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Estimates of three classical summations on the spaces Fp^α,q(R^n)(0〈p≤ 1).pdf

App1．Math．3．ChineseUniv

2014，29(3)：329—338

Estimatesofthreeclassicalsummationson thespaces

’g()(0

GAO Gui—lian ZHONGYong，3，

Abstract．Weobtaintheboundednesson审 ()forthePoissonsummationandGauss

summation．Theirmaximaloperatorsareprovedtobeboundedfrom毋，。()toLOO()．

Forthemxa imaloperatoroftheBochner—Rieszsummation，weprovethatitisboundedrfom

审，()to ，。。()．

§1 Introduction

LetN ≥1，thefollowingthreesummationsarecalledPoissonsummation，Gausssummation

andBochner—Rieszsummationrespectively,

PⅣf(x)=．，|e-Nl~l／(y)e2wix．Ydy

GNf(X)：豫／e-Nly[。()e2’Ydy，

竹

BN6f(X)： (～ ) 21ri~．y咖

Andtheirmaximaloperatorsaredefinedby

P,f(x)=suplPNf(x)l， Gf(x)=sup}GNf(x)I， B ，()=suplBN6f(x)

Ⅳ 1 Ⅳ ≥1 Ⅳ 1

It’Swell—knownthattheabovethreeoperatorsplay animportantrolein studyingspherical

summationofFourierseries．Inordertoobtaintheirconvergenceina(quasi一)BanachspaceX，

weneedtoshowtheirboundednessonX ．Moreover，ifwewanttoshow theiralmosteverywhere

convergenceforf∈X，theboundednessonXfortheirmaximaloperatorsmustbeconsidered．

Inthispaper，weconsiderthecasewhenXistheTriebel—Lizorkinspaces ，(R)(0

ThereareSOmanyworksthatofCUSontheboundednessofoperatorsonTriebel-Lizorkinspaces

，

Received：2012—11-3O．

MRSubjectClassification：42B08．42B35．

Keywords：Poissonsummation，Gau$ssummation，Bochner．Rieszsummation，Triebel-Lizorkinspace．

DigitalObjectIdentifier(DOI)：10．i007／si1766-014-3141．2．

SupportedbytheZh~iangPostdoctoralScienceFoundationofChina(BSH1302046)，theNationalNatural

ScienceFoundationofCh

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