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

TL;DR

This paper establishes sharp Fourier transform inequalities for the $l^p$-unit ball in two dimensions, revealing asymptotic behavior as p approaches 1, and applies these results to lattice point discrepancy bounds.

Contribution

It provides new sharp inequalities for the Fourier transform of the $l^p$-unit ball and derives asymptotic behavior as p approaches 1, with applications to lattice point discrepancy.

Findings

Sharp inequalities for Fourier transform of $l^p$-unit ball in $ ^2$

Asymptotic behavior of inequalities as p approaches 1

Bounds for lattice point discrepancy for dilates of $B_p$

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 $B_p$.