Blaschke condition and zero sets in weighted Dirichlet spaces

c 2011 by Institut Mittag-Leffler. All rights reserved

Blaschke condition and zero sets in weighted Dirichlet spaces

Dominique Guillot

Abstract. LetD(μ) be the Dirichlet space weighted by the Poisson integral of the positive measureμ. We give a characterization of the measuresμequal to a countable sum of atoms for which the Blaschke condition is a necessary and sufficient condition for a sequence to be a zero set forD(μ).

1. Introduction

The Dirichlet space D is the space of analytic functions on the unit disc D having a finiteDirichlet integral

D(f) :=1 π

D|f(z)|²dA(z).

It is a subspace ofH² that has been extensively studied, notably by Beurling and Carleson. Nevertheless, many natural questions aboutD are still open today. For example, there is still no complete characterization of the invariant subspaces of the shift operatorSf(z)=zf(z) onD. In this paper, we are interested in another impor- tant question: the characterization of the zero sets ofD. A sequence{z_n}^∞n=1⊂Dis azero set forDif there exists a functionf∈D,f≡0, such thatf(zn)=0. According to a theorem of Carleson [1], later improved by Shapiro and Shields [8], if

∞ n=1

log 1

1−rn

₋1

<∞

for some sequence of radii{rn}^∞n=1, then{rne^iθn}^∞n=1 is a zero set forD for any choice of arguments{θn}^∞n=1. On the other hand, by a result of Nagel, Rudin and

Work supported by scholarships from NSERC (Canada) and FQRNT (Quebec).

Shapiro [5], if the series is divergent, there exists a sequence of arguments such that {rne^iθn}^∞n=1 is not a zero set.

A complete characterization of the zero sets for D is still an open problem.

Contrary to the H² case, to find such a characterization, one must take into con- sideration the arguments of the pointsz_n, making the problem much more difficult.

For a review of the recent results, see [4] and the references cited therein.

In studying the invariant subspaces for the shift on D, Richter and Sund- berg [6], [7], have introduced weighted versions of D. For a positive Borel measure μonT, the spaceD(μ) is the space of analytic functions onDwith finiteμ-Dirichlet integral

Dμ(f) =1 π

D|f(z)|²P μ(z)dA(z), whereP μ(z) is the Poisson integral of μ:

P μ(z) =

T

1−|z|²

|e^it−z|²dμ(e^it).

The spaceD(μ) is a Hilbert space with the normf²μ=f2+Dμ(f). In particular, ifμ=0, we get the Hardy spaceH², and forμ=m, we obtain the Dirichlet spaceD.

The spaces D(μ) are interesting since they give intermediate examples of spaces between the well-known H² case and the difficult Dirichlet case. Moreover, the study of particular examples (like the case where μ is a Dirac measure) is often easier and gives good insights about what can happen in the most difficult cases.

In this paper, we study the zero sets of functions inD(μ) in the case where μis a countable sum of Dirac measuresμ=

n≥1cnδζ_n, where{ζn}^∞n=1⊂T. Recall that a sequence{z_n}^∞n=1is a zero set forH²if and only if the sequence satisfies the Blaschke condition

n≥1(1−|zn|)<∞. SinceD(μ)⊂H², the Blaschke condition is the minimum required for a sequence to be a zero set in D(μ). We give a char- acterization of the measures μ=

n≥1cnδζ_n for which this condition is necessary and sufficient. We also prove that, given a sequence of points{ζ_n}^∞n=1⊂T, we can always find a way to distribute a mass on{ζn}^∞n=1such that the Blaschke condition is necessary and sufficient to get a zero sequence inD(μ).

2. The Blaschke condition

Theorem 2.1. Let μ be a positive finite Borel measure on T. If there exists F∈D(μ)such that F=0 μ-a.e.,then

T

logV2μ(e^it)dt <∞,

whereV2μ is the Newtonian potential associated withμ:

V₂μ(e^it) =

T

dμ(ζ)

|e^it−ζ|².

Proof. SupposeF=0 μ-a.e. According to [6], Proposition 2.2,

D_μ(F) =

T

T

|F(e^it)|²

|e^it−ζ|² dt 2πdμ(ζ).

Thus, ifF∈D(μ), by applying Fubini’s theorem and Jensen’s inequality

∞>logDμ(F)≥

T

log

|F(e^it)|²

T

dμ(ζ)

|e^it−ζ|² dt

2π

=

T

2 log|F(e^it)| dt 2π+

T

log

T

dμ(ζ)

|e^it−ζ|² dt

2π. SinceF∈H², it follows that log|F|is integrable and so

T

log

T

dμ(ζ)

|e^it−ζ|²

dt <∞.

This completes the proof.

In the case whereμ is a countable sum of Dirac measures, the preceding con- dition is also sufficient.

Theorem 2.2. Letμ=_∞

n=1c_nδ_ζn with _∞

n=1c_n<∞,c_n≥0andζ_n∈T. Then the following are equivalent:

(1) There existsF∈D(μ)such that F=0 μ-a.e.;

(2) There existsF∈D(μ)such that zFμ=Fμ; (3) We have

T

logV2μ(e^it)dt <∞.

Proof. According to a result of Richter and Sundberg ([6], Corollary 2.3), for F∈D(μ),

zF²μ−F²μ=

T|F|²dμ.

This shows the equivalence between (1) and (2). We will prove that (1) is equivalent to (3). According to Theorem2.1, we have (1) implies (3). Now assume that (3) is true. DefineF by

F(z) = exp

−

T

e^it+z

e^it−zlogV2μ(e^it)dt 2π

.

According to (3),F is well defined. Moreover, remark that 1

|F(e^it)|=V2μ(e^it)≥¹₄μ(T) a.e.

As a consequence, |F(e^it)|≤4/μ(T) a.e. and so F∈H^∞. We claim that F(ζk)=0 for each k. To prove this claim, recall that the Poisson integral of a measureν has a radial limit equal to∞at e^itwhenever the symmetric derivative ofν ate^it,

Dν(e^it) = lim

ε→0⁺

1 2ε

t+ε t−ε

dμ(e^it)

exists and is equal to ∞. Notice that |F(z)|=exp(−P(logV2μ))(z). Therefore, to prove thatF(ζ_k)=0, it suffices to show that the symmetric derivative of logV₂μat ζ_k is∞. Letε>0 and let ζ_k=e^iα. Then

1 2ε

α+ε α−ε

logV2μ(e^it)dt≥ 1 2ε

α+ε α−ε

log ck

|e^it−ζk|²

dt

≥C ε

ε 0

log c_k

t²

dt

=Clogck−2C ε

ε 0

logt dt

=Clogck−2C

ε (εlogε−ε)

=Clogck+2C−2Clogε,

Let us computeDμ(F). We have Dμ(F) =

T

|F(e^it)|²

|e^it−ζ|² dt

2πdμ(ζ) =

T

T

1

|e^it−ζ|²dμ(ζ)|F(e^it)|² dt 2π

=

T|F(e^it)| dt 2π≤

T

4 μ(T)

dt 2π<∞. HenceF∈D(μ) andF(ζk)=0 for everyk.

Lemma 2.3. Let {ζn}^∞n=1⊂T. Then there exists a sequence {zn}^∞n=1⊂Dsuch that _∞

n=1(1−|zn|)<∞and ∞ n=1

1−|z_n|²

|ζk−zn|²=∞ for everyk.

Proof. We can easily construct such a sequence. For example, we may let r_n,k:=1−1/n²k² and consider the points {r_n,kζ_k}^∞_n,k=1. Let {z_m}^∞m=1 be an enu- meration of the points of this set. For eachk, we have

∞ m=1

1−|z_m|²

|ζk−zm|²=∞,

since an infinite number of the points of the sequence{zm}^∞m=1are on the ray from the origin toζk and

1−|rζk|²

|ζk−rζk|²→∞ asr→1⁻. Moreover,

∞ m=1

(1−|z_m|) = ∞ n=1

∞ k=1

1 n²

1 k² <∞.

Therefore, the sequence{z_m}^∞m=1 satisfies the conditions of the lemma.

Theorem 2.4. Letμ=_∞

n=1c_nδ_ζn wherec_n≥0,_∞

n=1c_n<∞andζ_n∈T. The following are equivalent:

(1) For each Blaschke sequence {zn}^∞n=1,there exists f∈D(μ)and f≡0,such thatf(zn)=0for every n;

(2) We have

T

logV2μ(e^it)dt <∞.

Proof. According to Lemma2.3, there exists a Blaschke sequence such that ∞

n=1

1−|zn|²

|ζk−zn|²=∞

for everyk, and, according to the hypothesis of the theorem, there exists f∈D(μ) such that f(zn)=0 for every n. We can decompose f as f=BSF, where B is the Blaschke product associated with the sequence {z_n}^∞n=1, S is a singular inner function andF is an outer function. According to a formula of Carleson [3] gener- alized by Richter and Sundberg [6], Theorem 3.1, BF∈D(μ) and so, without loss of generality, we can assume thatS=1. Also,

D_μ(BF) =D_μ(F)+

∞ k=1

c_k|F(ζ_k)|^2∞

n=1

1−|z_n|²

|ζk−zn|²<∞.

Since the right-hand sum is infinite for everyk, we must haveF(ζk)=0 for everyk.

SoF=0 μ-a.e. It follows from Theorem2.2 that

T

logV2μ(e^it)dt <∞.

Conversely, suppose that the preceding integral is convergent and let{z_n}^∞n=1 be a Blaschke sequence. Then there exists F∈D(μ) such that F=0 μ-a.e. IfB is the Blaschke product associated with the sequence{zn}^∞n=1, thenDμ(BF)=Dμ(F). So f=BF∈D(μ) andf(zn)=0 for everyn.

3. Carleson’s condition is not necessary Definition3.1. A closed setE⊂T is aCarleson set if

T

log 1

d(e^it, E)

dt <∞.

If E is a Carleson set, then according to a classical result of Carleson ([2], Theorem 1), there exists a holomorphic functionF such that F=0 onE and F is continuously differentiable on T. As a consequence, if {ζ_n}^∞n=1 is Carleson, there existsF∈D(μ) such thatF(ζn)=0 for everyn. Theorem2.2gives another proof of this result, as follows.

Corollary 3.2. Let μ=_∞

c δ , where c ≥0, _∞

c <∞ and ζ ∈T.

Proof. It suffices to remark that

T

logV2μ(e^it)dt=

T

log

T

1

|e^it−ζ|²dμ(ζ)

dt≤

T

log μ(T)

d(e^it, E)²dt.

Therefore, if E={ζ_n}^∞n=1 is a Carleson set, then every Blaschke sequence is a zero sequence in D(μ) if μ=

n≥1cnδζ_n for a summable sequence {cn}^∞n=1. The following theorem shows that we can give a mass to almost any sequence{ζn}^∞n=1in such a way that the Blaschke condition becomes a necessary and sufficient condition to have a zero sequence. Therefore, Carleson’s condition is not necessary.

Theorem 3.3. Let E:={ζn}^∞n=1⊂T. There exists a sequence {cn}^∞n=1 such thatc_n≥0, _∞

n=1c_n<∞and c_n>0 for an infinite number of n and

T

logV2μ(e^it)dt <∞, whereμ=_∞

n=1cnδζ_n. As a consequence,for this measureμ,the Blaschke condition is necessary and sufficient to have a zero sequence inD(μ).

Proof. Set ζn:=e^iαn and suppose, without loss of generality, that ζn→ζ and that 0<α1<α2<α3<...<2π(if this is not the case, extract such a sequence and let cn=0 for the other points ofE). Denote bymjthe middle point of the arc (ζj, ζj+1).

Letm₀ be the middle point of the arc contained between ζ and ζ₁ containing no other point ofE. Denote byJj the arc (mj−1, mj),j=1,2, ...,and letJ:= ^∞_j=1Jj

(see figure below).

Moreover, denote by d_j:=d(J_j, E\{ζ_j}), the distance between the arc J_j and the other points ofE. Consider the sequencec_n:=d²_n/2ⁿ.

We will show that this sequence satisfies the hypothesis of the theorem. First, we have_∞

n=1dn<∞sincedn=min(d(mn, ζn), d(ζn, mn−1))≤|Jn|and_∞

n=1|Jn|≤

2π since the arcs Jj are disjoint. Therefore, _∞

n=1cn<∞. Next, to estimate the integral, remark that

T\J

logV2μ(e^it)dt≤C+

T∩J

logV2μ(e^it)dt,

for a constant C >0, since the integral over B(ζ, ε)∩J is bigger than the integral overB(ζ, ε)∩J^c. Therefore, it is sufficient to bound the integral overJ:

J

logV2μ(e^it)dt=

J

log _∞

k=1

ck

|e^it−ζ_k|²

dt=

J

log _∞

k=1

d²_k/2^k

|e^it−ζ_k|²

dt

≤^∞

n=1

Jn

log _∞

k=1 k=n

1 2^k

+ d²_n/2ⁿ

|e^it−ζ_n|²

dt,

because|e^it−ζk|≥dk ife^it∈Jn and k=n. Hence,

J

logV2μ(e^it)dt≤^∞

n=1

Jn

log

1+ d²_n/2ⁿ

|e^it−ζn|²

dt≤^∞

n=1

Jn

log

1+ d²_n

|e^it−ζn|²

dt.

SetJ_n:=(α_n−a_n, α_n+b_n) witha_n, b_n∈[0,2π). Then

J

logV₂μ(e^it)dt^∞

n=1

αn+bn

αn−an

log

1+ d²_n (t−αn)²

dt

= ∞ n=1

bn

−a_n

log

1+d²_n t²

dt

= ∞ n=1

tlog

1+d²_n

t²

+2dnarctan t

d_n b_n

−a_n

≤ ∞ n=1

8πdn+ ∞ n=1

log

1+d²_n b²_n

bn 1+d²_n

a²_n an

.

Note that d_n=min(a_n, b_n), asm_n is the middle point of the arc (ζ_n, ζ_n+1). Thus,

The first sum is convergent since _∞

n=1dn<∞. For the second sum, notice that an+bn=|Jn|. As theJn are disjoint, it follows that_∞

n=1|Jn|≤2π. This completes the proof.

In particular, let E:={ζ_n}^∞n=1 be a monotone sequence such that ζ_n→ζ and such thatE is not a Carleson set. According to the theorem, we can choose masses cn>0 such that the Blaschke sequence is necessary and sufficient to have a zero sequence inD(μ) withμ=_∞

n=1cnδζ_n. So we have the following corollary.

Corollary 3.4. There exists a sequenceE:={ζ_n}^∞n=1 and a sequence {c_n}^∞n=1

such that E is not a Carleson set, c_n>0 for every n, _∞

n=1c_n<∞, and if μ=

_∞

n=1cnδζ_n,then the Blaschke condition is a necessary and sufficient condition for having a zero sequence inD(μ).

Acknowledgements. This work is part of the author’s doctoral dissertation.

The author would like to thank Professor Thomas Ransford for having read the paper and for his invaluable remarks and suggestions.

References

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2. Carleson, L., Sets of uniqueness for functions regular in the unit circle,Acta Math.

87(1952), 325–345.

3. Carleson, L., A representation formula for the Dirichlet integral,Math.Z.73(1960), 190–196.

4. Mashreghi, J.andShabankhah, M., Zero sets and uniqueness sets with one cluster point for the Dirichlet space,J.Math.Anal.Appl.357(2009), 498–503.

5. Nagel, A.,Rudin, W.andShapiro, J., Tangential boundary behavior of functions in Dirichlet-type spaces,Ann.of Math.116(1982), 331–360.

6. Richter, S.and Sundberg, C., A formula for the local Dirichlet integral,Michigan Math.J.38(1991), 355–379.

7. Richter, S.and Sundberg, C., Multipliers and invariant subspaces in the Dirichlet space,J.Operator Theory28(1992), 167–186.

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Dominique Guillot

D´epartement de math´ematiques et de statistique Universit´e Laval

Qu´ebec, QC G1V 0A6 Canada

dominique.guillot.1@ulaval.ca

Received April 26, 2010

published online August 27, 2011