<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0<\alpha <2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <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> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha /2<|s|\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo><</mo> <mo stretchy="false">|</mo> <mi>s</mi> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We obtain values of the Lebesgue exponent <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p=p(\gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\gamma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, for which the Fourier transform of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ E_{\alpha ,\beta }(e^{\dot{\imath }\pi s} |\cdot |^{\gamma })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> </mrow> <msup> <mi>e</mi> <mrow> <mover accent="true"> <mi>ı</mi> <mo>˙</mo> </mover> <mi>π</mi> <mi>s</mi> </mrow> </msup> <msup> <mrow> <mo stretchy="false">|</mo> <mo>·</mo> <mo stretchy="false">|</mo> </mrow> <mi>γ</mi> </msup> <mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is an <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^{p}(\mathbb {R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> function, using tools from the Littlewood-Paley theory. This question arises in the analysis of certain space-time fractional diffusion and Schrödinger problems and has been solved for the particular cases <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\beta =\alpha ,1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>=</mo> <mi>α</mi> <mo>,</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$s=-1/2,1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> via asymptotic analysis of Fox <i>H</i>-functions. The Littlewood-Paley theory provides a simpler proof that allows considering all values of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\beta ,\gamma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>,</mo> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$s\in (-1,1]\setminus [-\alpha /2,\alpha /2]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mo stretchy="false">[</mo> <mo>-</mo> <mi>α</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo>,</mo> <mi>α</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. This enabled us to prove various key estimates for a general class of nonlocal space-time problems.</p>

A Littlewood-Paley Approach to the Mittag-Leffler Function in the Frequency Space and Applications to Nonlocal Problems

期刊论文
发表日期：

2026年5月29日

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Ahmed A. Abdelhakim

摘要

Let $0<\alpha <2$ $0 < α < 2$ , $\beta >0$ $β > 0$ and $\alpha /2<|s|\le 1$ $α / 2 < | s | \leq 1$ . We obtain values of the Lebesgue exponent $p=p(\gamma )$ $p = p (γ)$ , $\gamma >0$ $γ > 0$ , for which the Fourier transform of $ E_{\alpha ,\beta }(e^{\dot{\imath }\pi s} |\cdot |^{\gamma })$ $E_{α, β} (e^{\dot{ı} π s} {| \cdot |}^{γ})$ is an $L^{p}(\mathbb {R}^d)$ $L^{p} (R^{d})$ function, using tools from the Littlewood-Paley theory. This question arises in the analysis of certain space-time fractional diffusion and Schrödinger problems and has been solved for the particular cases $\alpha \in (0,1)$ $α \in (0, 1)$ , $\beta =\alpha ,1$ $β = α, 1$ , and $s=-1/2,1$ $s = - 1 / 2, 1$ via asymptotic analysis of Fox H-functions. The Littlewood-Paley theory provides a simpler proof that allows considering all values of $\beta ,\gamma >0$ $β, γ > 0$ and $s\in (-1,1]\setminus [-\alpha /2,\alpha /2]$ $s \in (- 1, 1] \ [- α / 2, α / 2]$ . This enabled us to prove various key estimates for a general class of nonlocal space-time problems.