On the maximal operator associated with the free Schrödinger equation

Wang, Sichun

Wang, Sichun

Studia Mathematica, Tome 122 (1997), p. 167-182 / Harvested from The Polish Digital Mathematics Library

Résumé

For d > 1, let $(S_{d} f) (x, t) = ʃ_{ℝ^{n}} e^{i x \cdot ξ} e^{{i t | ξ |}^{d}} f ̂ (ξ) d ξ$ , $x \in ℝ^{n}$ , where f̂ is the Fourier transform of $f \in S (ℝ^{n})$ , 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 (*) $(ʃ_{| x | < R} | (S_{d} * f) (x) {|^{p} d x)}^{1 / p} \leq C_{R} ∥ f ∥_{H_{1 / 4}}$ 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.

Publié le : 1997-01-01

Zbl 0877.42007

EUDML-ID : urn:eudml:doc:216368

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     author = {Sichun Wang},
     title = {On the maximal operator associated with the free Schr\"odinger equation},
     journal = {Studia Mathematica},
     volume = {122},
     year = {1997},
     pages = {167-182},
     zbl = {0877.42007},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv122i2p167bwm}
}

Wang, Sichun. On the maximal operator associated with the free Schrödinger equation. Studia Mathematica, Tome 122 (1997) pp. 167-182. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv122i2p167bwm/

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