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.