@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 -
