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<p>For several weights based on lattice point constructions in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^d (d \geqslant 2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>⩾</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we prove that the sharp <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> weighted restriction inequality for the sphere is very different than the corresponding result for the paraboloid. The proof uses Poisson summation, linear algebra, and lattice counting. We conjecture that the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> weighted restriction is generally better for the circle for a wide variety of general sparse weights.</p>

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A distinction between the paraboloid and the sphere in weighted restriction

  • 期刊论文

  • 发表日期：

    2026年5月22日

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  • Alex Iosevich,
  • Ruixiang Zhang

摘要

For several weights based on lattice point constructions in \(\mathbb {R}^d (d \geqslant 2)\) R d ( d ⩾ 2 ) , we prove that the sharp \(L^2\) L 2 weighted restriction inequality for the sphere is very different than the corresponding result for the paraboloid. The proof uses Poisson summation, linear algebra, and lattice counting. We conjecture that the \(L^2\) L 2 weighted restriction is generally better for the circle for a wide variety of general sparse weights.