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On the maximal operator associated with the free Schrödinger equation

100%

Wang S.

Studia Mathematica

1997

tom 122

nr 2

167-182

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.

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

100%

Wang S.

Studia Mathematica

2006

tom 176

nr 2

95-112

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 $(S_{d}f)(x,t) = 1/(2π)ⁿ ∫_{ℝⁿ} e^{ix·ξ} e^{it|ξ|^{d}} f̂(ξ)dξ$, f ∈ 𝓢(ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, $(S**_{d}f)(x) = sup_{t∈ ℝ} |1/(2π)ⁿ ∫_{ℝⁿ} e^{ix·ξ} e^{it|ξ|^{d}} f̂(ξ)dξ|$, f ∈ 𝓢(ℝⁿ), x ∈ ℝⁿ, where f̂ is the Fourier transform of f and 𝓢(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 ∈ 𝓢(ℝⁿ), if n ≥ 3, 0 < d ≤ 2, and p ≥ 2n/(n-2), the maximal function estimate $(∫_{ℝⁿ} |(S**_{d}f)(x)|^{p} dx)^{1/p} ≤ C||f||_{H_{s}(ℝⁿ)$ holds for s > n(1/2 - 1/p) and fails for s < n(1/2 - 1/p), where $H_{s}(ℝⁿ)$ is the L²-Sobolev space with norm $||f||_{H_{s}(ℝⁿ)} = (∫_{ℝⁿ} (1+|ξ|²)^{s}|f̂(ξ)|²dξ)^{1/2}$. We also prove that for radial functions f ∈ 𝓢(ℝⁿ), if n ≥ 3, n/(n-1) < d < n²/2(n-1), then the estimate $(∫_{ℝⁿ} |(S**_{d}f)(x)|^{2n/(n-d)}dx)^{(n-d)/2n} ≤ C||f||_{H_{s}(ℝⁿ)}$ 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].

Ograniczanie wyników

2 Studia Mathematica

2 Wang S.

1 2006

1 1997

Wyniki wyszukiwania

On the maximal operator associated with the free Schrödinger equation

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