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On the maximal operator associated with the free Schrödinger equation. (English) Zbl 0877.42007

Fix \(d\) in \((1, \infty)\). For \(f\) in the Schwartz space \(S({\mathbb{R}}^n)\), denote by \(\widehat f\) the Fourier transform of \(f\), by \({\mathcal S}_tf(x)\) the inverse Fourier transform of \(\xi\mapsto\widehat{f}(\xi)e^{it|\xi|^d}\) and by \({\mathcal S}^\ast f\) the maximal function \(\sup_{0<t<1}|{\mathcal S}_tf|\). P. Sjölin [J. Aust. Math. Soc., Ser. A 59, No. 1, 134-142 (1995; Zbl 0856.42013)] showed that the \(L^p\)-norm of \({\mathcal S}^\ast f\) may be estimated locally by the Sobolev \(H^{1/4}\)-norm of \(f\) when \(p = 4n/(2n-1)\), provided that \(f\) is radial. In the paper under review, it is shown that this is false when the requirement that \(f\) is radial is no longer imposed. More general versions of this result are also established.
Reviewer: M.Cowling (Sydney)

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B15 Multipliers for harmonic analysis in several variables
35J10 Schrödinger operator, Schrödinger equation

Citations:

Zbl 0856.42013
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