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For d > 1, let , , where f̂ is the Fourier transform of , and its maximal operator. P. Sjölin ([11]) has shown that for radial f, the estimate (*) 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 and its associated global maximal operator by
, f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ,
, 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, 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...
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