Real and Complex Analysis (Rudin)

HARMONIC FUNCTIONS 249
As before, Loo(T) is the space of all (equivalence classes of) essentially
bounded functions on T, normed by the essential supremum norm, relative to
Lebesgue measure. For g E Loo(T), Ilglloo stands for the essential supremum of I g I.
11.32 Theorem To every f E H OO corresponds a function f* E Loo(n, defined
almost everywhere by
f*(e i8 ) = limf(re i8 ). (1)
r-+ 1
The equality 11111 "" = Ilf* II 00 holds.
If f*(e i8 ) = 0 for almost all e i8 on some arc leT, then f(z) = 0 for every
z E U.
(A considerably stronger uniqueness theorem will be obtained later, in
Theorem 15.19. See also Theorem 17.18 and Sec. 17.19.)
PROOF By Theorem 11.30, there is a unique g E Loo(n such that f = PEg]. By
Theorem 11.23, (1) holds withf* = g. The inequality Ilflloo :::;; IIf*lloo follows
from Theorem 11.16(1); the opposite inequality is obvious.
I n particular, iff * = 0 a.e., then II f * II 00 = 0, hence II f II 00 = 0, hence f = O.
Now choose a positive integer n so that the length of I is larger than
2nln. Let ex = exp {2niln} and define
n
F(z) = n f(exkz) (z E U). (2)
k=l
Then F E H OO and F* = 0 a.e. on T, hence F(z) = 0 for all z E U. If Z(f), the
zero set off in U, were at most countable, the same would be true of Z(F),
since Z(F) is the union of n sets obtained from Z(f) by rotations. But
Z(F) = U. Hencef = 0, by Theorem 10.18.
IIII
Exercises
1 Suppose U and v are real harmonic functions in a plane region n. Under what conditions is uv
harmonic? (Note that the answer depends strongly on the fact that the question is one about real
functions.) Show that u2 cannot be harmonic in n, unless u is constant. For which / E H(n) is 1/12
harmonic?
2 Suppose / is a complex function in a region n, and both / and /2 are harmonic in n. Prove that
either/orJis holomorphic in n.
3 If u is a harmonic function in a region n, what can you say about the set of points at which the
gradient of u is O? (This is the set on which Ux = u y = 0.)
4 Prove that every partial derivative of every harmonic function is harmonic.
Verify, by direct computation, that P,((} - t) is, for each fixed t, a harmonic function of rei'.
Deduce (without referring to holomorphic functions) that the Poisson integral P[dJl] of every finite
Borel measure Jl on T is harmonic in U, by showing that every partial derivative of P[dJl] is equal to
the integral of the corresponding partial derivative of the kernel.
5 Suppose / E H(n) and / has no zero in n. Prove that log 1/1 is harmonic in n, by computing its
Laplacian. Is there an easier way?