bloch-sobolev spaces and analytic composition operators

Computational Methods and Function TheoryVolume 5 (2005), No. 2, 381–393

Bloch-Sobolev Spacesand Analytic Composition Operators

Marko Kotilainen, Visa Latvala and Jie Xiao

(Communicated by Peter Duren)

Abstract. We introduce the so-called Bloch-Sobolev function spaces andshow that these spaces have nice closure properties. We also characterizethe boundedness and compactness of a composition operator Cφ (with an-alytic symbol φ between two subdomains Ω, Ω′ � R2) acting between twoBloch-Sobolev spaces. As a by-product we obtain a characterization of thoseanalytic mappings φ : Ω → Ω′, which are uniformly continuous with respectto the quasihyperbolic metrics in Ω and Ω′.

Keywords. Bloch-Sobolev spaces, composition operators, analytic mappings.

2000 MSC. Primary 31B05; Secondary 46E15, 47B38.

1. Introduction

The Schwarz-Pick lemma [6, p. 16] shows that if φ : D → D is analytic, then

(1.1) supz∈D

(1− |z|2)|(f ◦ φ)′(z)| ≤ supz∈D

(1− |z|2)|f ′(z)|

for all analytic functions f defined in the unit disc D. One can also express thisinequality by saying that the self-mappings of the unit disk do not increase thehyperbolic distance [6, p. 62]. The starting point of this paper was to understandthe invariance property (1.1) from a wider perspective. Such a study is partiallymotivated by the study of Bloch-invariance in the theory of several complexvalues, see [19]. The main novelty of our approach is that the invariance problemis studied for Bloch spaces which do not necessarily consist of smooth functionsonly. Such spaces are related to many classes of solutions of elliptic PDE’s.Another new feature in our presentation is that we consider arbitrary domains.

To be more precise, we study Bloch-invariance as a behaviour of the compositionoperator between two Bloch spaces which are complete with respect to a certainweak type Bloch norm. To keep the presentation simple, we only consider an

Received June 1, 2005, in revised form September 27, 2005.Jie Xiao was supported by NSERC (Canada).

ISSN 1617-9447/$ 2.50 c© 2005 Heldermann Verlag

382 M. Kotilainen, V. Latvala and J. Xiao CMFT

analytic composition mapping between two arbitrary subdomains. Note how-ever that certain extensions to a quasiregular case are possible, these are brieflycommented in Remark 3.4.

The starting point of our note is stated as follows: Let Ω, Ω′ � R2 be two givendomains with boundaries ∂Ω, ∂Ω′ and let φ : Ω → Ω′ be analytic. For whichmaps φ does the composition Cφf = f ◦ φ obey the inequality

(1.2) ess supx∈Ω

d(x, ∂Ω)|(∇Cφf)(x)| ≤ C ess supy∈Ω′

d(y, ∂Ω′)|∇f(y)|

for all f ∈ W 1,1loc (Ω′)? Here and henceforth, W 1,1

loc stands for the usual local firstorder Sobolev space with the integration exponent p = 1, d(x, E) stands for theEuclidean distance between the point x ∈ R2 and the set E ⊂ R2, and ∇ meansthe distributional gradient. It appears (Theorem 3.1) that the inequality (1.2)holds if and only if

supx∈Ω

d(x, ∂Ω)|φ′(x)|d(φ(x), ∂Ω′)

<∞.

The use of the essential supremum in the definition of Bloch seminorm for Sobolevfunctions looks simple but the real issue behind the definition is not trivial; thispaper heavily relies on the fact that our Bloch-Sobolev space

BS(Ω) =

{f ∈W 1,1

loc (Ω) : ‖f‖Ω := ess supx∈Ω

d(x, ∂Ω)|∇f(x)| <∞}

is a function space which is complete with respect to the given norm. Section 2 isdevoted to the study of the completeness properties (Lemmas 2.6, 2.7 and 2.9).These are based on Lipschitz continuity with respect to the quasihyperbolic met-ric (Theorem 2.3). Section 3 is devoted to the boundedness and compactnessresults for the analytic composition operator (Theorem 3.1 and 3.6). The gener-ality of the domains implies that the boundedness of Cφ does not always hold,i.e. (1.2) is not valid in general. As a by-product, the study of the boundednessallows us to characterize those analytic mappings φ : Ω → Ω′, which are uni-formly continuous with respect to the quasihyperbolic metrics in Ω and Ω′ (cf.Remark 3.3). This result is related to the theory developed in [20, Sections 11and 12]. To characterize the compactness of Cφ by means of condition

limd(x,∂Ω)→0

d(x, ∂Ω)|φ′(x)|d(φ(x), ∂Ω′)

= 0,

we need to assume that Ω′ is bounded and it does not have any isolated boundarypoints. The compactness problem has been previously studied in [12] in theclassical case where Ω and Ω′ coincide with the unit disc D ⊂ R2 and the studyhas been extended to the theory of several complex variables in [19].

5 (2005), No. 2 Bloch-Sobolev Spaces and Analytic Composition Operators 383

2. Bloch-Sobolev spaces: Lipschitz continuity and closurebehavior

Throughout this section, we assume that D � Rn is a domain with n ≥ 2. Wewrite d(x, ∂D) for the distance between x ∈ D and the boundary ∂D and denoteBx = B(x, 1

2d(x, ∂D)) for any x ∈ D. Here and elsewhere B(x, r) stands for the

Euclidean ball with center x and radius r > 0.

Lipschitz continuity of Bloch-Sobolev functions. To introduce our resulton the Lipschitz continuity of Bloch-Sobolev functions, we first give the followingdefinition.

Definition 2.1. A real-valued function f on D is called a Bloch-Sobolev func-tion, denoted f ∈ BS(D), if f belongs to the Sobolev spaces W 1,1

loc (D) and itsdistributional gradient ∇f satisfies

(2.1) ‖f‖D := ess supx∈D

d(x, ∂D)|∇f(x)| <∞.

As far as we know, Definition 2.1 is not standard. Usually one considers C1-functions satisfying (2.1). From our point of view, Definition 2.1 is natural sinceit creates a function space which has nice closure properties. Theorem 2.3 showsthat we could as well replace the assumption f ∈ W 1,1

loc (D) by the assumptionthat f is locally Lipschitz in Definition 2.1.

Along with the Bloch-Sobolev space BS(D) we also study the subspace consistingof harmonic Bloch-Sobolev functions; that is,

BSH(D) = {f ∈ BS(D) : f is harmonic in D}.The subspaces consisting of smooth or harmonic functions in BS(D) have beenpreviously studied in several papers, see for e.g. [4, 7]. For harmonic and moregenerally for p-harmonic functions one can equivalently replace the norm of thegradient by a certain averaged energy integral in the condition (2.1). This weaktype Bloch-Sobolev norm has appeared in the papers [1, 5, 8, 9, 10, 15, 16, 17].

Quasihyperbolic metric. It appears that BS(D) is contained in the famousfunction space BMO(D). To see this and to prove the Lip-continuity of Bloch-Sobolev functions, we introduce the notion of a quasihyperbolic metric. This isdefined for any x0, y0 ∈ D as a number

(2.2) kD(x0, y0) = infγ

∫γ

ds

d(x, ∂D),

where the infimum is taken over all rectifiable curves γ joining x0 to y0 in D. Thequasihyperbolic metric was introduced in [2] as a generalization of the classicalhyperbolic metric to arbitrary domains. We need the following auxiliary lemma,see [20, p. 34].


Lemma 2.2. If 0 < ρ < 1, x ∈ D, and y ∈ B(x, ρd(x, ∂D)), then

(2.3)1

C

|x− y|d(x, ∂D)

≤ kD(x, y) ≤ C|x− y|

d(x, ∂D)

for some constant C > 0 only depending on ρ and n.

The following result (extending [7, Proposition 1.2]) reveals that all Bloch-Sobolev functions are Lipschitz continuous relative to the quasihyperbolic metric.

Theorem 2.3. If f ∈ BS(D), then

(2.4) |f(x)− f(y)| ≤ ‖f‖DkD(x, y)

for all x, y ∈ D. Conversely, if

c(f, D) := supx,y∈D,x �=y

|f(x)− f(y)|kD(x, y)

<∞,

then there is a constant C = C(n) > 0 such that ‖f‖D ≤ Cc(f, D).

Proof. Assume that f ∈ BS(D). We first show that f is locally Lipschitz in Dby using a well-known result due to Morrey. For y ∈ D, let By = B(y, 1

2d(y, ∂D))

and let B(y′, r) ⊂ By. For any x ∈ B(y′, r), we clearly have

1

2d(y, ∂D) ≤ d(x, ∂D) ≤ 2d(y, ∂D).

Hence there is a constant C = C(n) > 0 such that∫B(y′,r)

|∇f(x)| dx ≤∫

B(y′,r)

‖f‖D

d(x, ∂D)dx ≤ Crn‖f‖D

d(y, ∂D).

By [13, Theorem 1.53], f is locally Lipschitz in By.

To prove the condition (2.4), let x, y ∈ D and let γD be the quasihyperbolicgeodesic joining x to y. It is known that the normal representation γ = γ(s) ofγD is smooth, see [14, Corollary 4.8]. By the absolute continuity of f ◦ γ, wehave

|f(x)− f(y)| = |(f ◦ γ)(l)− (f ◦ γ)(0)| ≤∫ l

0

|(f ◦ γ)′(s)| ds

=

∫ l

0

|∇f(γ(s)) · γ′(s)| ds ≤ ‖f‖D

∫ l

0

|γ′(s)|d(γ(s), ∂D)

ds

= ‖f‖DkD(x, y).

Here l denotes the length of γD.

Conversely, assume that c(f, D) <∞. Then, for fixed y ∈ D, we have

|f(x)− f(y)| ≤ Cc(f, D)|x− y|

d(y, ∂D)


whenever x ∈ By. Here the constant C comes from Lemma 2.2. This showsthat f is locally Lipschitz in D. By Rademacher’s Theorem [13, Corollary 1.73],f is differentiable a.e. in D. Suppose that f is differentiable at y ∈ D. Then theestimate |f(x)− f(y)|

|x− y| ≤ Cc(f, D)

d(y, ∂D), x ∈ By,

impliesd(y, ∂D)|∇f(y)| ≤ Cc(f, D).

Remark 2.4. Recall that f ∈ BMO(D) provided f ∈ L1loc(D) and

‖f‖BMO(D) = supB⊂⊂D

|B|−1

∫B

|f(x)− fB| dx < ∞.

Here the supremum is taken over all Euclidean balls B compactly contained in D,|B| stands for the Lebesgue measure of B, and fB = |B|−1

∫B

f dx is the integralaverage of f over B.

(a) Theorem 2.3, together with well-known results on BMO, implies the estimate‖f‖BMO(D) ≤ C‖f‖D for a constant C = C(n) > 0. Indeed, if f ∈ BS(D), thenfrom Lemma 2.2 and Theorem 2.3 it turns out that for some constant C = C(n)we have

sup f(By)− inf f(By) ≤ C‖f‖D

for all y ∈ D. The claim follows from [18, Corollary 2.26].

(b) If f ∈ BSH(D), then the converse inequality

‖f‖D ≤ C‖f‖BMO(D)

holds with a constant C = C(n) > 0. In other words, the Bloch-Sobolev semi-norm is equivalent to the BMO-seminorm in this case. This assertion holdsmore generally for p-harmonic functions with p > 1, see [8, Section 4] and [15,Theorem 3.8]. Hence BSH(D) equals to the set of harmonic BMO-functionson D.

Since the function x �→ kD(x, x0) is continuous for any fixed x0 ∈ D, we have:

Corollary 2.5. For each x0 ∈ D and every compact K ⊂ D there exists aconstant M > 0 only depending on x0, K and D such that if f ∈ BS(D), thenfor all x ∈ K

|f(x)| ≤ |f(x0)|+ M‖f‖D.

Convergence properties of Bloch-Sobolev functions. For analytic Blochfunctions in the unit disc D, the standard convergence properties are verifiedvia the fact that the uniform convergence of analytic functions implies uniformconvergence for the derivatives. We could use the same idea for harmonic Bloch-Sobolev functions but Theorem 2.3 offers a general simple argument for thispurpose. The convergence properties of Bloch-Sobolev functions rely on thefollowing key lemma.


Lemma 2.6. If fj ∈ BS(D) with supj∈N‖fj‖D < ∞ and the sequence (fj)

converges locally uniformly to f in D, then f ∈ BS(D). A similar assertionholds for the space BSH(D).

Proof. It suffices to show that f ∈ BS(D) if M := supj∈N‖fj‖D < ∞ and the

convergence fj → f is locally uniform.

Let x, y ∈ D and let ε > 0. By uniform convergence, we can find j ∈ N suchthat |fj(x)− f(x)| < ε and |fj(y)− f(y)| < ε. By Theorem 2.3,

|f(x)− f(y)| < |fj(x)− fj(y)|+ 2ε ≤ ‖fj‖DkD(x, y) + 2ε,

and hence|f(x)− f(y)| ≤MkD(x, y)

for all x, y ∈ D. The claim f ∈ BS(D) follows from Theorem 2.3.

Lemma 2.7. If fj ∈ BS(D) with supj∈N‖fj‖D <∞, then there is a subsequence

(fjk) which converges locally uniformly on D to a function f ∈ BS(D). A similar

assertion holds for the space BSH(D).

Proof. Again, it is enough to prove the first assertion. Corollary 2.5 impliesthat the family F = {fj

∣∣ j ∈ N} is locally equibounded and Theorem 2.3 impliesthat F is locally equicontinuous. Using the Ascoli-Arzela Theorem [6, p. 19]and the standard diagonalization argument we obtain a subsequence (fjk

) ⊂ F ,which converges locally uniformly to a function f in D. Lemma 2.6 implies thatf ∈ BS(D).

Bloch-Sobolev norm. In the forthcoming section we require a norm in BS(D).This is defined by ‖f‖D,x0 := |f(x0)|+ ‖f‖D, where x0 is any fixed point in D.

Remark 2.8. The choice of the fixed point x0 ∈ D can be made arbitrarily. Tosee this, let f ∈ BS(D) and let x1, x2 ∈ D, x1 �= x2. By Theorem 2.3,

‖f‖D,x1 = |f(x1)|+ ‖f‖D ≤ |f(x2)|+ ‖f‖D(kD(x1, x2) + 1)

≤ (kD(x1, x2) + 1)‖f‖D,x2 .

Changing the roles of x1 and x2 in the estimate we observe that the norms ‖f‖D,x1

and ‖f‖D,x2 are equivalent.

Lemma 2.9. (BS(D), ‖·‖D,x0) and (BSH(D), ‖·‖D,x0) are Banach spaces.

Proof. It is enough to prove that (BS(D), ‖·‖D,x0) is a Banach space. To do so,the only non-trivial step is to show that (BS(D), ‖·‖D,x0) is complete. Assumetherefore that (fj) is a Cauchy sequence in (BS(D), ‖·‖D,x0). Hence, for anyε > 0, there exists N ∈ N such that ‖fj − fk‖D,x0 < ε for all j, k ≥ N . If K ⊂ Dis compact, Corollary 2.5 implies the existence of a constant M > 0 such that

|fj(x)− fk(x)| ≤M‖fj − fk‖D,x0 < Mε

for every x ∈ K and for each j, k ≥ N . Hence (fj) converges locally uniformlyon D to a limit function f . The assertion follows from Lemma 2.6.


3. Composition operators: boundedness and compactness

In what follows Ω and Ω′ are supposed to be proper subdomains of R2. We alsorecall Cφ(f) := f ◦ φ, where f : Ω′ → R is locally Lipschitz in Ω′ and φ : Ω → Ω′

is analytic.

Boundedness of composition operators. We first characterize the bound-edness of the composition operator Cφ for Bloch-Sobolev functions. Recall thatthe points x0 ∈ Ω and x′

0 ∈ Ω′ are the fixed points needed for the definition ofthe Bloch-Sobolev norm above.

Theorem 3.1. Let Ω′ ⊂ R2 be bounded and let φ : Ω → Ω′ be analytic. Then thefollowing three statements are equivalent:

(a) the operator Cφ : BSH(Ω′) → BSH(Ω) is bounded,(b) the operator Cφ : BS(Ω′) → BS(Ω) is bounded,(c) the mapping φ satisfies the condition

(3.1) supx∈Ω

d(x, ∂Ω)|φ′(x)|d(φ(x), ∂Ω′)

<∞.

Proof. For convenience, let M be the supremum in (3.1). It is enough to provethe equivalence of (b) and (c) since the equivalence of (a) and (c) can be verifiedexactly the same way.

Let M < ∞ and let f ∈ BS(Ω′). By Theorem 2.3 and Rademacher’s Theorem(see [13, Corollary 1.73]) f is differentiable a.e. in Ω′. Hence by the chain rule

d(x, ∂Ω)|∇(f ◦ φ)(x)| ≤ d(x, ∂Ω)

d(φ(x), ∂Ω′)|φ′(x)| · d(φ(x), ∂Ω′)|∇f(φ(x))|

for a.e. x ∈ Ω. This implies ‖f ◦ φ‖Ω ≤ M‖f‖Ω′ and therefore

‖f ◦ φ‖Ω,x0 ≤ |f(φ(x0))|+ M‖f‖Ω′ .

By Corollary 2.5, there is a constant M ′ > 1 only depending on the choice of x′0

such that

|f(φ(x0))| ≤ |f(x′0)|+ M ′‖f‖Ω′ .

Thus

‖f ◦ φ‖Ω,x0 ≤ (M + M ′)‖f‖Ω′,x′0,

i.e. the condition (b) holds.

To prove the converse, assume on the contrary that M = ∞. Then there is asequence (xj) of points in Ω such that

limj→∞

d(xj, ∂Ω)

d(φ(xj), ∂Ω′)|φ′(xj)| = ∞.


For each j ∈ N pick aj ∈ ∂Ω′ such that d(φ(xj), ∂Ω′) = |φ(xj) − aj| and definefj(x) = ln |x− aj|. The functions fj are harmonic in Ω′ with the property

‖fj‖Ω′ = supx∈Ω′

d(x, ∂Ω′)|∇fj(x)| = supx∈Ω′

d(x, ∂Ω′)|x− aj| = 1.

Since Ω′ is bounded, supj∈N|fj(x

′0)| is finite, and therefore supj∈N

‖fj‖Ω′,x′0

<∞.On the other hand,

‖fj ◦ φ‖Ω ≥ d(xj, ∂Ω)|∇(fj ◦ φ)(xj)| = d(xj, ∂Ω)

d(φ(xj), ∂Ω′)|φ′(xj)|d(φ(xj), ∂Ω′)

|φ(xj)− aj| .

Since the right hand side tends to ∞ as j → ∞, we conclude that the condi-tion (a) does not hold.

As stated in Introduction, the Schwarz-Pick lemma guarantees that the condi-tion (3.1) always holds if Ω and Ω′ both coincide with the unit disc D. In generaldomains, however, the condition (3.1) is not necessarily true. The followingexample is [20, Example 11.4]:

Example 3.2. Let φ(z) = e(z+1)/(z−1) for z ∈ D. Then the map φ is analyticin D and φ(D) = D \ {0}. In this case

d(z, ∂D) = 1− |z|, z ∈ D,

d(φ(z), ∂D \ {0}) = min{|φ(z)|, 1− |φ(z)|}, z ∈ D.

Let (xj)j∈N be a sequence of real numbers such that xj → 1 as j → ∞. Thenφ(xj) → 0 as j →∞. A simple calculation shows that

supz∈D

d(z, ∂D)

d(φ(z), ∂(D \ {0})) |φ′(z)| ≥ 1− |xj|

|φ(xj)| |φ′(xj)| = 2

(1− |xj|) →∞

as j →∞.

Remark 3.3. An analytic function φ : Ω → Ω′ is uniformly continuous as amapping between the metric spaces (Ω, kΩ) and (Ω′, kΩ′) if and only if φ satisfies(3.1). This can be seen from Theorem 3.1 along with results of [8] and [10]. Tobe specific, the proof of Theorem 3.1 shows that the condition (3.1) holds if andonly if there is a constant C > 0 such that

(3.2) ‖f ◦ φ‖Ω ≤ C‖f‖Ω′

for all f ∈ BH(Ω′). Note that Ω′ does not have to be bounded for we consider theBloch-Sobolev seminorms in the inequality (3.2). Since the Bloch-Sobolev semi-norm is equivalent to the BMO-seminorm for harmonic functions (Remark 2.4),the claim follows from [10, Theorems 4.1 and 4.5].

Remark 3.4. For a reader who is interested in quasiregular mappings, we men-tion that Theorem 3.1 holds for all n ≥ 2 in the following form: let φ : Ω → Ω′ bea non-constant quasiregular mapping between two domains Ω � Rn and Ω′ � Rn.


Then the operator Cφ : BS(Ω′) → BS(Ω) is bounded if and only if the mapping φsatisfies the condition

(3.3) ess supx∈Ω

d(x, ∂Ω)|φ′(x)|d(φ(x), ∂Ω′)

<∞.

Here |φ′(x)| is defined almost everywhere in Ω by the formula

|φ′(x)| = max|h|=1

|φ′(x)h|,

see [3, Section 14] for details. The proof is the same as above. The point isthat the action of the chain rule takes essentially the required form (instead ofan equality we now have two inequalities with constants depending only on thedilatation of the mapping), see [3, Theorem 14.28]. For the other necessary prop-erties (almost everywhere differentiability and the behaviour of sets of measurezero), see [3, Lemma 14.22].

Compactness of composition operators. In [12], the authors characterizedthe compactness of a composition operator for analytic Bloch functions in theunit disc D. In what follows, we give a similar characterization in arbitraryplane domains for harmonic Bloch-Sobolev spaces. This requires an additionalassumption that ∂Ω′ does not contain any isolated points. In the light of Ex-ample 3.2 one more assumption on Ω′ is necessary. We remark that also in [20,Theorem 12.21] the connectedness of the boundary ∂Ω plays a role. We closethis note with an example which shows that the given characterization is nottrue for the operator Cφ : BS(Ω′) → BS(Ω).

We first recall a standard lemma which characterizes the compactness of a com-position operator.

Lemma 3.5. Let φ : Ω → Ω′ be an analytic mapping. Then the compositionoperator Cφ : BS(Ω′) → BS(Ω) is compact if and only if for any bounded sequence(fj) in BS(Ω′) such that fj → 0 locally uniformly on Ω′, the sequence ‖Cφfj‖Ω,x0

converges to zero. A similar assertion is true for Cφ : BSH(Ω′) → BSH(Ω).

Proof. We give a proof of the first assertion just to make the presentation self-contained. Let Cφ : BS(Ω′) → BS(Ω) be compact. If the desired conclusionis false, there is a bounded sequence (fj) in BS(Ω′) such that fj → 0 locallyuniformly on Ω′ and for some real number ε > 0 and a subsequence (fjk

) we have

(3.4) ‖Cφfjk‖Ω,x0 ≥ ε

for all k ∈ N. Since Cφ is compact we find a new subsequence (still denoted by(Cφfjk

)) and the limit function g ∈ BS(Ω) such that

(3.5) limk→∞

‖Cφfjk− g‖Ω,x0 = 0.

By Corollary 2.5, Cφfjk−g → 0 locally uniformly on Ω. Moreover, since fjk

→ 0locally uniformly on Ω′, we have Cφfjk

→ 0 locally uniformly on Ω just because


of the continuity of φ. Thus g = 0 and (3.5) yields that limk→∞ ‖Cφfjk‖Ω,x0 = 0.

This contradicts (3.4).

Conversely, let (fj) be a sequence in BS(Ω′) with supj∈N‖fj‖Ω′,x′

0< ∞. To

prove that Cφ is compact it is enough to show that the sequence (Cφfj) has anorm convergent subsequence. By Lemma 2.7, there is a subsequence (fjk

) whichconverges locally uniformly on Ω′ to a limit function f ∈ BS(Ω′). By assumption,

‖Cφfjk− Cφf‖Ω′,x′

0= ‖Cφ(fjk

− f)‖Ω′,x′0→ 0

as k →∞.

Theorem 3.6. Let φ : Ω → Ω′ be an analytic mapping. Suppose that Ω′ isbounded and the boundary ∂Ω′ does not have any isolated points. Then the fol-lowing two statements are equivalent:

(a) the operator Cφ : BSH(Ω′) → BSH(Ω) is compact,(b) for every ε > 0, there exists δ > 0 such that

d(x, ∂Ω)

d(φ(x), ∂Ω′)|φ′(x)| < ε

whenever d(φ(x), ∂Ω′) < δ.

Proof. First assume that (b) holds. Let (fj) be a sequence of functions inBSH(Ω′) such that supj∈N

‖fj‖Ω′,x′0

< ∞ and fj → 0 locally uniformly on Ω′.Let ε > 0 and let M := supj∈N

‖fj‖Ω′,x′0. By (b), there is δ > 0 such that

d(x, ∂Ω)

d(φ(x), ∂Ω′)|φ′(x)| < ε

M

whenever d(φ(x), ∂Ω′) < δ. It follows that

d(x, ∂Ω)|∇(fj ◦ φ)(x)| =d(x, ∂Ω)

d(φ(x), ∂Ω′)|φ′(x)| d(φ(x), ∂Ω′)|∇fj(φ(x))|

≤ Md(x, ∂Ω)

d(φ(x), ∂Ω′)|φ′(x)| < ε

whenever d(φ(x), ∂Ω′) < δ. Since fj → 0 locally uniformly on Ω′, we have|∇fj| → 0 locally uniformly on Ω′ by the well-known properties of two-dimensionalharmonic functions. Hence

limj→∞

(fj ◦ φ)(x0) = 0 and d(x′, ∂Ω)|∇fj(x′)| → 0

uniformly on the set {x′ ∈ Ω′ : d(x′, ∂Ω′) ≥ δ}. Notice here that Ω′ is assumedto be bounded. By the continuity of the function

x �→ d(x, ∂Ω)

d(φ(x), ∂Ω′)|φ′(x)|

there is a constant C > 0 such that

d(x, ∂Ω)|∇(fj ◦ φ)(x)| ≤ Cd(φ(x), ∂Ω′)|∇fj(φ(x))|


whenever d(φ(x), ∂Ω′) ≥ δ. It follows that supx∈Ω d(x, ∂Ω)|∇(fj ◦φ)(x)| < ε for jlarge enough. Thus limj→∞ ‖fj ◦ φ‖Ω,x0 = 0.

To prove the converse implication, assume that (b) fails. Choose ε > 0 and asequence (xj) in Ω such that

limj→∞

d(φ(xj), ∂Ω′) = 0 whiled(xj, ∂Ω)

d(φ(xj), ∂Ω′)|φ′(xj)| > ε

for all j ∈ N. By passing to a subsequence, we are free to assume that

wj := φ(xj) → w0 ∈ ∂Ω′

as j → ∞. Pick aj ∈ ∂Ω′ such that |aj − wj| = d(wj, ∂Ω′) and denote a0 = w0.Then clearly limj→∞ aj = w0. Suppose that there is a subsequence (wj) satisfying

(3.6) |w0 − wj| ≥ 2|aj − wj|for all j ∈ N. Define fj(x) = ln |aj − x| for j = 0, 1, 2, . . .. Then the sequence(fj) consists of harmonic Bloch-Sobolev functions in Ω′ and it converges to f0

locally uniformly on Ω′. On the other hand

‖Cφ(fj − f0)‖Ω,x0 ≥ d(xj, ∂Ω)|∇(fj ◦ φ)(xj)−∇(f0 ◦ φ)(xj)|= d(xj, ∂Ω)|φ′(xj)|

∣∣∣∣ 1

|aj − φ(xj)| −1

|w0 − φ(xj)|∣∣∣∣

= d(xj, ∂Ω)|φ′(xj)| ||w0 − wj| − |aj − wj|||aj − wj||w0 − wj|

≥ d(xj, ∂Ω)

2d(φ(xj), ∂Ω′)|φ′(xj)| > ε

2,

which implies that Cφ(fj − f0) does not converge to zero in the Bloch-Sobolevnorm. This contradicts the compactness of Cφ.

Assume then that (3.6) does not hold. Hence there is j0 ∈ N such that

|aj − wj| ≤ |w0 − wj| < 2|aj − wj|for all j ≥ j0. By hypothesis, w0 is not an isolated point of the boundary∂Ω′. Accordingly we may pick a sequence (bj) of points in ∂Ω′ \ {w0} suchthat limj→∞ bj = w0. We may also renumerate the points wj so that we have|bj − wj| ≥ 2|w0 − wj| for all j ∈ N. Now we can argue as above. Lettinggj(x) = ln |bj − x| for j = 0, 1, 2, . . . we have the estimate

‖Cφgj − Cφg0‖Ω,x0 ≥ d(xj, ∂Ω)|φ′(xj)|∣∣∣∣ 1

|bj − φ(xj)| −1

|w0 − φ(xj)|∣∣∣∣

= d(xj, ∂Ω)|φ′(xj)| ||w0 − wj| − |bj − wj|||bj − wj||w0 − wj|

≥ d(xj, ∂Ω)

4d(φ(xj), ∂Ω′)|φ′(xj)| > ε

4,


which contradicts the assumption.

Remark 3.7. In the first part of the proof of Theorem 3.6 we essentially usedthe fact that |∇fj| → 0 locally uniformly if fj → 0 locally uniformly. Our finalcomments emphasize the importance of this property.

(a) It is easy to find a counterexample which shows that Theorem 3.6 is nottrue for the composition operator Cφ : BS(Ω′) → BS(Ω). Let fj : D → [0, 1] bedefined by

fj(x) =

⎧⎪⎨⎪⎩

1

j− |x|, if |x| ≤ 1

j

0, if1

j< |x| < 1.

Then fj → 0 uniformly in D and the Bloch-Sobolev norms of fj are less orequal to 2. Choose φ(x) = x/2. Then φ trivially satisfies the condition (b) ofTheorem 3.6. However, by computing |∇(fj ◦ φ)| in the neighborhood of originone easily sees that ‖fj ◦φ‖D ≥ 1/4 for each j. Hence the condition of Lemma 3.5does not hold.

(b) Although Theorem 3.6 does not hold for the operator Cφ : BS(Ω′) → BS(Ω)in general, we can make some weaker conclusions from its proof. Indeed, the firstpart of the proof shows that under the condition (b) of Theorem 3.6 (Cφfj) → 0in the norm ‖ · ‖Ω,x0 whenever (fj) is a norm bounded sequence in BS(Ω′) suchthat fj → 0 locally uniformly and |∇fj| → 0 locally uniformly. We point outhere that for all p-harmonic functions fj (here p > 1) the locally uniform conver-gence fj → 0 implies the locally uniform convergence |∇fj| → 0. We leave theverification of this assertion as an exercise for interested readers.

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Marko Kotilainen E-mail: [email protected]: Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101Joensuu, Finland.

Visa Latvala E-mail: [email protected]: Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101Joensuu, Finland.

Jie Xiao E-mail: [email protected]: Department of Mathematics and Statistics, Memorial University of Newfoundland,St. John’s, NL, A1C5S7, Canada.

bloch-sobolev spaces and analytic composition operators

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