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<p>We consider Hardy operators, i.e., homogeneous Schrödinger operators consisting of the ordinary or fractional Laplacian in a half-space plus a potential given by a function which only depends on the appropriate power of the distance to the boundary of the half-space multiplied with a coupling constant. We compare the scales of homogeneous <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-Sobolev spaces generated by these Hardy operators with and without potential. To that end, we prove and use new square function estimates for operators whose heat kernels decay slowly and include singular weights. Our results hold for all admissible coupling constants in the local case and for repulsive potentials (positive coupling constants) in the fractional case, and extend those obtained recently in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. They also apply to attractive potentials (negative coupling constants) in the fractional case, provided the currently known heat kernel bounds for repulsive couplings extend to this regime.</p>

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Equivalence of Sobolev norms in Lebesgue spaces for Hardy operators in a half-space

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月17日

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  • The Anh Bui,
  • Konstantin Merz

摘要

We consider Hardy operators, i.e., homogeneous Schrödinger operators consisting of the ordinary or fractional Laplacian in a half-space plus a potential given by a function which only depends on the appropriate power of the distance to the boundary of the half-space multiplied with a coupling constant. We compare the scales of homogeneous \(L^p\) L p -Sobolev spaces generated by these Hardy operators with and without potential. To that end, we prove and use new square function estimates for operators whose heat kernels decay slowly and include singular weights. Our results hold for all admissible coupling constants in the local case and for repulsive potentials (positive coupling constants) in the fractional case, and extend those obtained recently in \(L^2\) L 2 . They also apply to attractive potentials (negative coupling constants) in the fractional case, provided the currently known heat kernel bounds for repulsive couplings extend to this regime.