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arXiv · 2208.07837

Fourier transform inequalities, lattice point discrepancy, asymptotic behavior, oscillatory integrals

Martin Lind
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Abstract

For $1<p\le 2$, we establish sharp inequalities for the Fourier transform of the characteristic function of the $l^p$-unit ball $B_p\subset\mathbb{R}^2$. We show that $$ \sup_{\boldsymbol{\omega} \in \mathbb{R}^2} \|\boldsymbol{\omega} \|_2^{3/2}|\widehat{\chi_{B_p}} (\boldsymbol{\omega})| \asymp (p-1)^{-1/2} \quad \text{as } p\rightarrow1+ $$ As an application, we obtain corresponding bounds for lattice point discrepancy inequalities for dilates of $B_p$.

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Martin Lind. 2022-08-16. Fourier transform inequalities, lattice point discrepancy, asymptotic behavior, oscillatory integrals. https://arxiv.org/abs/2208.07837

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