<p>In this paper, we study the mapping properties of the fractional integral operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$I_{\gamma }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mi>γ</mi> </msub> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ (0<\gamma <d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>γ</mi> <mo><</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> on the Hardy-amalgam spaces with variable exponents. More precisely, we show that if the measurable exponent functions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p\left( \cdot \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mfenced close=")" open="("> <mo>·</mo> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and <i>q</i> satisfy log-Hölder continuity conditions locally and at infinity, then the fractional integral operator of order <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$I_{\gamma }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mi>γ</mi> </msub> </math></EquationSource> </InlineEquation> maps <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {H}^{p\left( \cdot \right) ,q}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mrow> <mi>p</mi> <mfenced close=")" open="("> <mo>·</mo> </mfenced> <mo>,</mo> <mi>q</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> spaces continuously into <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {H}^{p_{1}\left( \cdot \right) ,q_{1}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mfenced close=")" open="("> <mo>·</mo> </mfenced> <mo>,</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> </msup> </math></EquationSource> </InlineEquation> spaces on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb {R}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> under the assumptions <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\frac{1}{ p_{1}\left( x\right) }=\frac{1}{p\left( \cdot \right) }-\frac{\gamma }{d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mfenced close=")" open="("> <mi>x</mi> </mfenced> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>p</mi> <mfenced close=")" open="("> <mo>·</mo> </mfenced> </mrow> </mfrac> <mo>-</mo> <mfrac> <mi>γ</mi> <mi>d</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\frac{1}{q_{1}}=\frac{1}{q}-\frac{\gamma }{d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <msub> <mi>q</mi> <mn>1</mn> </msub> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> <mo>-</mo> <mfrac> <mi>γ</mi> <mi>d</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the boundedness of the classical fractional integral operator in the variable exponent Hardy-amalgam spaces

期刊论文
发表日期：

2026年7月6日

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Lassane Traore,
Cihan Unal

摘要

In this paper, we study the mapping properties of the fractional integral operator $I_{\gamma }$ $I_{γ}$ $ (0<\gamma <d)$ $(0 < γ < d)$ on the Hardy-amalgam spaces with variable exponents. More precisely, we show that if the measurable exponent functions $p\left( \cdot \right) $ $p (\cdot)$ and q satisfy log-Hölder continuity conditions locally and at infinity, then the fractional integral operator of order $\gamma $ $γ$ , $I_{\gamma }$ $I_{γ}$ maps $\mathcal {H}^{p\left( \cdot \right) ,q}$ $H^{p (\cdot), q}$ spaces continuously into $\mathcal {H}^{p_{1}\left( \cdot \right) ,q_{1}}$ $H^{p_{1} (\cdot), q_{1}}$ spaces on $\mathbb {R}^{d}$ $R^{d}$ under the assumptions $\frac{1}{ p_{1}\left( x\right) }=\frac{1}{p\left( \cdot \right) }-\frac{\gamma }{d}$ $\frac{1}{p_{1} (x)} = \frac{1}{p (\cdot)} - \frac{γ}{d}$ and $\frac{1}{q_{1}}=\frac{1}{q}-\frac{\gamma }{d}$ $\frac{1}{q_{1}} = \frac{1}{q} - \frac{γ}{d}$ .