3. FUNCTION THEORETIC BACKGROUND 11
of conformal equivalence, the whole plane C and the rest of such domains. The
standard choice for the representative of the latter class is the unit disk D or the
upper half-plane C+. Correspondingly, most of complex function theory splits into
two main branches: the theory of entire functions and the theory of functions in D
or C+.
Function theory in D and C+ can be developed in parallel, due to their con-
formal equivalence. It is convenient, however, to keep some minor differences that
prove useful in applications. For instance, as the reader will see below, the Hardy
spaces in C+ are not mere conformal images of the Hardy spaces in D. At the same
time, the difference between the two versions of the same space is one dimensional
(i.e. H2 in the disk contains constants, while H2 in the half-plane does not) and all
the basic results are true for both spaces with minor adjustments. The same holds
for the model spaces, Toeplitz kernels, etc.
In the main part of the text, we will mostly need the spaces defined in the half-
plane. Hence, below we will only provide the C+-versions of some of the definitions.
3.1. Hardy spaces. For 0 p we denote by
Hp(D)
and
Hp(C+)
the
Hardy spaces of analytic functions in the unit disk D and the upper half-plane C+
correspondingly. These spaces are defined as follows.
By Hol(Ω) we denote the set of all functions holomorphic in a complex domain
Ω. For p ∞,
Hp(D)
= {f Hol(D) | sup0r1
|f(rξ)|pdm(ξ)
∞},
where m is the normalized Lebesgue measure on the unit circle T, m(T) = 1, and
Hp(C+)
= {f Hol(C+) | sup0y |f(x +
iy)|pdx
∞}.
For p 1 the supremum from the definition raised into the power 1/p presents
the norm in Hp. For 0 p 1, the supremum gives a metric in the corresponding
space. For p = both spaces are defined as spaces of bounded holomorphic
functions in the corresponding domains with sup-norms.
By the theorem of Fatou, all Hp functions possess non-tangential boundary
limits on the boundary of the domain. Via these limits, each Hp-function can be
uniquely identified with an Lp-function on the boundary. Moreover, by a version
of the maximum principle, if f
Hp(C+) (Hp(D))
for p 1 and f

are the
non-tangential boundary values of f on R (T) then ||f||Hp(C+) = ||f
∗||Lp(R,dx)
(||f||Hp(D) = ||f ∗||Lp(T,dm)). Similar equations hold for metrics for p 1.
This way every Hardy space
Hp
can be identified with a closed subspace of
Lp
on the boundary. In the case p = 2,
H2
becomes a Hilbert space with the inner
product inherited from the corresponding
L2:
f, g
H2(C+),
f, g
H2
= f
∗(x)¯∗(x)dx,
g
f, g
H2(D),
f, g
H2
= f
∗(ξ)¯∗(ξ)dm(ξ).
g
A standard approach is to think of f as of a function defined inside the domain
and almost everywhere on the boundary and write f instead of f

in the above
formulas and in similar situations.
Previous Page Next Page