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ESTIMATES FOR SCHRÖDINGER MAXIMAL OPERATORSALONG CURVE WITH COMPLEX TIME

  • Niu, Yaoming (Faculty of Mathematics Baotou Teachers' College of Inner Mongolia University of Science and Technology) ;
  • Xue, Ying (Faculty of Mathematics Baotou Teachers' College of Inner Mongolia University of Science and Technology)
  • Received : 2018.08.14
  • Accepted : 2019.10.23
  • Published : 2019.12.30

Abstract

In the present paper, we give some characterization of the L2 maximal estimate for the operator Pta,γf(Γ(x, t)) along curve with complex time, which is defined by $$P^t_{a,{\gamma}}f({\Gamma}(x,t))={\displaystyle\smashmargin{2}{\int\nolimits_{\mathbb{R}}}}\;e^{i{\Gamma}(x,t){\xi}}e^{it{\mid}{\xi}{\mid}^a}e^{-t^{\gamma}{\mid}{\xi}{\mid}^a}{\hat{f}}({\xi})d{\xi}$$, where t, γ > 0 and a ≥ 2, curve Γ is a function such that Γ : ℝ×[0, 1] → ℝ, and satisfies Hölder's condition of order σ and bilipschitz conditions. The authors extend the results of the Schrödinger type with complex time of Bailey [1] and Cho, Lee and Vargas [3] to Schrödinger operators along the curves.

Acknowledgement

Supported by : NSFC, Inner Mongolia University, natural science foundation of Inner Mongolia

References

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