Radial functions and maximal operators of Schrödinger type (Q2884645)

Let \(f\) belong to the Schwartz space and NEWLINE\[NEWLINE\begin{aligned} S_t f(x) &= \int_{{\mathbb R}^n}e^{ix\cdot \xi} e^{it|\xi|^a} \hat{f}(\xi) d\xi,\;\;\;x\in {\mathbb R}^n, t\in {\mathbb R}, a > 1,\\ S^{\star} f(x) &= \sup_{0<t<1} |S_t f(x)|, \;\;x\in {\mathbb R}^n.\\ S^{\star\star} f(x) &= \sup_{t>0} |S_t f(x)|, \;\;x\in {\mathbb R}^n.\end{aligned}NEWLINE\]NEWLINE Here \(\hat{f}\) denotes the Fourier transform of \(f\).NEWLINENEWLINEThe following estimates are under investigation NEWLINE\[NEWLINE\begin{aligned} \| S^{\star}f \|_{L^q ({\mathbb R}^n)} &\leq C \|f \|_{H_s},\\ \| S^{\star\star}f \|_{L^q ({\mathbb R}^n)} &\leq C \|f \|_{H_s},\end{aligned}NEWLINE\]NEWLINE where \( \| f \|_{H_s} = \left(\int_{{\mathbb R}^n} \left(1+|\xi|^2\right)^s |\hat{f}(\xi)|^2 d\xi\right)^{1/2} \) is the norm in the inhomogeneous Sobolev space, \(n\geq 2\), and \(f\) is radial.NEWLINENEWLINESharp and almost sharp results are obtained.