6

ADELE ZUCCHI

The function ^ ^ ^ ( C ^ o ) , C € T and zo G ft, is the analog for ft of the

Poisson kernel for D (Green's kernel). It is possible to show ([13] Proposition

1.6.6) that -^g{Cizo) 0 for £ G T. As an immediate consequence of this fact,

we have that duo is mutually absolutely continuous with respect to ds for all

zo e ft.

Each function / G

ifp(ft)

has nontangential boundary values /* almost

everywhere da, and the function /* defined by these limits is in

LP(T,UJ).

The

mapping / —• /* is an isometry from HP(Q) onto a closed subspace of

LP(T,LU).

Further if the boundary values /* vanish on a set of positive measure, so does

the function / .

A function / denned on ft is in

if00

(ft) if it is holomorphic and bounded.

if00

(ft) is a closed subspace of L°°(r,c;) and it is a Banach algebra if endowed

with the supremum norm. Finally, the mapping / —• /* is an isometry of H°° (ft)

onto a weak*-closed subalgebra of L°°(r,u;).

Let 1 p oo. We will follow the common practice of using the same letter

/ to denote both the function on ft and its boundary values. In this way Hp(ft)

can be viewed as a closed subspace of

LP(T,UJ).

In particular, i7°°(ft) can be

viewed as a weak*-closed subalgebra of L°°(r,u;), and H2(Q) is a Hilbert space.

Further, if ft = Z?, we denote

HP(D)

simply by

Hp.

The classes

Hp(fl)

are invariant under conformal transformations of ft. Let

0 be a one-to-one holomorphic mapping of a region ft onto a region

ft7.

Then

the map V : #

p

(ft') -

iJp(ft)

denned by Vf = f o (j), is a bounded invertible

operator. If the

HP(Q,')

norm is determined at the point j)(zo), then V is also

an isometry.

We recall that a sequence {fn}^Li of elements in if00 (ft) is weak*-convergent

if and only if it is boundedly pointwise convergent, i.e., it converges pointwise

and sup

n G N

||/n||oo is finite.

PROPOSITION 2.2.2. ([14])i?(ft) is sequentially boundedly weak*-dense in iJ°°(ft),

i.e., for any f G

if00(ft)

there exists a sequence { r n } ^ ! in R(fl) weak*-

convergent to f and such that max{|rn(z)| : z G ft} ||/||oo

for

aM

n-

We continue this section by reformulating (following [20]) the concepts of

inner and outer functions and of Blaschke product and singular inner function

in a manner suitable for function theory on multiply connected regions, so that

the classical factorization theorems remain true.

Let us first recall that if h is a real-valued harmonic function defined in ft

and d*h — (dh/dx)dy — (dh/dy)dx, then, for j = 1,... ,ra, the number Jr d*h

is called the period about Tj of the harmonic conjugate *h (or of the conjugate

differential d*h) of h. There are many times when it is highly advantageous to

"correct" these periods, that is, to add a harmonic function ho to h so that the

periods of *(ft + ho) are all zero, and thus *(h + ho) is single-valued. There are

several ways to do this (cf. [13, 19]). In particular for any harmonic function h on

ft there exists a harmonic function ho on ft which is continuous on ft, constant

on each component of T and such that h + ho has a single-valued harmonic

conjugate ([18] p.304).