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

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 $$\underset{\ddot{\mathbf{I}}‰\in {\mathbb{R}}^2}{\sup }\Vert \ddot{\mathbf{I}}‰{\Vert }_2^{3/2}|\widehat{\ddot{I}{‡}_{B_p}}(\ddot{\mathbf{I}}‰)|\asymp (p-1{)}^{-1/2}\text{as }p\to 1+$$ As an application, we obtain corresponding bounds for lattice point discrepancy inequalities for dilates of $B_p$.