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

Sichun Wang

Studia Mathematica, Tome 176 (2006) no. 2, p. 95.

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 / ( 2 π ) ⁿ ∫ ℝ ⁿ e i x · ξ e i t | ξ | d f ̂ ( ξ ) d ξ , f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, ( S * * d f ) ( x ) = s u p t ∈ ℝ | 1 / ( 2 π ) ⁿ ∫ ℝ ⁿ e i x · ξ e i t | ξ | d f ̂ ( ξ ) d ξ | , 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 ( ∫ ℝ ⁿ | ( S * * d f ) ( x ) | p d x ) 1 / p ≤ C | | f | | H s ( ℝ ⁿ ) h o l d s f o r s > n ( 1 / 2 - 1 / p ) a n d f a i l s f o r s < n ( 1 / 2 - 1 / p ) , w h e r e Hs(ℝⁿ) i s t h e L ² - S o b o l e v s p a c e w i t h n o r m | | f | | H s ( ℝ ⁿ ) = ( ∫ ℝ ⁿ ( 1 + | ξ | ² ) s | f ̂ ( ξ ) | ² d ξ ) 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 ( ∫ ℝ ⁿ | ( S * * d f ) ( x ) | 2 n / ( n - d ) d x ) ( n - d ) / 2 n ≤ C | | f | | H s ( ℝ ⁿ ) 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

Classification : 42B25, 35Q55
Mots-clés : free Schrödinger equation, maximal functions, spherical harmonics, oscillatory integrals