On the maximal operator associated with the free Schrödinger equation
Volume 122 / 1997
Studia Mathematica 122 (1997), 167-182
DOI: 10.4064/sm-122-2-167-182
Abstract
For d > 1, let $(S_{d}f)(x,t) = ʃ_{ℝ^n} e^{ix·ξ} e^{it|ξ|^d} f̂(ξ)dξ$, $x ∈ ℝ^n$, where f̂ is the Fourier transform of $f ∈ S (ℝ^n)$, and $(S_{d}*f)(x) = sup_{0 < t < 1} |(S_{d}f)(x,t)|$ its maximal operator. P. Sjölin ([11]) has shown that for radial f, the estimate (*) $(ʃ_{|x| < R} |(S_{d}*f)(x)|^p dx)^{1/p} ≤ C_{R}∥f∥_{H_{1/4}}$ 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.
