Complex Convexity in Lebesgue-Bochner Function Spaces

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 1, January 1996

COMPLEX CONVEXITY IN LEBESGUE-BOCHNER FUNCTION SPACES

PATRICK N. DOWLING, ZHIBAO HU, AND DOUGLAS MUPASIRI

Abstract. Complex geometric properties of continuously quasi-normed spaces are introduced and their relationship to their analogues in real Banach spaces is discussed. It is shown that these properties lift from a continuously quasi-normed space X to Lp(µ, X), for 0

1. Introduction There are many papers in the literature dealing with the problem of lifting a property of a Banach space X to the Lebesgue-Bochner function spaces Lp(µ, X), for 1 norm 1 element of Lp(µ, X), for 1

Received by the editors July 22, 1994. 1991 Mathematics Subject Classiﬁcation. Primary 28A05, 46E40. Key words and phrases. Quasi-normed spaces, complex extreme points, complex strongly extreme points, analytic denting points.

c 1996 American Mathematical Society

127

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spaces but for continuously quasi-normed spaces (see section 2). The complex analogue of strict convexity is complex strict convexity. Complex strict convexity was introduced by Thorp and Whitley [14] for complex Banach spaces, and is a measure of convexity. However, it was shown by Dilworth [2], that it can be viewed as a measure of subharmonicity. There are some lifting results known for these complex geometric properties. For example, if X is a continuously quasi-normed space which is uniformly PL- convex (respectively, uniformly H-convex), then Lp(µ, X) is uniformly PL-convex (respectively, uniformly H-convex), for 0

2. Preliminaries and definitions

A quasi-normed space is a pair (X, )whereXis a vector space and is a k·k k·k function on X with values in R+ satisfying (a) λx = λ x for all scalars λ and all x in X; (b)k therek exists| |kC>k 0 such that x + y C[ x + y ] for all x and y in X; and, k k≤ k k k k (c) x =0ifandonlyifx=0. k k The smallest C for which (b) holds is called the quasi-norm constant of (X, ). k·k A quasi-normed space (X, ) is said to be a continuously quasi-normed space if is uniformly continuous onk·k the bounded subsets of X. We shall call a complete continuouslyk·k quasi-normed space a continuous quasi-Banach space. A continuous quasi-Banach space (X, ) is said to be locally PL-convex if log is a pluri- subharmonic function onk·kX [1, Proposition 2.2]. k·k If X is a continuous quasi-Banach space, then BX will denote the unit ball of X and SX will denote the unit sphere of X. Throughout this paper, D will denote the open unit disc in the complex plane, and T will denote its boundary.

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Deﬁnition 1. Let X be a complex Banach space and let x SX .Thenxis called ∈ a complex extreme point of BX if for every non-zero y X,thereexistsz Dso that x + zy > 1. ∈ ∈ k k A Banach space is said to be complex strictly convex if every element of SX is a complex extreme point of BX . The following characterization of complex extreme points can be found in [8].

Theorem 2. Let X be a complex Banach space and let x SX . Then the following conditions are equivalent: ∈ (i) x is a complex extreme point of BX ; 2π iθ p dθ (ii) there exists 0 1; ∞ ∈ k k 2π R (iii) for each 0 1. ∞ ∈ 0 k k 2π From Theorem 2, we are naturally led to the following:R

Deﬁnition 3. Let X be a continuous quasi-Banach space. A point x SX is said ∈ to be a complex extreme point of BX if for each non-zero y X, ∈ 2π dθ x + eiθy > 1. k k 2π Z0

We say that X is complex strictly convex if every element of SX is a complex extreme point of BX . Remarks. 1. Clearly Deﬁnitions 1 and 3 are equivalent in Banach spaces because of The- orem 2. 2. Let X be a locally PL-convex continuous quasi-Banach space. A point x SX ∈ is a complex extreme point of BX if and only if for some 0 1. This follows from [1,Theorem∞ 2.4]. ∈ k k 3. Note that to obtain the result in RemarkR 2 one can relax the condition that X is locally PL-convex, by only insisting that the function log x + zy is a subharmonic in the complex variable z for each y X. However,k in mostk ap- plications one deals with locally PL-convex continuous∈ quasi-Banach spaces, so from now on we will restrict our attention to that class of spaces. A strengthening of the notion of complex extreme point is the following: Deﬁnition 4. Let X be a locally PL-convex continuous quasi-Banach space. A point x SX is said to be a complex strongly extreme point of BX if for each ε>0, ∈ 2π there is δ>0sothat x+eiθy dθ 1+δ, whenever y X and y ε. 0 k k 2π ≥ ∈ k k≥ Remarks. R 1. A point x SX is a complex strongly extreme point of BX if and only if for some 0 0, there is δp > 0sothat x+e y 0 k k 2π ≥ 1+δp, whenever y X and y ε. ∈ k k≥ R This follows from [1, Theorem 2.4].

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2. If X is a complex Banach space, then a point x SX is a complex strongly ∈ extreme point of BX if and only if for each ε>0, there is δ>0 such that for each y X with y ε,thereisz Dwith x + zy 1+δ. See [2, Proposition∈ 2.2; 6, Theoremk k≥ 7]. ∈ k k≥ 3. It is clear from Remark 2 that the notion of a complex strongly extreme point in a complex Banach space is the analogue of a strongly extreme point of a real Banach space. It should be noted that complex strongly extreme points are the local analogues of uniform PL-convexity [1]. The local analogue of uniform H-convexity (see [3]) is the following: Deﬁnition 5. Let X be a locally PL-convex continuous quasi-Banach space. A point x SX is said to be an analytic denting point of BX if for each ε>0, there ∈ 2π iθ dθ iθ n ikθ is δ>0sothat 0 P(e ) 2π > 1+δ,whereP(e )= k=0 ake ,ak X,for k k 2π iθ dθ ∈ 0 k nand n N,witha0=xand P (e ) x ε. ≤ ≤ R∈ 0 k − k 2Pπ ≥ Remarks. R 1. A point x SX is an analytic denting point of BX if and only if for some 0 0, there is δp > 0sothat 0 P(e ) 2π > 1+ iθ n ikθ k k δp, whenever P (e )= k=0 ake ,ak X,for0 k n R 1 2π iθ∈ p dθ ≤ ≤ and n N,witha0 =xand ( P (e ) x ) p ε. ∈ P 0 k − k 2π ≥ This follows by modifying a result of [16,R Theorem 2.1]. 2. For 0

x SX is an analytic denting point of BX if and only if for ∈ each ε>0, there is δ>0sothat f 1 1+δ whenever 1 ˆ k kˆ ≥ f H (T,X)withf(0) = x and f f(0) 1 ε. ∈ k − k ≥ A similar characterization of an analytic denting point can also be obtained in terms of Hp(T,X)-functions using Remark 1. 3. It is clear from the deﬁnitions that analytic denting points are complex strongly extreme points. The converse is not true as the following example shows. Example 6. The point x =(1,1,1, ...) is a complex strongly extreme point of

B`∞ , since it is a strongly extreme point of B`∞ . z+a On the other hand, if a D,letϕa be the M¨obius transformation ϕa(z)= . ∈ 1+¯az Then ϕa maps D onto D and T onto T.Also,

iθ sup ϕa(e ) a =1+ a 1. θ | − | | |≥ ∈R iθ That is, supθ ϕa(e ) ϕˆa(0) 1. ∈R | − |≥

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Let am m∞=1 be dense in T.Let rn n∞=1 be a sequence in (0, 1) with lim rn =1. { } { } n Deﬁne →∞

fn : D¯ ` by → ∞

1 ∞ f (z)= ϕ (z) . n r a rnam n m m=1

Note that for each z , f (z) = 1 , fˆ (0) = (1, 1, 1, ...)andf is analytic T n `∞ rn n n on D. ∈ k k 1 1 Hence fn H (T,`∞)and fn 1= , so limn fn 1 = 1. However ∈ k k rn →∞ k k

2π ˆ 1 iθ dθ fn fn(0) 1 = sup ϕrnam (e ) 1 k − k 0 m rnam − 2π Z ∈N

2π 1 iθ dθ = sup ϕrnam (e ) ϕˆrnam (0) 0 rn m − 2π Z ∈N

1 1, for all n N. ≥ rn ≥ ∈

Thus x is not an analytic denting point of BX .

3. Lifting theorems

Throughout this section (Ω, Σ,µ) will denote a positive, complete measure space. If X is a continuous quasi-Banach space and 0

1 p p f p = f(ω) dµ(ω) . k kL (µ,X) k k ZΩ

Lp(µ) denotes Lp(µ, X), when X is the scalar ﬁeld.

Theorem 7. Let X be a locally PL-convex continuous quasi-Banach space, let 0 < p< ,andletf SLp(µ,X). Then f is a complex strongly extreme (respectively, ∞ ∈ f(ω) analytic denting) point of BLp (µ,X) if and only if f(ω) is a complex strongly k k extreme (respectively, analytic denting) point of BX ,forµ-almost all ω supp f = ω Ω:f(ω)=0 . ∈ { ∈ 6 } Proof. We will prove the theorem for analytic denting points; the result for complex strongly extreme points follows in a similar manner. Suppose that for µ-almost all ω supp f,f(ω)/f(ω) is an analytic ∈ k k

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p p denting point of BX .LetGnn∞=1 be a sequence in H (T,L (µ, X)) with {} limn Gn + f p =1,andGn(0) = 0. To show that f is an analytic dent- →∞ k k ing point of BLp(µ,X) it suﬃces to show that limn Gn p =0.SinceGn(0) = c →∞ k k 0, Gn(0) (ω)=0forµ-almost all ω Ω. Hence ∈ c c 2π p iθ p dθ Gn( )+f = Gn(e )+f p k · kp k kL (µ,X) 2π Z0

2π iθ p dθ = Gn(e )(ω)+f(ω) dµ(ω) k k 2π Z0 ZΩ

2π iθ pdθ = Gn(e )(ω)+f(ω) dµ(ω). k k 2π ZΩ Z0

1 2π iθ pdθ p Deﬁne gn(ω)= Gn(e )(ω)+f(ω) . 0 k k 2π Note that gn(ω) f(ω) for µ-almost all ω Ωand ≥kR k ∈

p p Gn()+f = gn(ω) dµ(ω) k · kp ZΩ

1= f(ω) pdµ(ω). → k k ZΩ

Hence, there is a subsequence gnk k∞=1 of gn n∞=1 converging pointwise µ-almost everywhere to f . { } { } k k Note that 2π p iθ p dθ Gn = Gn (e ) p k k kp k k kL (µ,X) 2π Z0

2π iθ p dθ = Gn (e )(ω) dµ(ω) k k k 2π Z0 ZΩ

2π iθ p dθ = Gn (e )(ω) dµ(ω) k k k 2π ZΩ Z0

p = Gn ( )(ω) dµ(ω). k k · kp ZΩ k Also, since Gn ( )(ω)+f(ω) p =gn (ω) f(ω) and Gn (0)(ω)=0forµ- k k · k k −→k k k almost all ω Ω, Gnk ( )(ω) p 0forµ-almost all ω Ω, because f(ω)/ f(ω) ∈ k · k −→n ∈ k k is an analytic denting point of BX for µ-almost all ω supp df. ∈

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2π p iθ p dθ Gn ( )(ω) = Gn (e )(ω) k k · kp k k k 2π Z0

2π iθ pdθ = Gn (e )(ω)+f(ω) f(ω) k k − k 2π Z0

2π p iθ dθ C Gnk (e )(ω)+f(ω) + f(ω) ≤ 0 k k k k 2π Z h i (where C is the quasi-norm constant of X)

2π p iθ p p dθ (2C) Gnk (e )(ω)+f(ω) + f(ω) ≤ 0 k k k k 2π Z p p p =(2C) [gn (ω) + f(ω) ]. k k k

p p p 1 p Since (2C) [gnk ( ) + f( ) ] k∞=1 converges in L (µ) and since Gnk ( )(ω) p { · k · k } k · k −→k 0, for µ-almost all ω Ω, we have by a variation of the Lebesgue Dominated Convergence Theorem∈ that

p p Gnk p = Gnk ( )(ω) p dµ(ω) 0. k k k · k −→k ZΩ

By repeating this argument for each subsequence of Gn ∞ we conclude that { k }n=1 Gn p 0. Therefore, f is an analytic denting point of BLp(µ,X). k k −→n Conversely, suppose that

E = ω supp f : f(ω)/ f(ω) is not an analytic denting point of BX { ∈ k k }

is not a µ-null set. Then µ∗(E) > 0, where µ∗ is the outer measure of µ. Deﬁne

p En = ω Ω : there exists gk k∞=1 in H (T,X), withg ˆk(0) = 0, { ∈ { } 1 1 gk + f(ω) p f(ω) + , gk p , for all k N, and f(ω) n . k k ≤k k k k k ≥ n ∈ k k≤ }

Clearly, E = ∞ En and so there is n0 N such that µ∗(En0 ) > 0. Without n=1 ∈ loss of generality we can also assume that µ∗(En ) < . Therefore, there exists S 0 ∞ F Σ such that En F, µ(F)=µ∗(En )and f(ω) n0 for all ω F .Let ∈ 0 ⊂ 0 k k≤ ∈ ε0=µ∗(En0)/2. 1 Let C be the quasi-norm constant of Lp(T,X). Choose M>0sothat(1+x)p min 1 ,1 ≤ 1+Mx { p } for all x [0,ε0]. ∈ 1 1 min p ,1 If δ>0isgiven,choose0<ε1 < 6 so that M(3ε1µ∗(En0 )) { } <δ.SinceX is a continuous quasi-Banach space, so is Lp(T,X) [1, Proposition 2.1], and hence p and p are uniformly continuous functions on bounded subsets of X Lp(T,X) k·k p k·k p p and L (T,X) respectively. Therefore there is δ1 > 0sothat x+y x +ε1 k k ≤k k

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p p whenever x 2n0Cand y <δ1,and h+k p h p +ε1 whenever k k≤ k k k kL (T,X) ≤k kL (T,X) h p n0 +1and k p <δ1. k kL (T,X) ≤ k kL (T,X) Since f is measurable and µ(F ) >ε0, there exists pairwise disjoint, measur- k able subsets F1, ..., Fk of F each of positive µ-measure and j=1 µ(Fj ) >ε0 and diamf(Fj ) <δ,foreachj=1,2, ..., k. Clearly, µ∗(Fj E)=µ(Fj)>0foreach ∩ P p j=1, ..., k.Chooseωj Fj Efor each j =1, ..., k.Choosegj H(T,X)with 1 ∈ ∩ p p ∈ gˆj(0) = 0, gj p > , and gj + f(ωj) p < f(ωj) + ε1. Deﬁne k k n0 k k k k

p G : T L (µ, X) → by k iθ iθ χ G(e )= gj(e ) Fj . j=1 X Clearly, G Hp(T,Lp(µ, X)), Gˆ(0) = 0, and ∈

1 2π p iθ p dθ G p = G(e ) p k k k kL (µ,X) 2π Z0

1 2π dθ p = G(eiθ)(ω) pdµ(ω) k k 2π Z0 ZΩ

1 2π dθ p = G(eiθ)(ω) p dµ(ω) k k 2π ZΩ Z0

1 p 2π k iθ p dθ = gj (e ) χF (ω) dµ(ω)   k k j  2π  Ω 0 j=1 Z Z X    

1 k p p = gj χF (ω) dµ(ω)   k kp j   Ω j=1 Z X    

1 k p 1 1 p 1 1 p ε0 µ Fj ε = . ≥ n   ≥ n 0 n 0 j=1 0 0 [   On the other hand, for all ω Ω ∈

p k f(ω) ;if ω Ω Fj , G(eiθ)(ω)+f(ω) p = k k ∈ \ j=1 iθ p k k ( gj(e )+f(ω) ;ifω Fj. k k S∈

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k Hence, if ω Ω Fj , ∈ \ j=1 S 2π dθ G(eiθ)(ω)+f(ω) p = f(ω) p, k k 2π k k Z0 and if ω Fj, then ∈ 2π 2π iθ pdθ iθ pdθ G(e )(ω)+f(ω) = gj(e )+f(ω) k k 2π k k 2π Z0 Z0

p = gj + f(ω) p k kL (T,X)

p = gj + f(ωj)+(f(ω) f(ωj)) p k − kL (T,X)

p gj +f(ωj) p + ε1 ≤k kL (T,X)

p f(ωj) +2ε1 ≤k k

p = f(ω)+f(ωj) f(ω) +2ε1 k − k

p f(ω) +3ε1, ≤k k where the ﬁrst inequality follows from the fact that gj +f(ωj) p n0 +1 and k kL (T,X) ≤ f(ω) f(ωj) p <δ1, and the last inequality follows because f(ω) 2n0C k − kL (T,X) k k≤ and f(ω) f(ωj) <δ1. Therefore,k − k

2π iθ pdθ p G(e )(ω)+f(ω) f(ω) +3ε1χ k (ω). k k 2π ≤k k j=1Fj Z0 ∪ Consequently,

1 2π p iθ p dθ G( )+f p = G(e )(ω)+f(ω) dµ(ω) k · k 0 Ωk k 2π Z Z 1 2π dθ p = G(eiθ)(ω)+f(ω) p dµ(ω) Ω 0 k k 2π Z Z 1 p

  p   f(ω) +3ε1χ (ω) dµ(ω) ≤ k k k ZΩ      Fj     j=1     [     1   k p p = f +3ε1µ Fj k k   j=1 [   

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min 1 ,1 k { p } f +M(3ε1µ Fj ) ≤k k   j=1 [  min 1,1 f +M(3ε1µ(F )) { p } ≤k k f +δ. ≤k k p p It follows that for each n N,thereexistsGn H(T,L (µ, X)) such that 1 ∈ ∈ p ε0 1 Gn p , Gˆn(0) = 0 and Gn( )+f p < f + . k k ≥ n0 k · k k k n Hence f is not an analytic denting point of BLp(µ,X). Theorem 8. Let X be a locally PL-convex continuous quasi-Banach space, let 0 0, where µ∗ is the outer measure of µ. Deﬁne}

1 En = ω Ω: f(ω)

Then E = ∞ En. Hence there is n0 N such that µ∗(En0 ) > 0. Without n=1 ∈ loss of generality we may assume that µ∗(En ) < . Therefore, there exists F Σ S 0 ∞ ∈ such that En F, µ(F)=µ∗(En )and f(ω)

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1 By the deﬁnition of En , there exists a function g : En X such that < 0 0 → n0 2π iθ p p p g(ω)

Πn = Fα α ∆n ,Fα Fβif α, β ∆withα β, and there is { } ∈ ⊃ ∈ ≤ Aα Fα En such that µ∗(Aα)=µ(Fα),Aα Aβif α β,and ⊂ ∩ 0 ⊃ ≤ with both diam f(Fα) <δn and diam g(Aα) <δn, for all α ∆n. ∈

If α ∆n,andAα = ,chooseωα Aα. Deﬁne ∈ 6 ∅ ∈ χ gn = g(ωα) Fα .

α ∆n X∈ p 1 Clearly gn L (µ, X)andforallω F, < gn(ω)

2π dθ f(ω)+eiθh(ω) p f(ω) p. k k 2π ≥k k Z0

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Since n N was arbitrary, we therefore have ∈ 2π dθ f(ω)+eiθh(ω) p = f(ω) p k k 2π k k Z0 for all ω Ω. Hence ∈ 2π dθ 2π dθ f + eiθh p = f(ω)+eiθh(ω) pdµ(ω) k kp 2π k k 2π Z0 Z0 ZΩ

2π dθ = f(ω)+eiθh(ω) p dµ(ω) k k 2π ZΩ Z0

= f(ω) pdµ(ω) k k ZΩ

= f p. k kp p Thus since h =0inL (µ, X),f is not a complex extreme point of B p . 6 L (µ,X) Remark. Theorem 8 (b) does not hold if the restriction of X being separable is re- moved [8]. Theorem 8 immediately gives the following improvement of [2, Theorem 2.5]: Corollary 9. Let X be a locally PL-convex continuous quasi-Banach space and let 0

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Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056 E-mail address: [email protected]

Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056 Current address: Department of Mathematics, El Paso Community College, P.O. Box 20500, El Paso, Texas 79998 E-mail address: [email protected]

Department of Mathematics, University of Northern Iowa, Cedar Falls, Iowa 50614 E-mail address: [email protected]

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