1)/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>></mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$E_{\alpha ,\beta }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is the two-parameter Mittag-Leffler function. As an application, we obtain some values of the Lebesgue exponent <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p=p(\sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\sigma >(n-1)/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>></mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, for which the Fourier transform is in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L^{p}(\mathbb {R}^{n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Such values cannot be obtained via the well-known <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$L^{p}(\mathbb {R}^{n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> properties of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$E_{\alpha ,\beta }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and the Hausdorff-Young inequality, when <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\sigma \le n/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>≤</mo> <mi>n</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the asymptotic behaviour of the Fourier transform of the Mittag-Leffler function

期刊论文
发表日期：

2025年11月3日

查看全文

Ahmed A. Abdelhakim

摘要

Let $\alpha \in (0,2)$ $α \in (0, 2)$ and let $\beta >0$ $β > 0$ . Fix $-\pi <\varphi \le \pi $ $- π < φ \leq π$ such that $|\varphi |>\alpha \pi /2$ $| φ | > α π / 2$ . We obtain asymptotic upper bounds on the Fourier transform of the radially symmetric tempered distribution $\begin{aligned} \mathbb {R}^n\ni x\mapsto E_{\alpha ,\beta }(e^{\dot{\imath } \varphi } |x|^{\sigma }), \end{aligned}$ $\begin{matrix} R^{n} ∋ x \mapsto E_{α, β} (e^{\dot{ı} φ} {| x |}^{σ}), \end{matrix}$ for $\sigma >(n-1)/2$ $σ > (n - 1) / 2$ , where $E_{\alpha ,\beta }$ $E_{α, β}$ is the two-parameter Mittag-Leffler function. As an application, we obtain some values of the Lebesgue exponent $p=p(\sigma )$ $p = p (σ)$ , $\sigma >(n-1)/2$ $σ > (n - 1) / 2$ , for which the Fourier transform is in $L^{p}(\mathbb {R}^{n})$ $L^{p} (R^{n})$ . Such values cannot be obtained via the well-known $L^{p}(\mathbb {R}^{n})$ $L^{p} (R^{n})$ properties of $E_{\alpha ,\beta }$ $E_{α, β}$ and the Hausdorff-Young inequality, when $\sigma \le n/2$ $σ \leq n / 2$ .