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<p>We establish fractional Hardy inequality on bounded domains in <InlineEquation ID="IEq1"> <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> with inverse of distance function from smooth boundary of codimension <i>k</i>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k=2, \dots ,d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation>, as weight function. The case <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(sp=k\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mi>p</mi> <mo>=</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> is the critical case, where optimal logarithmic corrections are required. All the other cases of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(sp&lt;k\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mi>p</mi> <mo>&lt;</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(sp&gt;k\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mi>p</mi> <mo>&gt;</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> are also addressed.</p>

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Fractional Hardy inequality with singularity on submanifold

  • 期刊论文

  • 发表日期：

    2026年1月12日

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  • Adimurthi,
  • Prosenjit Roy,
  • Vivek Sahu

摘要

We establish fractional Hardy inequality on bounded domains in \(\mathbb {R}^{d}\) R d with inverse of distance function from smooth boundary of codimension k, where \(k=2, \dots ,d\) k = 2 , ⋯ , d , as weight function. The case \(sp=k\) s p = k is the critical case, where optimal logarithmic corrections are required. All the other cases of \(sp<k\) s p < k and \(sp>k\) s p > k are also addressed.