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On the maximal operator associated with the free Schrödinger equation

Sichun Wang — 1997

Studia Mathematica

For d > 1, let ( S d f ) ( x , t ) = ʃ n e i x · ξ e i t | ξ | d f ̂ ( ξ ) d ξ , x n , where f̂ is the Fourier transform of f S ( n ) , and ( S d * f ) ( x ) = s u p 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 d x ) 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

Sichun Wang — 2006

Studia Mathematica

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 ) 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 | 1 / ( 2 π ) 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...

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