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

Sichun Wang. "A radial estimate for the maximal operator associated with the free Schrödinger equation." Studia Mathematica 176.2 (2006): 95-112. <http://eudml.org/doc/286478>.

@article{SichunWang2006,
abstract = {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].},
author = {Sichun Wang},
journal = {Studia Mathematica},
keywords = {free Schrödinger equation; maximal functions; spherical harmonics; oscillatory integrals},
language = {eng},
number = {2},
pages = {95-112},
title = {A radial estimate for the maximal operator associated with the free Schrödinger equation},
url = {http://eudml.org/doc/286478},
volume = {176},
year = {2006},
}

TY - JOUR
AU - Sichun Wang
TI - A radial estimate for the maximal operator associated with the free Schrödinger equation
JO - Studia Mathematica
PY - 2006
VL - 176
IS - 2
SP - 95
EP - 112
AB - 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].
LA - eng
KW - free Schrödinger equation; maximal functions; spherical harmonics; oscillatory integrals
UR - http://eudml.org/doc/286478
ER -