<p>In this paper, we study an analog of Titchmarsh’s theorem for the Hartley–Bessel transform on the real line. Using the Hartley–Bessel operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Lambda _\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Λ</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>, the associated generalized translation, and suitable higher-order differences, we obtain integrability and decay estimates for the Hartley–Bessel transform in weighted <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> spaces. More precisely, we prove conditions ensuring that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {H}_\alpha (f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">H</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> belongs to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^\beta (\mathbb {R},d\mu _\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>β</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <mi>d</mi> <msub> <mi>μ</mi> <mi>α</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$1<p\le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We also introduce a Hartley–Lipschitz class defined by means of the operator <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Lambda _\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Λ</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> and establish a corresponding estimate for the tail integral of the transform. Finally, in the Hilbert space case <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we prove an equivalence between this Hartley–Lipschitz condition and a decay estimate for the Hartley–Bessel transform.</p>

An analog of Titchmarsh’s theorem for the Hartley–Bessel transform

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发表日期：

2026年8月5日

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Mohamed Laamri,
Abdellatif Akhlidj

摘要

In this paper, we study an analog of Titchmarsh’s theorem for the Hartley–Bessel transform on the real line. Using the Hartley–Bessel operator $\Lambda _\alpha $ $Λ_{α}$ , the associated generalized translation, and suitable higher-order differences, we obtain integrability and decay estimates for the Hartley–Bessel transform in weighted $L^p$ $L^{p}$ spaces. More precisely, we prove conditions ensuring that $\mathcal {H}_\alpha (f)$ $H_{α} (f)$ belongs to $L^\beta (\mathbb {R},d\mu _\alpha )$ $L^{β} (R, d μ_{α})$ , where $1<p\le 2$ $1 < p \leq 2$ . We also introduce a Hartley–Lipschitz class defined by means of the operator $\Lambda _\alpha $ $Λ_{α}$ and establish a corresponding estimate for the tail integral of the transform. Finally, in the Hilbert space case $p=2$ $p = 2$ , we prove an equivalence between this Hartley–Lipschitz condition and a decay estimate for the Hartley–Bessel transform.