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 $$\underset{\mathit{\omega }\in {\mathbb{R}}^2}{\sup }\mathrm{\parallel }\mathit{\omega }{\mathrm{\parallel }}_2^{3\mathrm{/}2}\mathrm{\mid }\widehat{{\chi }_{B_p}}(\mathit{\omega })\mathrm{\mid }\asymp (p-1{)}^{-1\mathrm{/}2}\text{as }p\to 1+$$ As an application, we obtain corresponding bounds for lattice point discrepancy inequalities for dilates of $B_p$.