Real and Complex Analysis (Rudin)

INTEGRATION ON PRODUCT SPACES 177
Suggestion: There is a measurable real function 6 so that dll = ei B d I Ill. Let A. be the subset of
E where cos (6 - ex) > 0, show that
Re [e-i·Jl(AJ] = Lcos+ (6 -
ex) dllll,
and integrate with respect to ex (as in Lemma 6.3).
Show, by an example, that l/n is the best constant in this inequality.
14 Complete the following proof of Hardy's inequality (Chap. 3, Exercise 14): Suppose f~ 0 on
(0, (0),/ Ell, 1 < p < 00, and
1 i~
F(x) = - f(t) dt.
x 0
Write xF(x) = J~ f(t)t·t-· dt, where 0 ~ ex < l/q, use Holder's inequality to get an upper bound for
F(x)P, and integrate to obtain
1" FP(x) dx ~ (1 - exq)l - P(exp)-l f' fP(t) dt.
Show that the best choice of ex yields
r FP(x) dx ~ C ~ J r fP(t) dt.
15 Put Ip(t) = 1 - cos t if 0 ~ t ~ 2n, !p(t) = 0 for all other real t. For - 00 < x < 00, define
f(x) = 1, g(x) = Ip'(x), h(x) = foo Ip(t) dt.
Verify the following statements about convolutions of these functions:
(i) (f. g)(x) = 0 for all x.
(ii) (g • h)(x) = (Ip * Ip)(x) > 0 on (0, 4n).
(iii) Therefore (f * g) * h = 0, whereasf. (g • h) is a positive constant.
But convolution is supposedly associative, by Fubini's theorem (Exercise 5(c». What went wrong?
16 Prove the following analogue of Minkowski's inequality, forf~ 0:
Supply the required hypotheses. (Many further developments of this theme may be found in [9].)