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AdvancesinAppliedMathematics
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PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.124187
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BasicEstimatesforaClassofDispersive
Semigroups
HuiwenHuang
DepartmentofMathematics,ZhejiangNormalUniversity,JinhuaZhejiang
Received:Mar.24
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Abstract
Inthispaper,westudytheproblemofone-dimensionalattenuationestimationofdis-
persionsemigroups
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Keywords
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This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2023.1241871806
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DOI:10.12677/aam.2023.1241871807
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[1]Strichartz,R.S.(1977) RestrictionsofFourier TransformstoQuadratic Surfacesand Decay of
SolutionsofWaveEquations.
DukeMathematicalJournal
,
44
,705-714.
https://doi.org/10.1215/S0012-7094-77-04430-1
[2]Ben-Artzi,M.andTreves,F.(1994)UniformEstimatesforaClassofEvolutionEquations.
JournalofFunctionalAnalysis
,
120
,264-299.https://doi.org/10.1006/jfan.1994.1033
[3]Ben-Artzi,M.andSaut,J.-C.(1999)UniformDecayEstimatesforaClassofOscillatory
IntegralsandApplications.
DifferentialIntegralEquations
,
12
,137-145.
https://doi.org/10.57262/die/1367265625
[4]Cui,S.andTao,S.(2005)StrichartzEstimatesforDispersiveEquationsandSolvabilityof
theKawaharaEquation.
JournalofMathematicalAnalysisandApplications
,
304
,683-702.
https://doi.org/10.1016/j.jmaa.2004.09.049
[5]Kenig,C.E.,Ponce,G.andVega,L.(1991)OscillatoryIntegralsandRegularityofDispersive
Equations.
IndianaUniversityMathematicsJournal
,
40
,33-69.
https://doi.org/10.1512/iumj.1991.40.40003
[6]Kenig, C.E.,Ponce, G.and Vega, L.(1989)Onthe(Generalized)Korteweg-deVriesEquations.
DukeMathematicalJournal
,
59
,585-610.https://doi.org/10.1215/S0012-7094-89-05927-9
DOI:10.12677/aam.2023.1241871808
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[7]Wang,B.,Huo,Z.,Hao,C.andGuo,Z.(2011)HarmonicAnalysisMethodforNonlinear
EvolutionEquations,I.WorldScientific,Singapore,Hackensack,NJ.
https://doi.org/10.1142/8209
[8]Basu,S.,Guo,S.,Zhang,R.andZorin-Kranich,P.(2021)AStationarySetMethodfor
EstimatingOscillatoryIntegrals.arXivpreprintarXiv:2103.08844
DOI:10.12677/aam.2023.1241871809
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