REMARK ON GENERALIZED CLARKSON,S INEQUALITIES FOR EXTREME CASES

REMARK ON GENERALIZED CLARKSON,S INEQUALITIES FOR EXTREME CASES

By

Mikio KATo and Ken-ichi MiyAzAKi*

(Received November 30, 1993)

Introduction

As a `global' and high dimensional version of classical Clarkson's inequalities ([2]) the generalized Clarkson inequalities (for L,, 1 ÅqpÅq oo) were given in Kato [5] (see also [12], [8], [9]), which include as special cases Boas' ([1]) and Koskela's ([7]) inequalities generalizing Clarkson's. In the recent paper [8] Maligranda and Persson gave a more generalized inequality ([8], Theorem 4.1 and also Corollary 4.2).

We here confine ourselves to the inequalities in [5] for the rest cases p=1 and oo, where equality-attainedness is mainly discussed. The same are considered for some spaces of continuous functions. In other words, we observe the behavior of the

operator norm of the Littlewood matrices between l,2" (E)-spaces for these Banach spaces E (cf. [10], [5]). As a straightforward application the von Neumann-Jordan

constant (Clarkson [3]) for these spaces is determined.

1. Preliminaries

Let L, = L.(M, 2], pt), 1 SpS oo, be the usual L,-space on an arbitrary measure space (M, E, pt). Fora Banach space E, let l,N(E),1 =Åqr$ oo, be the space ofE-valued sequences {xj} of length N with the norm

U{)cj} 11,(E) =

N {Z II x,Hr.}'i" if 1 s. rÅq oo, j=1

irp.iEXN 11 ;'CjllE if r.., ...

Let A. = (6ij) be the Littlewood matrices, i.e.,

"i= (1 -l)• An+i=(,A". -ll:) (n=i•2• )

*i The authors are supported in part by the Grant-in-Aid for Scientific Research from the Ministry

of Education, Science and Culture (05640203, 1993).

28 Mikio KATo and Ken-ichi MiyAzAKi

Now, the generalized Clarkson inequalities are stated as

GENERALIzED CLARKsoN's INEQuALmEs (Kato [5], Theorem 1;cf. [12], [8], [9]).

Let 1 ÅqpÅq oo and 1 :ll r, s S. oo. Then, for an arbitrary positive integer n and for all fi,f2,•..,f2n in L,,

.

2n 2n 2n (1) {Z ll 2 6,,f, llb"}'!s ;.l{ 2"c(r•s;p){Z il f,"r.}i!r,

i:=1 ,j=1 j=1

where

c(r, s; p) =

1; + -1- - min(1 ,-1, År (f min (p, p') srs oo,

p/

1 S. s S. max (p, p'),

1

- if 1 S- r S. min (p, p'),

1 S. s Åq= r',

r if s' S.rS. oo,

1

max (p, p') S. s$ oo

and 1/p + 1/p' == 1/r + 1/r' = 1/s + 1/s' == 1. Equality is attained in (1) for all 1 $ r, s ms{; oo, where fbr the .first case, the underl.ying measure space (M, X, itt) is ctssumed to be `big enough', which means the existence qf- 2" mutually dis.1'oint measurcthle sets qf .finitcJ positive measure: Consequently,

(1 ') 11 A.: l,2 'i (L,)-t,,2"(L,) Ii = 2nc(r-s;p).

REMARK. For complex l,2'i-spaces,

Il A.: l,2"-t,2"ll = 2"C("S;2) (1 S. r, s ;.:S oo)

(cf. Pietsch [10] and Kato [5]).

The von Neumann-Jordan constant (Clarkson [3]) for a Banach space E, we denote it by CNJ(E), is defined as the smallest constant C satisfying

(2) i} ;ilg ll j'c;(tiv.11i,++11tiylfi/;,;Ii2 ;:;;c

for all x and y in E with 11 )c l12 + 11 jy ll2 iE O. As is easily g. een, if C is best possible in the right-hand side inequality of (2), so is 1/C in the left. For any Banach space E, 1 :S C.,(E) :;i 2; and it is a Hilbert space if and only if C.,(E)=1 (Jordan and von Neumann [4]). For E = L, (Clarkson [3]), l,N(L,) and t,(L,) (Kato and Miyazaki [6]), CNJ(E)=22MaX(iiP•ifP')mi. (Note that (1') with n=1 and r=s--2 means

Clarkson's result just stated.)

For a locally compact Hausdorff space X, let C,(X), Co(X) and Cb(X) be the spaces of continuous functions on X which have compact support, vanish at infinity, and are bounded, respectively; and equipped with the sup-norm 11•11 (cf. e.g,, [11]).

2. Generalized Clarkson's inequalities for extreme cases

THEoREM 1. Let p=1 or oo and let1:S r, s:!i oo. Then, for all fi,,f2,...,f2n in

L p)

2n 2n 2n

(3) {Z 11 Z e,,,f,, llSi}'fs Åq.. 2"('i"''iis){Z "f,"r,}i!r,

i-- l j= 1 j= 1

where 1/r + 1/r' = 1. Equality is attained fbr 1 $ r, s5 oo if the underlying measure space (M, E, ") is `bi.q enough': Therefbre, forp=1 or oo

(3') ll A.: t,2"(L,) .l,2, "(L,)"= 2n(ilr'+ifs).

PRooF. For E=L,, p=1 or co, clearly 11 A.: l,2"(E) -l,2."(E) II = 1.

Hence we have for all 1Sr sf{ oo

-) -

11 An: lr2"(E) - l,2"(E) 11

;Sl Ii I : l,2"(E) - l?"(E) 11 ii A. : l?"(E) . II(E) ll U I : lk" (E) . I,2, "(E) 11 :$ 2n(1-1!r) . 1 . 2nfs = 2n(1!r'+1!s),

where I denote the identity operators. This implies (3). (Note that the above argument, and therefore the inequality (3) is valid for any Banach space.)

For the latter assertion, take measurable sets Ej (j = 1,...,2") with O Åq pt(Ej) Åq oo and Ejn E, = e (1' # k). If p = 1, put fj = "(Ej)-ixj, where zj = x., denotes the

characteristic function of Ej. Then, since 11fj II, = 1 (1 S. J' Åq.. 2") and

2n

11 2 ÅíijL- IIi=2" for all 1 :sl i .sl 2", j--1

we have

2rt 2n

{2 11 Z s,jfj fils,}'fs .. 2n(i+ifs)

i-1 j--1

2n

., 2n(i!r'+i!s){ Z "fj 11r,}iir, J'=1

as desired. In the case where p = co, let gi =Z,2';, sjkxk. Then, 11gj11. = 1 (1 ;.:{ ]' .fl 2")

and

30 Mikio KATo and Ken-ichi MiyAzAKi

2n 11 2 sijgjll.= 2" -for all 1 i:s i .s 2"

j=1

since 2i:i6i,•gj =Z,2"=,(Z,2•:,6ijsjk)xk = 2"xi. (Note that A.2 = 2"E., where E. is the unit matrix.) Hence we have

2n 2n 2n

{2 ll Z Åí,jgjllS.}iiS=2"('fr''ifs){Z ilgi"r,.}iir

i=l j=1 j=1

as the preceding case, which completes the proof.

THEoREM 2. Let X be a locall.v compact Hausdonyff space and let E be one qf C,(X), C,(X), and C,(X). Let 1 S. r, sS. oo. Then, for all f,,1År,...,fÅr. in E,

2n 2n 2n (4) {2 H 2 6,,f, IIs}'!s ;s 2n(i!r'+iis){2 li f, "r}i!r,

i=1 j=1 ,j=1

where 1/r + 1/r' == 1. Equality is attained.for1 ;.:{ r,s:-:{ oo if X is `big enough', which means the eJcistence qf 2" non-empty mutually di.sjoint open sets: Consequoitly,

(4') ll A.: l,2"(E)-l,2"(E)"=2n(ifr'+i!s).

PRooF. We have only to show that equality is attained in (4) for C,(X). By our assumption and Urysohn's lemma there exist .f)EC,(X) (1 S.]' Åq.. 2") such that llfjll =1 and their supports are mutually disjoint (cf. e.g., [11], Theorem 2.7 and 2.12). Put gj =Z,2"=,Åíj,fkEC,(X). Then, gj's attain equality in (4) (the proof is the same as in Theorem 1; esp., the case p= oo).

By Theorems 1 and 2 we have

CoRoLLARy. Let E be one of L,,L., C,(X), Co(X), and Cb(.X). Then, the von Neumann-Jordan constant fbr E is 2, i,e.,

(5) C.,(E) -= 2,

where the underlying measure space M resp. Iocally compact Hctusdo(ff space X are assumed to be `non-trivial', which means the eJyistence qf two distl'oint, measurable sets ' of=.finite positive measure resp. non-empty open sets.

Indeed, by (3') and (4') we have for such E, 11A,:l,2(E)-l,2(E)ll-2,

which implies (5).

References

[1] R. P. Boas, Some uniformly convex spaces, Bull. Amer. Math. Soc. 46 (1940), 304-311.

[2] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396-414.

[3] J. A. Clarkson, The von Neumann-Jordan constant for the Lebesgue spaces, Ann. Math. 38 (1937), 114-I15.

[4] P. Jordan and J. von Neumann, On inner products in linear metric spaces, Ann Math. 36 (1935), 719-723.

[5] M. Kato, Generalized Clarkson's inequalities and the norms of the Littlewood matrices, Math.

Nachr. Il4 (1983), 163-170.

[6] M. Kato and K. Miyazaki, The von Neumann-Jordan constant for l:(L.)-spaces, Bull. Kyushu Inst.

Tech., Math. Natur. ScL 40 (1993), 23-27.

[7] M. Koskela, Some generalizations of Clarkson's inequalities, Univ. Beograd. PubL EIektrotechn.

Fak. Ser. Mat. Fiz. No. 634-677 (1979), 89-93.

[8] L. Maligranda and L. E. Persson, On Clarkson's inequalities and interpolation, Math. Nachr, 155 (1992), 187--197.

[9] K. Miyazaki and M. Kato, On a vector-valued interpolation theoretical proof of the generalized Clarkson inequalities, to appear in Hiroshima Math. J.

[10] A. Pietsch, Absolutely-p-summing operators in L,-spaces II, Sem. Goulaouic-Schwartz, Paris, 1970/1971.

[11] W. Rudin, Real and complex aiialysis, 3rd. Ed., McGraw-Hill, New York, 1986.

[12] A.Tonge, Random Clarkson inequalities and L,-versions of Grothendieck's inequality, Math.

Nachr. 131 (1987), 335-343.

Department of Mathematics

Kyushu Institute Qf Technology