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For d > 1, let $\left(S_{d}f\right)(x,t)={ʃ}_{{ℝ}^n}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi$, $x\in {ℝ}^n$, where f̂ is the Fourier transform of $f\in S\left({ℝ}^n\right)$, and $(S_d*f)\left(x\right)=su{p}_{0<t<1}\left|\left(S_{d}f\right)(x,t)\right|$ its maximal operator. P. Sjölin ([11]) has shown that for radial f, the estimate (*) $({ʃ}_{\left|x\right|<R}|(S_d*f)\left(x\right){{|}^{p}dx)}^{1/p}\le C_R\parallel f{\parallel }_{H_{1/4}}$ holds for p = 4n/(2n-1) and fails for p > 4n/(2n-1). In this paper we show that for non-radial f, (*) fails for p > 2. A similar result is proved for a more general maximal operator.

Let d > 0 be a positive real number and n ≥ 1 a positive integer and define the operator $S_d$ and its associated global maximal operator $S*{*}_d$ by $\left(S_{d}f\right)(x,t)=1/\left(2\pi \right)ⁿ{\int }_{ℝⁿ}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi$, f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, $(S*{*}_{d}f)\left(x\right)=su{p}_{t\in ℝ}|1/\left(2\pi \right)ⁿ{\int }_{ℝⁿ}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi |$, f ∈ (ℝⁿ), x ∈ ℝⁿ, where f̂ is the Fourier transform of f and (ℝⁿ) 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 ∈ (ℝⁿ), if n ≥ 3, 0 < d ≤ 2, and p ≥ 2n/(n-2), the...