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

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 $B_p$.