Science China Mathematics

, Volume 55, Issue 10, pp 2109–2123 | Cite as

Estimates for wave and Klein-Gordon equations on modulation spaces



We prove that the fundamental semi-group \(e^{it\left( {m^2 I + \left| \Delta \right|} \right)^{1/2} }\) (m ≠ 0) of the Klein-Gordon equation is bounded on the modulation space Mp,qs (ℝn) for all 0 < p, q and s ∈ ℝ. Similarly, we prove that the wave semi-group \(e^{it\left| \Delta \right|^{1/2} }\) is bounded on the Hardy type modulation spaces µp,qs (ℝn) for all 0 < p, q, and s ∈ ℝ. All the bounds have an asymptotic factor tn|1/p−1/2| as t goes to the infinity. These results extend some known results for the case of p ⩾ 1. Also, some applications for the Cauchy problems related to the semi-group \(e^{it\left( {m^2 I + \left| \Delta \right|} \right)^{1/2} }\) are obtained. Finally we discuss the optimum of the factor tn|1/p−1/2| and raise some unsolved problems.


Klein-Gordon equation wave equation modulation space 


42B37 42B35 35L05 


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Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaChina
  2. 2.Department of MathematicsUniversity of Wisconsin-MilwaukeeMilwaukeeUSA

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