Page 103 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Making use of Hölder’s inequality, we get
1 1
p q
|fg|d |f| d |g| d
q
p
X X X
= f g … (3)
p q
From (2) and (3) it follows that
|F (f)| f g .
p q
F|f|
Hence sup : f Lp(X) and f 0 g
f q
p
F g (Using definition of the norm) … (4)
q
q–1
Further, let f = |g| sgn g … (5)
Since sgn g = 1, we get

p
|f| = |g| p(q–1) = |g| q ( p (q – 1) = q)
1 1
p q
Thus, f L (X) and |f| d = |g| d … (6)
q
p
p
X X
1
p
q /p
q
p
But |g| d |g| d = g q
X X
which implies on using (6) that
q /p
f = g … (7)
p q

Now F (f) = fg d |g| q 1 g sgng d
X X

q
= |g| d g q
q
X
q
Hence g g = F (f) F f .
q p
and this on using (7) yields that

q q/p
g = F (f) F g
q q
q q /p
g = g F … (8)
q q
( g 0)

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