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<p>In this paper, we prove a generalization of the Stein–Weiss inequality for the fractional integral operator in variable Lebesgue spaces with nonstandard power weights of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(w(x)=|x|^{\gamma (x)}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. As applications, we prove Hardy–Sobolev and related inequalities with these weights.</p>

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A Stein–Weiss Inequality with Nonstandard Power Weights and Applications

  • 期刊论文

  • 发表日期：

    2026年7月7日

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  • Alexandre Almeida,
  • David Cruz-Uribe,
  • Humberto Rafeiro

摘要

In this paper, we prove a generalization of the Stein–Weiss inequality for the fractional integral operator in variable Lebesgue spaces with nonstandard power weights of the form \(w(x)=|x|^{\gamma (x)}\) w ( x ) = | x | γ ( x ) . As applications, we prove Hardy–Sobolev and related inequalities with these weights.