<p>In this manuscript, we extend our previous work on the Riemann-Liouville fractional integral of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^p(t_0,t_1;X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>;</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In particular, we establish its boundedness for the case <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha > 1/p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1< p < \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, as well as the case <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p = \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we present a comprehensive summary of the results obtained so far.</p>

The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces IV

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2026年2月11日

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Paulo Mendes Carvalho Neto,
Renato Fehlberg Júnior

摘要

In this manuscript, we extend our previous work on the Riemann-Liouville fractional integral of order $\alpha > 0$ $α > 0$ in $L^p(t_0,t_1;X)$ $L^{p} (t_{0}, t_{1}; X)$ . In particular, we establish its boundedness for the case $\alpha > 1/p$ $α > 1 / p$ , with $1< p < \infty $ $1 < p < \infty$ , as well as the case $\alpha >0$ $α > 0$ with $p = \infty $ $p = \infty$ . Moreover, we present a comprehensive summary of the results obtained so far.