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Let d > 0 be a positive real number and n ≥ 1 a positive integer and define the operator Sd and its associated global maximal operator S ∗∗ d by (Sdf)(x, t) = 1 (2π)n \ Rn ee d f̂(ξ) dξ, f ∈ S(R), x ∈ R, t ∈ R, (S d f)(x) = sup t∈R ∣∣∣∣ 1 (2π)n \ Rn ee d f̂(ξ) dξ ∣∣∣∣, f ∈ S(R ), x ∈ R, where f̂ is the Fourier transform of f and S(R) is the Schwartz class of rapidly decreasing functions. If d = 2, Sdf 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 ∈ S(R), if n ≥ 3, 0 < d ≤ 2, and p ≥ 2n/(n − 2), the maximal function estimate ( \ Rn |(S d f)(x)| p dx )1/p ≤ C‖f‖Hs(Rn) holds for s > n(1/2−1/p) and fails for s < n(1/2−1/p), where Hs(R ) is the L-Sobolev space with norm ‖f‖Hs(Rn) = ( \ Rn (1 + |ξ|)|f̂(ξ)| dξ )1/2 . We also prove that for radial functions f ∈ S(R), if n ≥ 3, n/(n− 1) < d < n/2(n− 1), then the estimate ( \ Rn |(S d f)(x)| 2n/(n−d) dx )(n−d)/2n ≤ C‖f‖Hs(Rn) 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]. 2000 Mathematics Subject Classification: Primary 42B25; Secondary 42A45.

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