Real and Complex Analysis (Rudin)

CHAPTER
SEVENTEEN
This chapter is devoted to the study of certain subspaces of H(U) which are
defined by certain growth conditions; in fact, they are all contained in the class N
defined in Chap. 15. These so-called HP-spaces (named for G. H. Hardy) have a
large number of interesting properties concerning factorizations, boundary
values, and Cauchy-type representations in terms of measures on the boundary of
U. We shall merely give some of the highlights, such as the theorem of F. and M.
Riesz on measures J.l whose Fourier coefficients jJ.(n) are 0 for all n < 0, Beurling's
classification of the invariant subspaces of H2, and M. Riesz's theorem on conjugate
functions.
A convenient approach to the subject is via subharmonic functions, and we
begin with a brief outline of their properties.
Subharmonic Functions
17.1 Definition A function u defined in an open set n in the plane is said to
be subharmonic if it has the following four properties.
(a)
- 00 ~ u(z) < 00 for all zen.
(b) u is upper semicontinuous in n.
(c) Whenever i5(a; r) c: n, then
1 f"
(d) None ofthe integrals in (c) is - 00.
u(a) ~ -2 u(a + rei') dO.
n _"
335