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A note on $l^p$ norms of weighted mean matrices

We present some results concerning the $l^p$ norms of weighted mean matrices. These results can be regarded as analogues to a result of Bennett concerning weighted Carleman's inequalities.

8

002 guA 25 ]AF.htam[ 1v1923.808:0viXraANOTEONlpNORMSOFWEIGHTEDMEANMATRICES

PENGGAO

Abstract.Wepresentsomeresultsconcerningthelpnormsofweightedmeanmatrices.TheseresultscanberegardedasanaloguestoaresultofBennettconcerningweightedCarleman’sin-equalities.

1.Introduction

Supposethroughoutthatp=0,1=1.Forp≥1,letlpbetheBanachspaceofallcomplex

sequencesa=(an)n≥1withnorm

q

||a||p:=(

∞|an|p)1/p<∞.

n=1

ThecelebratedHardy’sinequality([8,Theorem326])assertsthatforp>1,

(1.1) ∞n=1

1

p ∞|an|pp 1.n=1

Hardy’sinequalitycanberegardedasaspecialcaseofthefollowinginequality:

(1.2)

C·a pp

=

∞ n=1 ∞cn,kak p≤Uk=1

∞|an|p,n=1

inwhichC=(cn,k)andtheparameterp>1areassumed xed,andtheestimateistoholdforall

complexsequencesa∈lp.ThelpoperatornormofC||C||p,p=sup||a||p=1

isthende nedas

C·a

p

.

Itfollowsthatinequality(1.2)holdsforanya∈lpwhenU1/p≥||C||p,pandfailstoholdforsomea∈lpwhenU1/p<||C||p,p.Hardy’sinequalitythusassertsthattheCes´aromatrixoperatorC,

givenbycn,k=1/n,k≤nand0otherwise,isboundedonlpandhasnorm≤p/(p 1).(Thenorm

isinfactp/(p 1).)

WesayamatrixA=(atriangularmatrixAisasummabilityn,k)isalowertriangularmatrixamatrixifamatrixAisaweightedmeanmatrixifitsentriesn,k≥0andsatisfy: ifn

n,k=0forn<kandalowerk=1an,k=1.Wesayasummability(1.3)

an,k=λ nk/Λn,1≤k≤n;Λn=

λi,λi≥0,λ1>0.

i=1

Hardy’sinequality(1.1)nowmotivatesonetodeterminethelpoperatornormofanarbitrary

summabilityorweightedmeanmatrixA.In[7],theauthorprovedthefollowingresult:

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