Improved bounds for Stein's square functions (Q2894580)

Let the Bochner-Riesz means of order \(\lambda > 0\) and Stein's square function be defined by NEWLINE\[NEWLINE\widehat{S_t^{\lambda}f}(\xi) = \left( 1- \frac{|\xi|^2}{t^2} \right)_{+}^{\lambda} \widehat{f}(\xi), NEWLINE\]NEWLINE and NEWLINE\[NEWLINEH^{\lambda}f(x) = \left(\int_{0}^{\infty}\left|\frac{\partial}{\partial t} S_t^{\lambda}f(x) \right|^2 t \, dt \right)^{1/2}NEWLINE\]NEWLINE on \(\mathbb{R}^d\), respectively.NEWLINENEWLINEThe authors improve the known results for Stein's square function by proving the following.NEWLINENEWLINELet \(d \geq 2\) and \( \frac{2(d+2)}{d}\leq p < \infty\). If \(\lambda > d \big(\frac{1}{2}- \frac{1}{p}\big)\), then \(H^{\lambda}\) is bounded on \(L^{p}(\mathbb{R}^d)\).NEWLINENEWLINEThe authors also prove a weighted norm inequality for the maximal Bochner-Riesz operator and Stein's square function, \(L^{p}(\mathbb{R}^d)\)-bounds on classes of radial Fourier multipliers for \(p \geq 2+\frac{4}{d}\), \(d \geq 2\), and space-time regularity results for wave and Schrödinger operators.