Real and Complex Analysis (Rudin)

HARMONIC FUNCTIONS 241
The regions nIX expand when IX increases. Their union is U, their intersection
is the radius [0, 1).
Rotated copies of n", with vertex at elf, will be denoted by eitn".
11.19 Maximal Functions If 0 < IX < 1 and u is any complex function with
domain U, its nontangential maximal function N" u is defined on T by
Similarly, the radial maximal function of u is
(N"u)(eit) = sup {I u(z) I : Z E eitn,,}. (1)
(Mrad u)(ei~ = sup {I u(rei~ I: 0 :::;; r < I}. (2)
If u is continuous and A. is ~ positive number, then the set where either of
these maximal functions is :::;; A. is a closed subset of T. Consequently, N" u and
Mradu are lower semicontinuous on T; in particular, they are measurable.
Clearly, Mradu:::;; N"u, and the latter increases with IX. Ifu = P[dJ.L], Theorem
11.20 will show that the size of N" u is, in turn, controlled by the maximal function
MJ.L that was defined in Sec. 7.2 (taking k = 1). However, it will simplify the
notation if we replace ordinary Lebesgue measure m on T by u = m/2n. Then u is
a rotation-invariant positive Borel measure on T, so normalized that u(T) = 1.
Accordingly, MJ.L is now defined by
The supremum is taken over all open arcs leT whose centers are at ei9, including
T itself (even though T is of course not an arc).
Similarly, the derivative DJ.L of a measure J.L on T is now
ill\ 1· J.L(I)
(DJ.L)(e J = 1m u(I) , (4)
as the open arcs leT shrink to their center ei9, and ei9 is a Lebesgue point of
f E IJ(T) if
lim U;I) i I f - f(ei~ I du = 0, (5)
where {I} is as in (4).
If dJ.L = f du + dJ.L. is the Lebesgue decomposition of a complex Borel
measure J.L on T, where f E Ll(T) and J.L. 1. u, Theorems 7.4, 7.7, and 7.14 assert
that
that almost every point of T is a Lebesgue point of J, and that DJ.L = J, DJ.L. = 0
a.e. [u].
(3)
(6)