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

Sichun Wang

Studia Mathematica, Tome 173 (2006), p. 95-112 / Harvested from The Polish Digital Mathematics Library

Résumé

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 ∈ (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 ∈ (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 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 $Hs(ℝⁿ)$ 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 ∈ (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].

Détail

Publié le : 2006-01-01

Zbl 1106.42014

EUDML-ID : urn:eudml:doc:286478