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
has nontangential boundary values /* almost
everywhere da, and the function /* defined by these limits is in
mapping / —• /* is an isometry from HP(Q) onto a closed subspace of
Further if the boundary values /* vanish on a set of positive measure, so does
the function / .
A function / denned on ft is in
(ft) if it is holomorphic and bounded.
(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
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
simply by
The classes
are invariant under conformal transformations of ft. Let
0 be a one-to-one holomorphic mapping of a region ft onto a region
the map V : #
(ft') -
denned by Vf = f o (j), is a bounded invertible
operator. If the
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
there exists a sequence { r n } ^ ! in R(fl) weak*-
convergent to f and such that max{|rn(z)| : z G ft} ||/||oo
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).
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