Abstracts ofInternational Workshop on Function Theory

CONTENTS

Boundary estimates ofp-harmonic functions in a metric measure spaceHiroaki Aikawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Equivalent norms on weighted Lipschitz spaces on the unit ballHong Rae Cho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Certain operators on Bergman spaces on strongly pseudoconvexdomains that improve integrability

Zeljko Cuckovic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5U-invariant Hilbert modules andK-homology

Kunyu Guo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6Minimum moduli of weighted composition operators on algebrasof analytic functions

Takuya Hosokawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7TBA

Junyun Hu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Rank one commutators on invariant subspaces of the Hardy spaceon the bidisk

Keiji Izuchi∗ and Kou Hei Izuchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Equivalence between canonical domains for doubly connectedplanar domains

Moonja Jeong∗, Jong-Won Oh and Masahiko Taniguchi . . . . . . . 10Another look at average formulas of Nevanlinna counting functionsof holomorphic self-maps of the unit disk

Hong Oh Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Norm estimates of the pre-Schwarzian derivatives in the univalentfunction theory

Yong Chan Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Zero Products of Toeplitz operators with harmonic symbols

Boorim Choe and Hyungwoon Koo∗ . . . . . . . . . . . . . . . . . . . . . . . . . 13On a generalized Bergman projection

Ern Gun Kwon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Generalized interpolation in Hardy spaces

Daniel H. Luecking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Existence of tangential limits forα-harmonic functions on half spaces

Yoshihiro Mizuta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

2

Schatten-Herz type Toeplitz operators on the harmonic Bergman spaceKyunguk Na . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Some integration operators on analytic functions in the unit discTakahiko Nakazi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Compact operator in the Toeplitz algebraKyesook Nam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Products of composition and differentiation between Hardy spacesShuichi Ohno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Characterization of the weighted Bergman spaces in terms ofderivatives

Jong-Do Park . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Brennan’s conjecture for weighted composition operators

Wayne Smith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Dual Toeplitz operators

Karel Stroethoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Invariant differential operators and their applications

Toshiyuki Sugawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Quasiconformal maps and the ideal boundary of a Riemann surface

Masahiko Taniguchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25The essential norm of a weighted composition operator on Bergmantype space in several variables

Seiichiro Ueki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26On Mobius invariant QK spaces

Hasi Wulan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27The integral operator with the closed range

Rikio Yoneda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28Quantum Douglas Algebras–Toeplitz algebras

Dechao Zheng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3

Boundary estimates ofp-harmonic functionsin a metric measure space

Hiroaki Aikawa(Shimane University)

Let (X, d, µ) be a metric measure space equipped with a doubling mea-sure supporting a(1, p)-Poincare inequality for1 < p < ∞. The notionof p-harmonicity is defined onX and a number of properties of classicalharmonic functions have been extended top-harmonic functions inX.

In this talk, we present two of such extensions. One is a Carleson typeestimate forp-harmonic functions on a uniform domain or a John domainin X; the other is a characterization of domains inX for whichp-harmonicextensions of Holder continuous boundary data are globally Holder contin-uous. These results are new even ifX is the Euclidean space. This talk isbased on the joint works with N. Shanmugalingam [1] and [2].

REFERENCES

[1] H. Aikawa and N. Shanmugalingam,Carleson-type estimates forp-harmonic func-tions and the conformal Martin boundary of John domains in metric measure spaces,Michigan Math. J.,53, 1 (2005), 165–188.

[2] H. Aikawa and N. Shanmugalingam,Holder estimates ofp-harmonic extension oper-ators, J. Differential Equations, to appear.

4

Equivalent norms on weighted Lipschitz spaceson the unit ball

Hong Rae Cho(Pusan National University)

A continuous functionω : (0,∞) → (0,∞) with lim supt→0+ ω(t) = 0will be called a majorant ifω(t) is increasing andω(t)/t is nonincreasingfor t > 0. If, in addition, there is a constantC(ω) > 0 such that

∫ δ

0

ω(t)

tdt + δ

∫ ∞

δ

ω(t)

t2dt ≤ C(ω) · ω(δ).

whenever0 < δ < 1, then we sayω is a regular majorant.For E ⊂ RN , the Lipschtz-type spaceΛω(E) = f : ‖f‖Λω(E) < ∞

consists of the functionsf : E → RN satisfying

‖f‖Λω(E) = supE|f |+ sup|f(z1)− f(z2)|

ω(|z1 − z2|) : z1, z2 ∈ E, z1 6= z2 < ∞.

Let B be the unit ball inCn. In this chapter we introduce several equiv-alent norms onΛω(B), each of them depending only on the modulus of thefunction in question.

5

Certain operators on Bergman spaces on strongly pseudoconvexdomains that improve integrability

Zeljko Cuckovic(University of Toledo)

We study two types of operators that improve integrability. First, wecharacterize bounded and compact weighted composition operators actingon Bergman spaces on strongly pseudoconvex domains inCn. We also giveestimates on their essential norms (joint work with Ruhan Zhao). Thenwe study the mapping properties of Toeplitz operators onLp on stronglypseudoconvex domains whose symbol is a positive power of the distance tothe boundary (joint work with Jeff McNeal).

6

U-invariant Hilbert modules and K-homology

Kunyu Guo(Fudan University)

This talk mainly concerns homogeneous submodules of essentially nor-mal U-invariant Hilbert modules. When a homogeneous submodule is es-sentially normal, its spectrum, essential spectrum are completely describedby zero variety of the submodule. In dimensionsd = 2, 3, it is shownthat theC∗-extension determined by the corresponding quotient module isnot trivial if a homogeneous submodule is essentially normal. In dimensiond = 2, and in the case of finite multiplicity, it is proved that each homo-geneous submodule isp-essentially normal forp > 2. The talk will alsogives an explicit expression forK-homology invariant defined in the caseof dimensiond = 2.

7

Minimum moduli of weighted composition operators onalgebras of analytic functions

Takuya Hosokawa(Nippon Institute of Technology)

We study the minimum moduli of weighted composition operators on thedisk algebra and the space of bounded analytic functions.

8

TBA

Junyun Hu(Jiaxing University)

TBA

9

Rank one commutators on invariant subspaces ofthe Hardy space on the bidisk

Keiji Izuchi ∗ and Kou Hei Izuchi(Niigata University)

A closed subspaceM of L2(Γ2) is called invariant ifLzM ⊂ M andLwM ⊂ M . For an invariant subspaceM of L2(Γ2), we have two naturaloperatorsRz = PMLz andRw = PMLw onM . Write [Rw, R∗

z] = RwR∗z −

R∗zRw on M . Mandrekar showed that forM of H2(Γ2), [Rw, R∗

z] = 0if and only if M = ϕH2(Γ2) for an inner functionϕ. So, this conditioncharacterizes Beurling type invariant subspaces inH2(Γ2). Nakazi posedthe following conjecture: if[Rw, R∗

z] = [Rw, R∗z]∗, then[Rw, R∗

z] = 0. Ohnoand the first author showed the following:

Let M be an invariant subspace ofL2(Γ2). Then the following conditionsare equivalent.

(i) [Rw, R∗z] = [Rw, R∗

z]∗ and [Rw, R∗

z] 6= 0.

(ii) M = ϕ(H2(Γ2)⊕

( ∞∑j=0

⊕C · zj w

1− rzw

)),

whereϕ is a unimodular function onΓ2 andr is a real number with0 <|r| < 1.

After that, Nakazi asked: is there an invariant subspaceM in H2(Γ2)having the form in (ii) ? We answer his question affirmatively. It is notso difficult to see thatrank [Rw, R∗

z] = 1 for an invariant subspaceMof the form in (ii). Until now, there are not so many studies on invariantsubspaces ofH2(Γ2) with rank [Rw, R∗

z] = 1. We are interested in suchinvariant subspaces, and discuss on this subject. Yang essentially showedthatrank [Rw, R∗

z] = 1 for invariant subspaces of the form:

ϕ(q1(z)H2(Γ2) + q2(w)H2(Γ2)

),

whereϕ is inner, andq1(z), q2(w) are one variable inner.Let

(1) M1 = ϕ(H2(Γ2)⊕

( ∞∑j=0

⊕ zj span wi

1− azwn ; 1 ≤ i ≤ n))

.

Theorem. Let M1 be an invariant subspace ofH2(Γ2) given in (1). Thenrank [Rw, R∗

z] = 1 for M1.

10

Equivalence between canonical domains for doublyconnected planar domains

Moonja Jeong∗ (University of Suwon)Jong-Won Oh (Yonsei University)

Masahiko Taniguchi (Kyoto University)

S. Bell in [1] made a conjecture on a new canonical domain of the non-degeneraten-connected planar domains and his conjecture was solved in[2]. The coefficient body of a new canonical domain forn = 2 was studiedin [3].

Now letA(r) = z ∈ C : |z + 1/z| < r , r > 2.

The annulus and the above domain are canonical domains forn = 2 and weshow the equivalence of them in an explicit formula. It is done by using theproperty of the Ahlfors map in [4] and [5]. Also we express it in anotherway by using the Teichmuller domain and the theta constants, and derivesome identity.

REFERENCES

[1] S. Bell, Finitely generated function fields and complexity in potential theory in theplane, Duke Math. J.98 (1999), 187–207.

[2] M. Jeong and M. Taniguchi,Bell representation of finitely connected planar domains,Proc. AMS.131(2003), 2325–2328.

[3] M. Jeong and M. Taniguchi,Algebraic kernel functions and representation of planardomains, J. Korean Math. Soc.40 (2003), 447–460.

[4] B.A.Mair and S. McCullough,Invariance of Extreme harmonic functions on an an-nulus; Applications to theta functions, Houston J. Math.20, (1994), 453-473.

[5] T. Tegtmeyer and A. Thomas,The Ahlfors map and Szego kernel for an annulus,Rocky Mountain J. Math.29, (1999), 709–723.

11

Another look at average formulas of Nevanlinna counting functions ofholomorphic self-maps of the unit disk

Hong Oh Kim(KAIST)

For a holomorphic self-mapϕ of the unit diskD on the complex plane,the Nevanlinna counting functionNϕ is defined by

Nϕ(w) =

∑ϕ(z)=w

log1

|z| , if w ∈ ϕ(U)

0 , if w /∈ ϕ(U)

It plays a very important role in the holomorphic change of variables byw = ϕ(z) in the integral representation and in the study of the compositionoperatorCϕ(f) = fϕ. The averages formulasNϕ(w) around a circle and adisk are given and exploited to the explicit representation of the Nevanlinnacounting functions of Rudin’s orthogonal functions. We also add anotherapplication of the average formulas to characterize a special class of innerfunctions.

12

Norm estimates for the Alexander transforms ofconvex functions of order alpha

Yong Chan Kim(Yeungnam University)

The hyperbolic sup norm of the pre-Schwarzian derivative of a locallyunivalent function on the unit disk measures the deviation of the functionfrom similarities. We present sharp norm estimates of the pre-Schwarzianderivatives for subclasses of univalent functions. We also consider theAlexander transforms in connection with pre-Schwarzian derivatives.

13

Zero products of Toeplitz operators with harmonic symbols

Boorim Choe and Hyungwoon Koo∗

(Korea University)

On the Bergman space of the unit ball inCn, we solve the zero-productproblem for two Toeplitz operators with harmonic symbols that have localcontinuous extension property up to the boundary. In the case where sym-bols have additional Lipschitz continuity up to the boundary, we solve thezero-product problem for multiple products with the number of factors de-pending on the dimensionn of the underlying space; the number of factorsis n+3. We also prove a local version of this result but with loss of a factor.

14

On a generalized Bergman projection

Ern Gun Kwon(Andong National University)

For−1 < α < ∞ and for a holomorphic self mapϕ of the unit discD,we consider a generalized Bergman projectionPϕ, α defined by

Pϕ, αf(z) =

∫

D

f(w) dAα(w)

(1− wϕ(z))2+α , z ∈ D.

The boundedness of the projection fromL∞(D) into Hardy families areexpressed in terms of the growth rates ofϕ.

15

Generalized interpolation in Hardy spaces

Daniel H. Luecking(University of Arkansas)

The usual definition of interpolating sequences requires a particular defi-nition of a norm on the sequence space being interpolated. While this normis a natural one, I will discuss another one that seems to be just as natural.One characterization of interpolation sequences forHp is that they must beuniformly discrete in the hyperbolic metric and a certain discrete measureassociated to the sequence is a Carleson measure. With the new norm, thecharacterization is almost the same: the measure must still be a Carlesonmeasure, but the sequence need not be uniformly discrete. Equivalently, thesequence must be a finite union of the usual interpolating sequences.

If time permits, I will show how this relates to other characterizations ofsuch finite unions, and to corresponding ideas in Bergman spaces.

16

Existence of tangential limits forα-harmonic functionson half spaces

Yoshihiro Mizuta(Hiroshima University)

Riesz [3] defined the notion ofα-harmonic functions on a domainΩin the n-dimensional Euclidean spaceRn, as solutions of the fractionalLaplace operators. We know that the Riesz potential of orderα, 0 < α ≤ 2,

Uαµ(x) =

∫|x− y|α−ndµ(y)

for a nonnegative measureµ on Rn is α-superharmonic inRn and α-harmonic outside the support ofµ; for this, see Riesz [3] and Landkof [2].

In the half spaceH = x = (x′, xn) ∈ Rn−1 ×R : xn > 0, consider

Pαf(x) = cα

∫

Rn\H

(xn

|yn|)α/2

|x− y|−nf(y)dy

for a measurable functionf onRn, where0 < α < 2 andcα = Γ(n/2)π−n/2−1 sin(πα/2). Then it is seen thatPαf is α-harmonicin H. Recently, Bass and You [1] have shown the existence of nontengen-tial limits for Pαf with f ∈ Λp,∞

β (Rn), which is the space ofLp Holdercontinuous functions of orderβ.

Our aim in this talk is to prove the existence of tangential limits forPαf ,as an improvement of their result.

REFERENCES

[1] R. F. Bass and D. You,A Fatou theorem forα-harmonic functions, Bull. Sci. Math.127(2003), 635–648.

[2] N. S. Landkof,Foundations of modern potential theory, Springer-Verlag, 1972.[3] M. Riesz,Integrales de Riemann-Liouville et potentiels, Acta Szeged9 (1938), 1–42.

17

Schatten-Herz type Toeplitz operators onthe harmonic Bergman space

Kyunguk Na(Korea University)

Motivated by a recent work of Loaizaet al.[9] for the holomorphic caseon the disk, we introduce and study the notion of Schatten-Herz type Toeplitzoperators acting on the harmonic Bergman space of the ball. We obtaincharacterizations of positive Toeplitz operators of Schatten-Herz type interms of averaging functions and Berezin transforms of symbol functions.Our characterization in terms of Berezin transforms settles a question posedby Loaizaet al.

This is a joint work with Boorim Choe and Hyungwoon Koo.

REFERENCES

[1] S. Axler, P. Bourdon and W. Ramey,Harmonic function theory, 2nd ed., Springer-Verlag, New York, 2001.

[2] B. Choe, Y. Lee and K. Na,Toeplitz operators on harmonic Bergman spaces, NagoyaMath. J. 174 (2004), 165–186.

[3] B. Choe, Y. Lee and K. Na,Positive Toeplitz operators from a harmonic Bergmanspace into another, Tohoku Math. J. 56(2004), 255-270.

[4] B. Choe, H. Koo and H. Yi,Positive Toeplitz operators between the harmonicBergman spaces, Potential Analysis 17(2002), 307-335.

[5] R. R. Coifman and R. Rochberg,Representation theorems for holomorphic and har-monic functions, Asterisque 77(1980), 11-65.

[6] H. Handenmalm, B. Koremblum and K. Zhu,Theory of Bergman spaces, SpringerVerlag, New York, 2000.

[7] E. Herenandez and D. Yang,Interpolation of Herz spaces and applications, Math.Nachr. 205(1999), 69–87.

[8] H. Kang and H, Koo,Estimates of the harmonic Bergman kernel on smooth domains,J. Funct. Anal. 185 (2001), 220–239.

[9] M. Loaiza, M. Lopez-Garcia and S. Perez-Esteva,Herz classes and Toeplitz operatorsin the disk, Publications Preliminares del Instituto de Matematicas, UNAM, 2003.

[10] J. Miao,Reproducing kenkels for harmonic Bergman spaces of the unit ball, Monatsh.Math. 125(1998), 25–35.

[11] K.Stroethoff,Harmonic Bergman spaces, Holomorphic spaces, MRSI Publications33(1998), 51–63.

[12] K. Zhu,Positive Toeplitz operators on weighted Bergman spaces of bounded symmet-ric domains, J. Operator Theory 20(1988), 329–357.

[13] K. Zhu, Operator theory in function spaces, Marcel Dekker, New York and Basel,1989.

18

Compact operator in the Toeplitz algebra

Kyesook Nam(Hanshin University)

m-Berezin transforms are introduced for bounded operators on the Bergmanspace of the polydisk. We show several properties ofm-Berezin transformand use them to show that a radial operator in the Toeplitz algebra is com-pact iff its Berezin transform vanishes on the boundary of the polydisk.

This is a joint work with Dechao Zheng.

19

Some integration operators on analytic functionsin the unit disc

Takahiko Nakazi(Hokkaido University)

Let ϕ be an analytic function on the open unit discD in the complexplane. Put

(Mϕf)(z) = ϕ(z)f(z),

(Iϕf)(z) =

∫ z

0

f ′(ζ)ϕ(ζ)dζ

and

(Jϕf)(z) =

∫ z

0

f(ζ)ϕ′(ζ)dζ

for a holomorphic functionf on D. In this lecture we study these opera-tors on Hilbert spaces, for example, weighted Hardy spaces and weightedDirichlet spaces. In particular, we study whenMϕ, Iϕ or Jϕ is a Fredholmoperator on a weighted Dirichlet space.

20

Products of composition and differentiationbetween Hardy spaces

Shuichi Ohno(Nippon Institute of Technology)

We will discuss boundedness and compactness of the products of com-position and differentiation between Hardy spaces.

REFERENCES

[1] C. C. Cowen and B. D. MacCluer,Composition Operators on Spaces of AnalyticFunctions, CRC Press, Boca Raton, 1995.

[2] P.L. Duren,Theory ofHp Spaces, Academic Press, New York, 1970.[3] M. Essen, D.F. Shea and C.S. Stanton,A value-distribution criterion for the class

L log L, and some related questions, Ann. Inst. Fourier (Grenoble)35(1985), 127–150.

[4] J. H. Shapiro,Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993.

[5] W. Smith,Composition operators between Bergman and Hardy spaces, Trans. Amer.Math. Soc.348(1996), 2331–2348.

[6] C.S. Stanton,Counting functions and majorization for Jensen measures, Pasific J.Math.125(1986), 459–468.

[7] K. Zhu,Operator Theory in Function Spaces, Marcel Dekker, New York, 1990.

21

Characterization of the weighted Bergman spacesin terms of derivatives

Jong-Do Park(Sogang University)

Let Bn be the unit ball inCn and dνs is defined bydνs(z) = (1 −|z|2)sdν(z), wheredν is the normalized Lebesgue measure onBn. We studysome properties of the operatorAl,s : L2

a(Bn, dνs) → R defined by

Al,s(f) =∑

|J |=l

‖(1− |z|2)lDJf(z)‖2L2

a(Bn,dνs),

whereJ is a multi-index with lengthl ≥ 0. The following is the basicformula. For anym ≥ 0, we have

∑

|J |=l

∫

Bn

(1− |z|2)m|DJzk|2dν(z) =k!n!Γ(m + 1)

Γ(|k|+ m + n− l + 1)P (|k|, l),

whereP (q, r) := r!(

qr

). Using this formula, we give an elementary proof of

the fact that for anys > −1,A0,s(f) is finite if and only ifAN,s(f) is finitefor any positive integerN .

We also construct the inner product formula for the Bergman space inthe unit ballBn: the inner product of two holomorphic functions can berepresented as the sum of different weighted inner products of derivatives.Using this formula, we study bounded Toeplitz products on the Bergmanspace in the unit ballBn in Cn.

22

Brennan’s conjecture for weighted composition operators

Wayne Smith(University of Hawaii)

Brennan’s conjecture concerns integrability of the derivative of a confor-mal mapτ of the unit diskD. The conjecture is that, for all suchτ ,∫

D(1/|τ ′|)pdA < ∞

holds for−2/3 < p < 2. This is known for−2/3 < p ≤ 1.421.We show Brennan’s conjecture is equivalent to a statement about weighted

composition operators. Letτ be as above and letϕ be an analytic self-mapof D. Define, forf analytic onD,

(Aϕ,pf)(z) =

(τ ′(ϕ(z))

τ ′(z)

)p

f(ϕ(z)).

There are always choices ofϕ that makeAϕ,p boundedon the BergmanspaceL2

a(D). We are interested in the set ofp for which there is a choiceof ϕ (depending onτ ) that makesAϕ,p compacton L2

a(D). We show thishappens if and only if(1/τ ′)p ∈ L2

a(D). Thus Brennan’s conjecture isequivalent to such a choice ofϕ existing for the range−1/3 < p < 1, andthis is known for−1/3 < p ≤ .7105.

23

Dual Toeplitz operators

Karel Stroethoff(University of Montana)

A dual Toeplitz operator is defined to be multiplication followed by pro-jection onto the orthogonal complement of a Bergman space in the spaceof square integrable functions on the domain. We will discuss properties ofdual Toeplitz operators.

24

Invariant differential operators and their applications

Toshiyuki Sugawa(Hiroshima University)

Various invariant differential operators have been studied by many au-thors. Particularly important are differential operators associated with spher-ical, Euclidean and hyperbolic metrics. According to the casesε = +1, 0,−1,those metrics are described byλε(z)|dz| = |dz|/(1 + ε|z|2) on Cε, whereC+1 = C (the Riemann sphere),C0 = C (the complex plane) andC−1 =D = |z| < 1 (the unit disk). For example, given a holomorphic functionf : D = C−1 → Cε, the invariant derivatives associated with(λ−1, λε) up tothe third order are given by

D1f(z) =(1− |z|2)f ′(z)

1 + ε|f(z)|2 ,

D2f(z) =(1− |z|2)2f ′′(z)

1 + ε|f(z)|2 − 2z(1− |z|2)f ′(z)

1 + ε|f(z)|2 − 2(1− |z|2)2f(z)f ′(z)2

(1 + ε|f(z)|2)2,

D3f(z) =(1− |z|2)3f ′′′(z)

1 + ε|f(z)|2 − 6(1− |z|2)3f(z)f ′(z)f ′′(z)

(1 + ε|f(z)|2)2

− 6z(1− |z|2)2f ′′(z)

1 + ε|f(z)|2 +6z2(1− |z|2)f ′(z)

1 + ε|f(z)|2

+12z(1− |z|2)2f(z)f ′(z)2

(1 + ε|f(z)|2)2+

6(1− |z|2)3f(z)2f ′(z)3

(1 + ε|f(z)|2)3.

In the present talk, we recall the definition of these operators of an arbi-trary order and then give some nontrivial relations between them. Finally,as applications, we give norm estimates of higher-order derivatives of holo-morphic maps of the unit disk intoCε. Most of this talk will be a part ofon-going joint research with Seong-A Kim.

25

Quasiconformal maps and the ideal boundary of a Riemann surface

Masahiko Taniguchi(Kyoto University)

It is well-known that quasiconformal maps of a Riemanm surface to an-other can be extended to a homeomorphism between the Royden compact-ifications of them. We consider the essential part of the Royden boundary,and discuss the Teichmuller space of it.

REFERENCES

[1] C. Constantinescu and A. Cornea,Ideale Rander Riemannscher Flachen,Springer-Verlag (1963).

[2] F. P. Gardiner and N. Lakic, Quasiconformal Teichmuller Theory, Mathematicalsurveys and Mono., AMS, (2000).

[3] Y. Imayoshi and M. Taniguchi,An Introduction to Teichmuller Spaces,Springer,Tokyo, 1992.

[4] M. Nakai, Existence of quasiconformal mappings between Riemann surfaces,Hokkaido Math. J.10 (1981), 525–530

[5] L. Sario and M Nakai,Classification Theory of Riemann Surfaces,Springer-Verlag(1970).

[6] M. Taniguchi,The Teichmuller space of the ideal boundary,preprint.

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The essential norm of a weighted composition operator on Bergmantype space in several variables

Seiichiro Ueki(Nippon Institute of Technology)

In this talk, we estimate the essential norm of a weighted compositionoperator on the Bergman type space defined in the complexn-dimensionalEuclidean spaceCn.

REFERENCES

[1] Zeljko Cuckovic and R. Zhao,Weighted composition operators on the Bergmanspace, J. London Math. Soc., 70 (2004), 499–511.

[2] S. Ueki,Weighted composition operators between weighted Bergman spaces in theunit ball ofCn, preprint.

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On Mobius invariant QK spaces

Hasi Wulan(Shantou University)

We give a general theory of QK spaces of functions analytic in the unitdisk, including the relationship between QK and some well-known func-tion spaces. This talk contains some results about QK spaces in terms ofCarleson type measures, boundary values, inner factors and the higher or-der derivatives. The compactness of composition operator from the Blochspace to QK is also given.

The integral operator with the closed range

Rikio Yoneda(Otaru University of Commerce)

Let g be an analytic function on the open unit diskD in the complexplaneC. We study the following integral operators

Jg(f)(z) :=

∫ z

0

f(ζ)g′(ζ)dζ,

Ig(f)(z) :=

∫ z

0

f ′(ζ)g(ζ)dζ

on weighted Bloch space and weighted Dirichlet spaces. Then we study theresult with respect to ” When do the integral operators have the closed range?”

REFERENCES

[1] R. Yoneda,The Integration operators on weighted Bloch spaces and weighted Dirich-let spaces with the closed range, preprint.

2000Mathematics Subject Classification.30D55.Key words and phrases.Function theory, Operator theory.

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29

Quantum Douglas Algebras–Toeplitz algebras

Dechao Zheng(Vanderbilt University)

Let B be a Douglas algebra and letB be the algebra on the disk generatedby the harmonic extensions of the functions inB. First we will show thatthe the theorem analogous to Chang-Marshall theorem holds for the diskalgebraB.

For eachα > −1, the “quantum Douglas algebra”Bα is the Toeplitzalgebra generated by Toeplitz operators (on the weighted BergmanspaceL2

a((1− |z|2)αdA(z))) with symbols inB.We will show that the quantum Douglas algebraBα has a canonical de-

compositionS = TBαS +R for someR in the commutator idealCBα ; andSis in CBα iff the Berezin transformBαS vanishes identically on the union ofthe maximal ideal space of the Douglas algebraB and the setM1 of trivialGleason parts. This extends the McDonald-Sundberg Theorem and answersa question of Davidson and Douglas. Some of results are my recent jointwork with S. Axler.