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<p>We use high-low frequency methods developed in the context of decoupling to prove sharp (up to <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>ε</mi> </msub> <msup> <mi>R</mi> <mi>ε</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$C_{\varepsilon }R^{\varepsilon }$</EquationSource> </InlineEquation>) square function estimates for the moment curve <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>t</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(t,t^{2},\ldots ,t^{n})$</EquationSource> </InlineEquation> in <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{R}^{n}$</EquationSource> </InlineEquation>. Our inductive scheme incorporates sharp square function estimates for auxiliary conical sets, which allows us to fully exploit lower dimensional information.</p>

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A Sharp Square Function Estimate for the Moment Curve in \(\mathbb{R}^{n}\)

  • 期刊论文

  • 发表日期：

    2026年3月11日

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  • Larry Guth,
  • Dominique Maldague

摘要

We use high-low frequency methods developed in the context of decoupling to prove sharp (up to C ε R ε $C_{\varepsilon }R^{\varepsilon }$ ) square function estimates for the moment curve ( t , t 2 , … , t n ) $(t,t^{2},\ldots ,t^{n})$ in R n $\mathbb{R}^{n}$ . Our inductive scheme incorporates sharp square function estimates for auxiliary conical sets, which allows us to fully exploit lower dimensional information.