Convexity plurisubharmonicity and the strong maximum modulus principle in Banach spaces Anne-Edgar Wilke

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Convexity, plurisubharmonicity and the strong

maximum modulus principle in Banach spaces

Anne-Edgar Wilke

Abstract. – In this article, we ﬁrst try to make the known analogy between convexity

and plurisubharmonicity more precise. Then we introduce a notion of strict plurisubhar-

monicity analogous to strict convexity, and we show how this notion can be used to study

the strong maximum modulus principle in Banach spaces. As an application, we deﬁne a

notion of

direct integral of a family of Banach spaces, which includes at once Bochner

spaces,

ℓp

direct sums and Hilbert direct integrals, and we show that under suitable

hypotheses, when

p<∞

, an

direct integral satisﬁes the strong maximum modulus

principle if and only if almost all members of the family do. This statement can be consid-

ered as a rewording of several known results, but the notion of strict plurisubharmonicity

yields a new proof of it, which has the advantage of being short, enlightening and uniﬁed.

1 Introduction

1.1 Convexity and plurisubharmonicity

Plurisubharmonic functions (psh, for short) were introduced independently by Oka [

p. 40] and Lelong [

, p. 306, déf. 1]. Since the origins of the theory, it has been observed

that there is a certain analogy between convex functions and psh functions; in fact, Oka

called the latter pseudoconvex functions. To make this analogy apparent, Bremermann [

pp. 34-38] collected a list of properties satisﬁed by convex functions, and showed that for

each of them, there is a corresponding property satisﬁed by psh functions.

Here are Bremermann’s main ideas, slightly reformulated. A continuous function

f:R→R

is said to be convex, or sublinear, if for all compact intervals

I⊂R

and for

all aﬃne functions

α: I →R

, the inequality

f≤α

holds on

as soon as it holds on the

boundary of

. A continuous function

f:Rn→R

is said to be convex, or plurisublinear, if

for all aﬃne maps γ:R→Rn, the composition f◦γis convex.

In the same way, an upper semicontinuous function

f:C→R∪ {−∞}

is said to

be subharmonic if for all connected, smoothly bounded compact sets

K⊂C

and for all

functions

α: K →R

continuous and harmonic in the interior of

, the inequality

f≤α

holds on

as soon as it holds on the boundary of

. An upper semicontinuous function

f:Cn→R∪ {−∞}

is said to be plurisubharmonic if for all aﬃne maps

γ:C→Cn

, the

composition f◦γis subharmonic.

arXiv:2210.14087v2 [math.CV] 7 Sep 2023

As Bremermann remarks, convexity and plurisubharmonicity can be deﬁned more

quickly in the following way. A continuous function

f:Rn→R

is convex if and only if for

all aﬃne maps γ:R→Rn,

f(γ(0)) ≤f(γ(−1)) + f(γ(1))

2.(1)

This amounts to asking that the value of

f◦γ

at the centre of the unit ball of

be less than

its mean on the sphere. Similarly, an upper semicontinuous function

f:Cn→R∪ {−∞}

is psh if and only if for all aﬃne maps γ:C→Cn,

f(γ(0)) ≤1

2πZ2π

f(γ(eit )) dt,(2)

which amounts to asking that the value of

f◦γ

at the centre of the unit ball of

be less

than its mean on the sphere.

From the above discussion, it is tempting to conclude, as does Bremermann, that the

analogy between convex functions and psh functions is obtained by replacing

with

real aﬃne maps with complex aﬃne maps, and sublinearity conditions with subharmonicity

conditions. These ideas permeate much of the literature on the subject; see for instance

[

, p. 225]. Yet we will show that this dictionary misses an essential point and therefore

is unsatisfactory.

Indeed, a fundamental result, due to Lelong [

, p. 325, n°17], states that psh

functions are stable under composition with a holomorphic map, from which one can

deﬁne the notion of psh function on a holomorphic manifold. This result does not appear

in Bremermann’s list, which is understandable, since from his point of view, there is no

analogous result for convex functions.

In fact, the natural domain of a convex function is a real aﬃne space, while the natural

domain of a psh function is a holomorphic manifold, or even a complex analytic space

Therefore, real aﬃne maps do not correspond to complex aﬃne maps, but rather to

holomorphic maps.

Thus it is preferable to deﬁne the notion of psh function in the following way: if

a holomorphic manifold, an upper semicontinuous function

f: X →R∪ {−∞}

is said

to be psh if the inequality

(2)

holds for all holomorphic maps

γ:D→X

, where

D⊂C

the closed unit disc, with the understanding that a map deﬁned on

is holomorphic if it

extends to a holomorphic map on a neighbourhood of D.

In the case

X=Cn

, if

satisﬁes this inequality for all aﬃne maps

γ:D→Cn

, then

psh: this is the contents of Lelong’s result. But this fact is best seen, not as a deﬁnition,

but rather as a characterisation, valid in a special case, and which is far from obvious. We

will not use it in this article.

The approach taken in this article would allow one to deﬁne the notion of convex function more generally

on any topological space

, as soon as one chooses a class of continuous maps

γ:[−1; 1]→X

playing the

role of aﬃne maps. An important case is when

is a Riemannian manifold and the maps

are geodesic

segments. In the same way, one could deﬁne the notion of psh function on any topological space

equipped

with a class of maps γ:D→Xplaying the role of holomorphic maps.

Beyond the aesthetic aspect, a good understanding of the analogy between convexity

and plurisubharmonicity enables one to obtain certain non-trivial results about psh func-

tions by adapting the proofs of the corresponding, usually easier, statements concerning

convex functions. We hope that the results presented in this article will serve to illustrate

this phenomenon.

1.2 Strict plurisubharmonicity

A continuous function

f:Rn→R

is strictly convex if the inequality

(1)

holds strictly

for all non-constant aﬃne maps

γ:R→Rn

. According to Bremermann’s dictionary, this

suggests the following deﬁnition: an upper semicontinuous function

f:Cn→R∪{−∞}

strictly psh if the inequality

(2)

holds strictly for all non-constant aﬃne maps

γ:C→Cn

This is, with a diﬀerent formulation, Carmignani’s deﬁnition [

, pp. 285-286, def. 1.1

and 1.2]

. However, this deﬁnition is very unsatisfactory: indeed, we will see an example

showing that the class of functions thus obtained is not stable under composition with a

biholomorphism.

It is therefore preferable, according to the principles explained above, to say that an

upper semicontinuous function

f: X →R∪ {−∞}

on a holomorphic manifold

is strictly

psh if the inequality

(2)

holds strictly for all non-constant holomorphic maps

γ:D→X

. We

will see several results showing that strict plurisubharmonicity, thus deﬁned, is a natural

notion, analogous to strict convexity.

In the real case, there exists a stronger notion than strict convexity: a function

f:Rn→

is said to be strongly convex if it can be written locally as the sum of a convex function

and a

function

such that

d2ϵ

is a positive deﬁnite symmetric bilinear form at every

point.

The analogous notion in the complex case is the following: a function

f: X →R∪{−∞}

is said to be strongly psh if it can be written locally as the sum of a psh function and a

function ϵsuch that ∂∂ϵis a positive deﬁnite hermitian form at every point.

Unfortunately, it is a common practice in the literature to call strongly psh functions

strictly psh. This situation is unhappy, because strong plurisubharmonicity is analogous to

strong convexity, and not to strict convexity.

1.3 The strong maximum modulus principle in Banach spaces

-Banach space

is said to be strictly convex if every aﬃne map

γ:[−1; 1]→E

whose

image is contained in the unit sphere is constant.

By analogy, it might be tempting to say that a

-Banach space

is strictly convex in the

complex sense if every aﬃne map

γ:D→E

whose image is contained in the unit sphere is

constant. This deﬁnition was strongly suggested by Thorp and Whitley [

], and explicitly

given by Globevnik [14, p. 175, def. 1].

After changing Carmignani’s deﬁnition 1.1 so that strictly subharmonic functions are assumed to be

ﬁnite on a dense subset, and correcting deﬁnition 1.2, which erroneously omits the hypothesis w̸=0.

However, the analogy turns out to be more satisfying if one asks instead that every

holomorphic map

γ:D→E

whose image is contained in the unit sphere be constant. In

this case, in order to keep the terminology consistent, we will say that Eis strictly psh.

The main result of Thorp and Whitley [

, p. 641, th. 3.1], slightly reformulated, is

that both deﬁnitions are actually equivalent, that is, that

is strictly psh if and only if

every aﬃne map

γ:D→E

whose image is contained in the unit sphere is constant. But

this fact is best seen, not as a deﬁnition, but rather as a non-trivial characterisation. We

will not use it in this article.

Beyond the analogy with strictly convex spaces, the importance of strictly psh spaces

lies in the fact that a

-Banach space is strictly psh if and only if it satisﬁes the strong

maximum modulus principle, that is, if and only if every holomorphic map from a connected

manifold Xto Ewhose norm has a local maximum is constant.

Strict convexity of an

-Banach space can be characterised in the following way:

(E, ∥·∥)

is strictly convex if and only if for every (or for one) increasing, strictly convex map

ψ:R+→R

, the composition

ψ◦ ∥·∥

is strictly convex. The analogous statement for a

-Banach space is the following:

(E, ∥·∥)

is strictly psh if and only if for every (or for one)

strictly convex map

ψ:R∪ {−∞} → R∪ {−∞}

, the composition

ψ◦log∥·∥

is strictly psh.

These two results will give us simple characterisations of strict convexity and strict

plurisubharmonicity of Lpdirect integrals, that we will now present.

1.4 Direct integrals

The notion of direct integral used in this article is rather basic, but suﬃcient to include at

once Bochner

spaces,

ℓp

direct sums and Hilbert direct integrals. One may consult [

pp. 61-62] and [9, pp. 683-686] for a more elaborate theory.

Let

(S, Σ,µ)

be a measure space, let

E= (Es)s∈S

be a measurable family, in a sense that

we will deﬁne, of real or complex Banach spaces, and let

p∈[1; ∞]

. A section of

is an

element of the product

Qs∈SEs

. Given a section

satisfying an appropriate measurability

condition, let ∥σ∥pbe the p-norm of the function s7→ ∥σ(s)∥Es; explicitly,

∥σ∥p=









ZS

∥σ(s)∥p

Esdµ(s)1

if p<∞,

ess sup

s∈S

∥σ(s)∥Esif p=∞.

(3)

Then

is said to be

-integrable if

∥σ∥p<∞

, and the

direct integral of the family

is deﬁned to be the space of

-integrable sections, up to equality almost everywhere,

equipped with the norm ∥·∥p. It is a Banach space, denoted by Lp(E).

If the family

is constant, equal to a Banach space

, then

Lp(E)

is the Bochner space

Lp(S, Σ,µ; E). In the case where Ehas dimension 1, one recovers Lebesgue Lpspaces.

Suppose that the

-algebra

is discrete, that

is the counting measure and that the

are pairwise distinct. Then

Lp(E)

is essentially the

ℓp

direct sum of the family

, denoted

ℓp(E)

. More precisely,

Lp(E)

is the closed subspace of

ℓp(E)

whose elements are the

sections with countable support; this subspace coincides with

ℓp(E)

except when

p=∞

and the set of those s∈Ssuch that Esis non-zero is uncountable.

Finally, Hilbert direct integrals correspond to the case where

p=2

and each

is equal

to one of the spaces ℓ2

n, for n∈N, or to ℓ2

∞.

Here are now the results promised. For conciseness purposes, the statements are slightly

less general than what will be proved in the article.

Theorem 1. Suppose that

-ﬁnite, that

is a discrete measurable family of

-Banach

spaces, and that

1<p<∞

. The direct integral

Lp(E)

is strictly convex if and only if

strictly convex for almost all s.

Theorem 2. Suppose that

-ﬁnite, that

is a discrete measurable family of

-Banach

spaces, and that

1≤p<∞

. The direct integral

Lp(E)

is strictly psh if and only if

strictly psh for almost all s.

The notion of discrete family which appears in these statements is, as we will see, insignif-

icant in practice: indeed, when

-ﬁnite, the existence of a non-discrete measurable

family cannot be proved in ZFC.

Even if we will prove Theorems 1 and 2 through entirely parallel methods, it is important

to note that the statements themselves are not rigorously analogous. Indeed, deﬁnition

(3)

is problematic in this respect, because the true complex analogue of

∥·∥Es

log∥·∥Es

, and

not

∥·∥Es

. An examination of the proofs reveals that this discrepancy is the origin of

the diﬀerence between the hypothesis

1<p<∞

in the real case and the hypothesis

1≤p<∞in the complex case.

Let us now give a few immediate consequences of Theorems 1 and 2.

Corollary 3. Suppose that

is non-zero and

-ﬁnite and that

1<p<∞

, and let

-Banach space. The Bochner space

Lp(S, Σ,µ; E)

is strictly convex if and only if

is. In

particular, the Lebesgue space Lp(S, Σ,µ;R)is strictly convex.

Corollary 4. Suppose that

is non-zero and

-ﬁnite and that

1≤p<∞

, and let

be a

-Banach space. The Bochner space

Lp(S, Σ,µ; E)

is strictly psh if and only if

is. In particular,

the Lebesgue space Lp(S, Σ,µ;C)is strictly psh.

Corollary 5. Suppose that

is countable, that

1<p<∞

, and that the

are

-Banach

spaces. Then ℓp(E)is strictly convex if and only if Esis strictly convex for all s.

Corollary 6. Suppose that

is countable, that

1≤p<∞

, and that the

are

-Banach

spaces. Then ℓp(E)is strictly psh if and only if Esis strictly psh for all s.

It is not diﬃcult to see that Corollaries 5 and 6 imply the same statements without the

countability hypothesis on

. Taking this remark into account, it turns out that Theorem 1

is a consequence of Corollaries 3 and 5, and that Theorem 2 is a consequence of Corollaries

4 and 6: indeed, we will see that

direct integrals are in fact

ℓp

direct sums of Bochner

spaces. Thus one sees that the notion of direct integral is a means to state and prove

results about ℓpdirect sums and Bochner spaces in a uniﬁed way.

Corollary 5 was proved by Day [

, p. 314] [

, p. 520, th. 6] through a simple and direct

method, which can also give Corollary 3, as the author remarks [

, p. 521]. This method

can actually be used to prove Theorems 1 and 2 when

1<p<∞

, but probably not in the

general case, as we will see in the appendix.

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Convexity,plurisubharmonicityandthestrongmaximummodulusprincipleinBanachspacesAnne-EdgarWilkeAbstract.–Inthisarticle,wefirsttrytomaketheknownanalogybetweenconvexityandplurisubharmonicitymoreprecise.Thenweintroduceanotionofstrictplurisubhar-monicityanalogoustostrictconvexity,andweshowhowthisnotioncan...

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Convexity plurisubharmonicity and the strong maximum modulus principle in Banach spaces Anne-Edgar Wilke

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