Image Areas of Functions in the Dirichlet Type Spaces and their Möbius Invariant Subspaces

U NI VERSITATIS MARIAE C U RI E - S K L 0 D 0 W S K A LUBLIN - POLONIA

VOL. L, 23 SECTIO A 1996

KARL J. WIRTHS (Braunschweig) and JIE XIAO* (Beijing)

Image Areas of Functions in the Dirichlet Type Spaces and their Mobius

Invariant Subspaces

Abstract. For p € (0, +oo) let Dp be the Dirichlet type space of func­

tions f analytic in the unit disk i/ = {z:|z|<l} for which

ll/IlL, == Jl/'O)|2(l - \z\2)p dxdy < oo.

Furthermore let Qp be the Mobius invariant subspace of Dp consisting of those f € Dp with sup ||/o <pw ||n < oo, where ipw(z) = (w — z)/(l — wz).

weu

In particular, let QPto = {/ € Qp : lim|w|_1 ||/o v>w||Dp = 0}. In this paper we investigate the image areas of functions in Dp , QP and Qp,o •

1. Introduction. Let U = {z : |2| < 1} and dU = {z : |z| = 1} denote the unit disk and the unit circle, respectively, and dm(z) the Lebesgue measure on U . For z, w 6 U , let

1 - wz g{z, w) = log ---

* This work was done during this author’s visit at the Technical University Braun­

schweig Germany. It was supported partially by a grant from the National Science Foundation of China.

1991 Mathematics Subject Classification. Primary 30D50, secondary 30D45.

be the Green function of U with pole at w. Throughout this paper we shall use A as a symbol for the class of functions analytic on U. We are interested in the Dirichlet-type spaces Dp, p G (0, oo) and their subspaces Qp invariant under analytic automorphisms of U .

Definition. Let pG[0,oo) and <pw(z) = (w - z)/(l - wz) . a) For f G A we say that f € Dp if

II/IIÎ., :=

- If)’*»« <oo.

b) For f G A we say that f G Qp if

II/IIqp = SUP H/oV’wIlLp < 00 • wet/

c) We say that f G Qpfi if lim|w|_i |\f o ç>w11Dp = 0 .

Obviously, the spaces , Qp and Qpfi increase with increasing p.

For special values of p these spaces may be identified as follows: Do is the Dirichlet space D, D\ is the Hardy space H2, D2 is the Bergman space B2,Qo is D , Qo,o is the set of constant functions, Qi is the space of analytic functions with bounded mean oscillation on dU, i. e. BMOA, Qip is the space of analytic functions of vanishing mean oscillation on dU , i. e. VMOA, Q2 is the Bloch space B and Q2 0 is the little Bloch space

Bo.

Furthermore

QP =(/:/€ A and sup ff |/'(z)|2 gp(z,'w) dm(z) < 00

1 weuJ Ju

and

Qpfl = : f 6 A and ^irrij J[ \f'(z)\2gp(z, w) dm(z) = 0 j .

As references concerning these identifications, cf. [1], [2], [3], [4], [13], [14]

and [15].

In this paper, we mainly study the characterization of functions / be­

longing to Dp , Qp and QPto resp. by the area of the image domains f(U).

Here, we would like to thank Prof. Ch. Pommerenke for his helpful suggestions.

2. Results. For f € A, w € U and r € (0,1] let t/r(w) = {z £ U :

|<pw(z)| < r}, in particular Ur = Ur(0) • If we denote by A(/(l7r(w)))= /7 |/'(z)|2 dm(z)

J JUr(w)

the area of /(f/r(w)) on the Riemann surface f(U), we get immediately that f £ D = Do if and only if sup0<r<1 A(/(f/r)) < oo. A similar characterization of the functions in Dp is delivered by

Theorem 1. Let p € (0,oo) and f £ A. Then f 6 Dp if and only if (1) i A(/(f/r))(l - r)p-1dr < oo .

Jo

Proof. Using the representation /(z) = anzn we get

f1 °° n2 f1

J A(/(i/r))(l-r)p-1dr = 2tt —|an|2^ r2n(l - r)p_1dr

and

[I |/'(z)|2(l-|z|2)pdm(z) = 27r£n2|an|2 f r2"’^ - r2)pdr,

JJu n=1 do

fl ~2n i yl

/ _(i _ ry-'dr = - / rin-1(l-r)pdr.

Jo P Jo

The inequalities

r2n~\l - r)pdr < [ r2n~l(1 - r2y>dr < 2P I r2n~\l - r)pdr

Jo Jo Jo

immediately show that the desired equivalence is valid.

Remarks.

1) The case p = 1 of Theorem 1 is the case A = 2 of Theorem 1 in [10].

2) Furthermore, applying Corollary 1 in [11], which says that for

OO

5(x) = 52 bnXn ’ bn - 0 ’ x e X) ’ n=0

the inequalities

i (1 — x)p-1<7(a:)da; < oo and n~^p+1' f ] < oo

n=l \fc=l /

are equivalent, we just find that f G Dp with /(z) = anzn if and only if

£n_(P+1) < 00 •

n=l \fc=l /

Note that the special cases p = 1 and p > 1 were given in [10, Corollary 2]

and [6, Proposition 2.21], respectively.

Next, we denote by a(/(i/r(w))) the area of the projection of /({7r(w)) from the Riemann surface into the complex plane, i.e.

<z(/(i/r(w))) = l[ dm(z).

JJf(ur(wy)

It is trivial that a(/({7r(w))) < A(/(Z7r(w))) and hence f € Dp implies (2) f a(/(t/r))(l- ry-1 dr < oo.

Jo

We will see below that the converse is not valid for any p G (0, oo).

Example 2. For p G (0,1) this is a consequence of the fact that there exist functions f G A , continuous on the closure of U such that

(3) |/'(z)|dm(z) = oo,

as proved by Rudin in [12]. The Schwarz inequality and (3) imply oo = JJu l-f'MIO - kl2>p/2(l - |i|2)-’/2<im(2)

< l/WIP - 12Pz) • y^(l - |i|2)~p<hn(z)

which proves that f Dp. On the other hand, the continuity of f on the closure of U implies the boundedness of a(/(i/)) and thus (2).

Example 3. Let Z — {m + in : (m,n) G Z2} and f G A be such that f(U) = C \ Z . Since f(U') is a Bloch domain i. e. C \ Z does not contain arbitrarily large euclidean disks, / is a Bloch function, that is (c. f. [8])

sup(l - |x|2)|/'(z)| < oo.

This implies that there exists a constant C such that

|/(z)|<Clogrl-p ztU.

From this we deduce

y a(/(t/r))i/r < C27r y ^log dr < oo .

On the other hand, we see that Z has zero capacity, so f H2 = D\ . So (2) in the case p = 1 is valid for this function f not in Di.

Example 4. By modifying a bit the proof of Lemma 2 in [11] one may show that for any 7 € (0,00) there exists a constant A\ such that

00

2n7r2“ < A’i | log r|-7 , r 6 (0,1).

n=0

Hence for the functions /(z) = 2n7z2" we get

00

sup |/(z)| < y'2nV < A’i| log r|~7 .

I»l<»- „=0

If we choose 7 < p/2 we derive

i (1 — r')p~1a(f(Ur'))dr < K2t i (1 — r)p_1| logr|-27dr < 00 .

Jo Jo

Considering the criterion for f to be a member of Dp given in Remark 2 above, we see that there is a constant K2 such that in our case

00 / n \ 00

52n_(p+1) 52 k\ ak \ 2 ) -A’ 2 52

n=l \fc=l / n=0

This sum is divergent for 7 > (p - l)/2 . So choosing 7 G ((p — l)/2, p/2), f 0 Dp, but (2) is valid.

Since we have seen in the proof of Theorem 1 that

p2pyy A(/(i/r))(l - rf'dr > H/||2Dp > Py' A(/(t/r))(l - ry-1 dr,

using the identities t/r(w) = = <pw({/r), we may formulate the following corollary to Theorem 1.

Corollary 5. Let p € (0,oo) and f G A.

a) f € Qp if and only if

sup i A(/(f/r(w)))(l - r)p-1dr < oo , wet/ Jo

b) f G Qpfl if and only if

lim i A(/(f/r(w)))(l — r)p~idr = 0.

H—i Jo

As in the discussion after Theorem 1 we see that for f € Qp (4) sup i a(f(UT(w)))(l - r)p~idr < oo

weuJo

and for f G Qp,o

(5) lim i a(f(Ur(w)))(l - r)p 1dr = 0.

M—i Jo

So far as the converse in the case p G (0,1) is concerned, the function f(z) = exp ) £ H°° \ Qp cf- [7, Corollary 4.2] delivers a counterexam­

ple.

For the case p = 1 the universal covering map (see [9]) f from U onto the universal covering surface of C \ Z (see Example 2) belongs to B \ BMOA = B \Qi , and as in Example 2 we see that (4) holds.

For p G (0,1] we don’t know whether (5) implies f G Qp,o •

For p G (l,oo) (4) implies f G Qp = B and (5) implies f G Qp,o — Bo • This is easily seen remarking that for fixed r G (0,1) (4) implies

sup a(/(i/r(w))) < 00 weu

and (5) implies

lim a(/(£Zr(w))_) = 0.

|w|—1

This according to Theorem 1 and Theorem 2 in [5] implies f G Qp resp.

/ € QP,o (compare [16], too).

References

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Institut für Analysis Technische Universität

D-38106 Braunschweig, Germany

received October 14, 1996

Department of Mathematics Peking University

Beijing 100871, Beijing, China

U NI VERSIT ATIS MARIAE C U RIE - S K Ł O D O W S K A LUBLIN - POLONIA

VOL. XLIX SECTIO A 1995

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11. M. Królikowska: Nonlinear Mappings with an Almost Sparse Jacobian Matrix 12. D. Partyka and K. Sakan: Harmonic and Quasiconformal Mappings which Agree

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13. H. Renelt: Generalized Powers and Extensions of Analytic Functions 14. V. Starkov: Harmonic Locally Quasiconformal Mappings

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