The Schwarz Lemma

By Seán Dineen

The Schwarz lemma is among the simplest results in complex analysis that capture the rigidity of holomorphic functions. This self-contained volume provides a thorough overview of the subject; it assumes no knowledge of intrinsic metrics and aims for the main results, introducing notation, secondary concepts, and techniques as necessary. Suitable for advanced undergraduates and graduate students of mathematics, the two-part treatment covers basic theory and applications.
Starting with an exploration of the subject in terms of holomorphic and subharmonic functions, the treatment proves a Schwarz lemma for plurisubharmonic functions and discusses the basic properties of the Poincaré distance and the Schwarz-Pick systems of pseudodistances. Additional topics include hyperbolic manifolds, special domains, pseudometrics defined using the (complex) Green function, holomorphic curvature, and the algebraic metric of Harris. The second part explores fixed point theorems and the analytic Radon-Nikodym property.

• Language: English
• Publisher: Dover Publications
• Release date: Apr 6, 2016
• ISBN: 9780486810973

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Index

Part I

1

THE CLASSICAL SCHWARZ LEMMA

In this course we discuss intrinsic¹ metrics and distances on complex manifolds. The underlying areas are complex analysis (of one, several, and infinitely many variables), functional analysis (Banach space theory), differential geometry, and potential theory. We assume some familiarity with complex analysis and Banach space theory but assume no knowledge of differential geometry or potential theory. It is our hope that this course will lead to a better understanding of the interconnections between some of the many different aspects of complex analysis and stimulate further research.

+ are the set of real numbers and positive real numbers respectively. A domain is a connected open subset of a Banach space and we assume, unless otherwise stated, that all complex manifolds are connected. X and Y complex manifolds modelled on X and Y will denote the set of all holomorphic mappings denote the vector space of all continuous linear mappings from the Banach space X into the Banach space Y. If Y we write Xand if X = Y .

are domains in X and Yif and only if f is continuous and for each one-dimensional affine subspace E of X and each ϕ ∈ Υis a holomorphic function of one complex variable. There are many other equivalent definitions (e.g. use Taylor series expansions or Cauchy–Riemann equations) of holomorphic mapping between Banach spaces. To the reader unfamiliar with infinite dimensional holomorphy (i.e. the study of holomorphic mappings between locally convex spaces and, in particular, Banach spaces) we suggest, at least on a first reading, that you treat everything in Part I as finite dimensional but keep in mind that in moving to infinite dimensions we lose local compactness and Lebesgue measure.

1.1The Schwarz lemma and the Schwarz–Pick lemma

We begin our study by discussing the classical Schwarz lemma of one complex variable. We shall use the Schwarz lemma to define intrinsic distances and these distances will in turn lead us to new versions of the Schwarz lemma. The Schwarz lemma itself consists of four hypotheses and three conclusions, each of which will be modified in later chapters.

is a domain in a Banach spacedenote the set of all biholomorphic automorphisms if

if bijective and holomorphic,

(b)f–1 is holomorphic.

For finite dimensional spaces (a) ⇒ (b), but it is not known if the implication is true in infinite dimensions (Suffridge [1]).

Classical Schwarz Lemma. If and f(0) = 0 then

(i) |f(z)| ≤ |z| for all ,

(ii) |f′(0)| ≤ 1.

Moreover, if |f(z)| = |z| for some non-zero or |f′(0)| = 1 then f(z) = eiθz for some θ and all z in .

Proof. Since fand f(z) = zh(z. By the maximum modulus theorem

for all r, 0 < r < 1. Since |f(zwe get, on letting r . Hence |f(z)| ≤ |z. This completes the proof of (i). Moreover, if |f(z0)| = |zthen |h(z0)| = 1 and, by the maximum modulus theorem, h is a constant function of modulus 1, i.e. there exists θ such that f(z) = zh(z) = eiθz .

and f(0) = 0 it follows that

If |f′(0)| = |h(0)| = 1 then, by the maximum modulus theorem, h is a constant function of modulus 1 and, as before, this implies f(z) = eiθz for some θ .

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We next prove a generalization of this lemma — the Schwarz–Pick lemma — but in order to do this we require the form of the biholomorphic automorphisms . We first define the Möbius transformations ϕa, and ϕ–a ∘ ϕa(z) = z . For θ ,

and ϕa whenever |a| < 1.

and ψ(0) = 0. By the Schwarz lemma |ψwe also have

Hence |ψ′(0)| = 1 and by the Schwarz lemma ψ(z) = eiθz for some θ . Hence f = ϕ–f(0) ∘ ψ are the composition of a rotation and a Möbius transformation.

Schwarz–Pick Lemma. If then

(i) for all z,

(ii) for all .

We have equality in (i) and (ii) if . If equality holds in (i) for one pair of points z ≠ w or if equality holds in (ii) at one point z then .

Proof. Let g = ϕf(w) ∘ f ∘ ϕ–w. Then

. In particular, g(0) = ϕf(w)(f(w)) = 0. By Schwarz’s lemma |g(ξ)| ≤ |ξand hence

i.e. |ϕf(w)(f(z))| ≤ |ϕw(z)| and thus

This proves (i). If equality holds for some pair z ≠ w then |g(ϕw(z))| = |ϕw(z)| and by the Schwarz lemma g(z) = eiθz for some θ and

.

By(i)

This proves (ii). Now suppose we have equality in (ii). Letting z = w we see that

and

.

then (i) applied to f and f–1 gives

and we have equality. Similarly for (ii) we have

.

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Remark. If f(0) = 0 then part (i) of the Schwarz–Pick lemma with w = 0 gives the Schwarz lemma.

The move from one to several complex variables frequently leads to fundamental problems and more technical proofs, and naturally one expects that a further generalization to infinite dimensions should lead to even more difficulties. This is not always the case. Perhaps it is because, in finite dimensions, a first approach is usually via coordinates while, in infinite dimensions, a coordinate-free approach is more natural and the coordinate approach — which assumes the existence of a Schauder basis — is tried later. When the coordinate-free approach is successful, the final proofs are often less technical and the estimates sharper (in the sense that we know they are dimension free). Without coordinates the Hahn–Banach theorem and its corollaries play a more central role (for instance, the Hahn–Banach theorem is used to prove the Cauchy inequalities which are used in the next proposition). We illustrate these points by moving directly to infinite dimensions and giving some generalizations of the results already proved.

First we recall some versions of the Hahn–Banach theorem and introduce some notation.

Hahn–Banach Theorem. Let X be a Banach space.

(a)If C is a closed convex subset of X and x0 ∉ C then there exists ϕ ∈ X′ such that

(b)If x ∈ X then there exists ϕ ∈ X′, ||ϕ|| = 1, such that ϕ(x) = ||x||

(c)If x ∈ X then .

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If f : A → (Y. If A is a Banach space we let A0 denote the open unit ball of A and we write ||f|| in place of ||f||A0. If X and Y are Banach spaces and ϕ : X → Y is a linear mapping then we say ϕ is an isometry if ϕ(Χ) = Y and ||ϕ(x)|| = ||x|| for all x ∈ X. If X0 and Y0 are the open unit balls of X and Y respectively then it is easily seen that an injective linear mapping ϕ : X → Y is an isometry if and only if ϕ(Χ0) = Y0.

Proposition 1.1 (Harris [1], Phillips [1].). If X is a Banach space with open unit ball X0, h ∈ Aut(X0); and h(0) = 0 then h is the restriction to X0 of a linear isometry of X onto X.

Proof. By the Cauchy inequalities ||h′(0)|| ≤ 1 and ||(h–1)′(0)|| ≤ 1.³ Since (h–1)′(0) = (h′(0))–1 we have for all x ∈ X

and hence ||h′(0)(x)|| = ||x||. Since h is invertible we have h′(0)(X) = X and hence h′(0) is a linear isometry of X onto X. Let f = (h′(0))–1 ∘ h. Then f ∈ Aut(X0), f(0) = 0, and f′(0)x = x all x ∈ X. Let f′(0) + Pk + ... be the Taylor series⁴ expansion of f at 0 with Pk denoting the first non-zero term of degree ≥ 2. Then

is the Taylor series expansion of

at 0. Since fn ∈ H(X0, X0) the Cauchy inequalities imply that ||nΡk|| ≤ 1 for all n and hence Pk = 0. Hence f(x) = x for all x ∈ X0 and h(x) = h′(0) ∘ f(x) = h′(0)(x) for all x ∈ X0.

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The technique used above is known as the Cartan iteration trick and can also be used to prove the following:

If is a bounded domain in a Banach space and fsatisfy a = f(a) = g(a) and f′(a) = g′(a) for some point a in then f ≡ g.

can be endowed with the structure of a Lie group and that complete holomorphic vector fields on arbitrary bounded circular domains are polynomials of degree less than or equal to two (see chapter 9).

The proof given above is due to Harris but it is not his original proof. The original proof used a different one-dimensional extension of the Schwarz lemma and complex extreme points and, although more technical, had several other consequences including a simple proof of the strong maximum modulus theorem (proposition 6.19).

In the proof of proposition 1.1 we used the fact that h ∈ Aut(X0) to show that h′(0) was a linear isometry from X onto X. Conversely if h ∈ H(X0, X0) and h′(0) is a linear isometry of X onto X . This result, when h(0) = 0, is a consequence of the following continuous form of the Schwarz lemma which we state without proof and which is also due to Harris [2].

Proposition 1.2. If X is a Banach space with open unit ball X0, h ∈ H(X0, X0), and h(0) = 0 then

for all x ∈ X0, where U is the set of all linear isometries from X onto X and d is the distance in the operator norm from h′(0) to U.

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. Similar results hold in an arbitrary Banach space and, without being very precise, we may say the following:

If X0 is the open unit ball of a Banach space X, a ∈ X0, and there exists an h ∈ Aut(X0) such that h(a) = 0 then there exists a ‘Möbius-type’ transformation ϕ such that ϕ(a) = 0.

In some cases we can write down explicit formulae for the Möbius transformations.

Example 1.3. Let H (the bounded linear operators from H into itself with the operator norm). For each A ∈ X0, the open unit ball of X, the mapping MA defined by

is called a Möbius transformation⁵. We have MA ∈ Aut(X0), MA(0) = A .

In general if we place on Aut(X0), X0 the open unit ball of the Banach space X, the topology of local uniform convergence⁶ (see chapter 9 for details) and let Aut⁰(X0) denote the connected component containing the identity mapping then a result of Kaup–Upmeier [1] states that

where Aut0(X0) = {ϕ ∈ Aut(X0) : ϕ(0) = 0}. Aut0(X0) consists, in view of proposition 1.1, of the restriction to X0 of linear isometries of X onto X and is a closed subgroup of Aut(X0) (called the isotropy subgroup of Aut(X0) at the origin).

1.2A Schwarz lemma for subharmonic functions

We begin this section by rapidly reviewing the basic properties of subharmonic functions ² by means of z = x + iy → (x, y).

Subharmonic functions are the complex analogue of convex functions and it is instructive to look first at three equivalent definitions of convex functions of one real variable.

A real-valued measurable function u of a real variable x is convex on an interval (a, b) if for all x, x′ in (a, b) we have

If for each point x in (a, b) and each r > 0 such that a < x – r < x + r < b we let μx,r denote the ‘invariant’ probability measure on {x – r, x + r} given by

then from (1.1) it follows that u is convex if and only if

for all x ∈ (a, b) and all appropriate r.

function u on (a, bon (a, bmay be discussed for arbitrary continuous functions if we use the theory of distributions to define derivatives and positivity.

With this interpretation it can be shown that an arbitrary continuous function u on (a, b) is convex if and only if

in the sense of distributions. In the present context this just means that

function ϕ of compact support in (a, b).

Finally we consider the functions on (a, b. These are just the continuous affine functions. If u is a function on (a, b) then u is convex if and only if for any x, y ∈ (a, b), x ≤ y, and any affine ϕ

implies u(t) ≤ ϕ(t) for x ≤ t ≤ y.

Thus we have a mean inequality criterion (1.2), a positivity criterion (1.3), and a maximum principle (1.4) each of which characterizes convex functions of one real variable. If we move to one complex variable we replace μx,r where

A simple calculation shows that

it is, however, necessary to place some regularity condition on the functions involved as, otherwise, the integrals or distributions involved would not be well defined. A weakened form of continuity — upper semicontinuity — is the required condition.

A function ϕ : X → [–∞, +∞), X a topological space, is upper semi-continuous if for all c the set {x ∈ X; ϕ(x) < c} is open.

It is easy to show that ϕ is upper semicontinuous if and only if

for all x0 ∈ X.

function u is said to be harmonic if Δu = 0. .

Definition 1.4. Let be a domain in and an upper semicontinuous function which is not identically –∞. Then u is said to be subharmonic if any one of the following equivalent conditions hold:

(a) for all and all r > 0 such that

we have

(b) in the sense of distributions (and in the usual sense if u is a function)

(c) if G is a relatively compact domain in and υ is harmonic on G and continuous on then u ≤ υ on ∂G ⇒ u ≤ υ on G.

Condition (c) is known as the principle of the harmonic majorant and shows how the term subharmonic arose. From (c) the following maximum principle can easily be deduced for subharmonic functions (i.e. they cannot, unless they are constant, achieve a maximum at an interior point):

Proposition 1.5. If u is subharmonic on a domain and

then u ≤ M on and u(z0) = M at some if and only if u ≡ M on .

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On combining (1.5) and (1.6) we see that

for any subharmonic function u .

We may remove part of the set over which the limit is taken in (1.7) and still obtain the same equality. The set removed cannot be too large and this leads to the concept of thin set

Definition 1.6. A subset S of is thin at p if either or there is an open neighbourhood V of p and u subharmonic on V such that

Example 1.7. A line segment of positive length is not thin at its end points.

Proof (S. Gardiner). It suffices to show (0, 1) is not thin at 0. Suppose otherwise. Then there exists a subharmonic function u on {z; |z| < 2} such that u(z) ≤ 0 for all z .

Then u(x) < δ for 0 < x ≤ α and without loss of generality we can suppose α = 1. Let Ω = {z; |z| < 1}\{x ∈ R, 0 ≤ x < 1} and let

Then w ∈ H(Ω) and

is a harmonic function on Ω. For ∊ > 0 the function

is subharmonic on Ω and, by considering each part of the boundary in turn, we can show that

By the maximum principle

and, since ∊ was arbitrary, we have

Hence

Since δ < u(0) this is impossible.

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There are various situations in which subharmonic functions may be extended. We confine ourselves to the following simple result whose proof is immediate.

Proposition 1.8. If p is a point in a domain , u is subharmonic on , and there exists ∊ > 0 such that and sup{u(z) : 0 < |z – p| < ∊} < ∞ then the function

is subharmonic on .

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Finally we shall need the following result.

Lemma 1.9. If u is subharmonic on a domain then eu is also subharmonic.

Proof. Since the exponential function is convex and increasing

for all x . If u is subharmonic and a > 0 then au + b is subharmonic for any real number bthen, either u(z0) = –∞, or

Since eu is easily seen to be upper semicontinuous and eu ≥ 0 this implies the desired result.

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Proposition 1.10 (Sibony [2]). Let u be a function defined on and of class on a neighbourhood of the origin. Suppose 0 ≤ u ≤ 1, u(0) = 0 and log u is subharmonic in . Then

(i) u(z) ≤ |z|² for with equality at one point different from 0 if and only if u(z) is equal to |z|² for all z in ,

(ii) Δu(0) ≤ 4 with equality if and only if u(z) = |z|² for all z in .

Proof. Since u(0) = 0 and u ≥ 0 it follows that u has a local minimum at 0. Hence grad(ulet υ(z) = u(z)|z|–2. Since log υ(z) = log u(z) – log |z|² it follows that log υ . By lemma 1.9, υ = elog υ . For z = x + iy near 0 we have, by Taylor’s theorem,

is bounded from above on a neighbourhood of 0 and, by proposition 1.8, υ . Since

. Hence u(z) ≤ |z. If we have equality at one point then by