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Abstract

Let $d>0$ be a positive real number and $n\geq 1$ a positive integer and define the operator $S_d$ and its associated global maximal operator $S_d^{**}$ by $$\eqalign{ (S_d f)(x,t)&= \frac{1}{(2\pi)^n}\int_{{\mathbb R}^n} e^{ix\cdot\xi} e^{it|\xi|^d} \widehat{f}(\xi)\, d\xi,\quad\ f\in {\cal S}({\mathbb R}^n),\ x\in {\mathbb R}^n,\, t\in {\mathbb R},\cr (S_d^{**}f)(x)&=\sup_{t\in {\mathbb R}}\left|\frac{1}{(2\pi)^n} \int_{{\mathbb R}^n} e^{ix\cdot\xi} e^{it|\xi|^d} \widehat{f}(\xi)\, d\xi\right|,\quad\ f\in {\cal S}({\mathbb R}^n),\ x\in {\mathbb R}^n,\cr}$$ where $\widehat{f}$ is the Fourier transform of $f$ and ${\cal S}({\mathbb R}^n)$ is the Schwartz class of rapidly decreasing functions. If $d=2$, $S_d f$ is the solution to the initial value problem for the free Schrödinger equation (cf. (1.3) in this paper). We prove that for radial functions $ f\in {\cal S}({\mathbb R}^n)$, if $n\geq 3$, $0< d\leq 2$, and $p\geq 2n/(n-2)$, the maximal function estimate $$ \bigg(\int_{{\mathbb R}^n}|(S_d^{**}f)(x)|^p\,dx\bigg)^{1/p} \leq C\|f\|_{H_s({\mathbb R}^n)} $$ holds for $s>n(1/2-1/p)$ and fails for $s< n(1/2-1/p)$, where $H_s({\mathbb R}^n)$ is the $L^2$-Sobolev space with norm $$ \|f\|_{H_s({\mathbb R}^n)}=\bigg(\int_{{\mathbb R}^n}(1+|\xi|^2)^s|\widehat{f}(\xi)|^2\, d\xi\bigg)^{1/2}. $$ We also prove that for radial functions $ f\in {\cal S}({\mathbb R}^n)$, if $n\geq 3$, $n/(n-1)< d< n^2/2(n-1)$, then the estimate $$ \bigg(\int_{{\mathbb R}^n}|(S_d^{**}f)(x)|^{2n/(n-d)}\,dx\bigg)^{(n-d)/2n} \leq C\|f\|_{H_s({\mathbb R}^n)} $$ holds for $s>d/2$ and fails for $s< d/2$. These results complement other estimates obtained by Heinig and Wang [7], Kenig, Ponce and Vega [8], Sjölin [9]–[13], Vega [19]–[20], Walther [21]–[23] and Wang [24].