Improved Stein inequalities for the Fourier transform (Q6658104)

The Stein (or Hardy-Littlewood-Stein) inequality states for a function \(f \in L_{p,q}(\mathbb R^n)\) (the classical Lorentz space) with some \(p \in (1,2)\), \(p' = p / (p-1)\) and \(0 < q \le \infty\) that \(\| \widehat f \|_{L_{p', q}(\mathbb R^n)} \le c \| f \|_{L_{p,q}(\mathbb R^n)}\) where \(\widehat f\) denotes the Fourier transform of \(f\). In this paper, the authors provide some sharper and more general versions of this result, in particular allowing to also choose \(p \ge 2\).