<p>For <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$1< p\le 2$</EquationSource> </InlineEquation>, we establish sharp inequalities for the Fourier transform of the characteristic function of the <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <msup> <mi>l</mi> <mi>p</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$l^{p}$</EquationSource> </InlineEquation>-unit ball <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>p</mi> </msub> <mo>⊂</mo> <msup> <mi mathvariant="double-struck">R</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$B_{p}\subset \mathbb{R}^{2}$</EquationSource> </InlineEquation>. We show that <Equation ID="Equa"> <EquationSource Format="MATHML"><math> <munder> <mo movablelimits="false">sup</mo> <mrow> <mi mathvariant="bold-italic">ω</mi> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mn>2</mn> </msup> </mrow> </munder> <msubsup> <mrow> <mo stretchy="false">∥</mo> <mi mathvariant="bold-italic">ω</mi> <mo stretchy="false">∥</mo> </mrow> <mn>2</mn> <mrow> <mn>3</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">|</mo> <mover accent="true"> <msub> <mi>χ</mi> <msub> <mi>B</mi> <mi>p</mi> </msub> </msub> <mo>ˆ</mo> </mover> <mo stretchy="false">(</mo> <mi mathvariant="bold-italic">ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mo>≍</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mspace width="1em" /> <mtext>as </mtext> <mi>p</mi> <mo stretchy="false">→</mo> <mn>1</mn> <mo>+</mo> </math></EquationSource> <EquationSource Format="TEX">$ \sup _{\boldsymbol{\omega}\in \mathbb{R}^{2}}\|\boldsymbol{\omega}\|_{2}^{3/2}| \widehat{\chi _{B_{p}}}(\boldsymbol{\omega})|\asymp (p-1)^{-1/2} \quad \text{as }p\rightarrow 1+ $</EquationSource> </Equation> As an application, we obtain corresponding bounds for lattice point discrepancy inequalities for dilates of <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>p</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$B_{p}$</EquationSource> </InlineEquation>.</p>

Sharp Fourier inequalities and lattice point discrepancy for $l^{p}$-balls

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2026年4月16日

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Martin Lind

摘要

For $1 < p \leq 2$ $1< p\le 2$ , we establish sharp inequalities for the Fourier transform of the characteristic function of the $l^{p}$ $l^{p}$ -unit ball $B_{p} \subset R^{2}$ $B_{p}\subset \mathbb{R}^{2}$ . We show that $sup_{ω \in R^{2}} {∥ ω ∥}_{2}^{3 / 2} | \hat{χ_{B_{p}}} (ω) | ≍ {(p - 1)}^{- 1 / 2} as p \to 1 +$ $ \sup _{\boldsymbol{\omega}\in \mathbb{R}^{2}}\|\boldsymbol{\omega}\|_{2}^{3/2}| \widehat{\chi _{B_{p}}}(\boldsymbol{\omega})|\asymp (p-1)^{-1/2} \quad \text{as }p\rightarrow 1+ $ As an application, we obtain corresponding bounds for lattice point discrepancy inequalities for dilates of $B_{p}$ $B_{p}$ .

Sharp Fourier inequalities and lattice point discrepancy for \(l^{p}\)-balls

摘要