A radial estimate for the maximal operator associated with the free Schrödinger equation

Sichun Wang¹

¹ Defence Research and Development Canada–Ottawa 3701 Carling Avenue Ottawa, Ontario, Canada K1A 0Z4

Studia Mathematica, Tome 176 (2006) no. 2, pp. 95-112

Résumé

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].

Détail

DOI : 10.4064/sm176-2-1

Keywords: positive real number geq positive integer define operator its associated global maximal operator ** eqalign frac int mathbb cdot widehat quad cal mathbb mathbb mathbb ** sup mathbb frac int mathbb cdot widehat right quad cal mathbb mathbb where widehat fourier transform cal mathbb schwartz class rapidly decreasing functions solution initial value problem schr dinger equation paper prove radial functions cal mathbb geq leq geq n maximal function estimate bigg int mathbb ** bigg leq mathbb holds fails where mathbb sobolev space norm mathbb bigg int mathbb widehat bigg prove radial functions cal mathbb geq n n estimate bigg int mathbb ** n d bigg n d leq mathbb holds fails these results complement other estimates obtained heinig wang kenig ponce vega lin vega walther wang