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<p>We prove weighted versions of the 2D Restriction Conjecture for the unit sphere in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {R}}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. Our results involve the weight functions <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((1+|x|)^\alpha (1+|y|)^\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <mi>α</mi> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">|</mo> <mi>y</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <mi>β</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((1+|x|+|y|)^\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo>+</mo> <mo stretchy="false">|</mo> <mi>y</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <mi>γ</mi> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\alpha ,\beta ,\gamma \ge 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Optimal Weighted Fourier Restriction Estimates for the Sphere in 2D

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  • 发表日期：

    2025年11月21日

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  • Rainer Mandel

摘要

We prove weighted versions of the 2D Restriction Conjecture for the unit sphere in \({\mathbb {R}}^2\) R 2 . Our results involve the weight functions \((1+|x|)^\alpha (1+|y|)^\beta \) ( 1 + | x | ) α ( 1 + | y | ) β and \((1+|x|+|y|)^\gamma \) ( 1 + | x | + | y | ) γ with \(\alpha ,\beta ,\gamma \ge 0\) α , β , γ ≥ 0 .