Page 74 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 6: Minkowski Inequalities

|f + g| f + |g| Notes
f + g a.e.
f + g f + g
Hence the result follows in this case also. Thus, we now assume that 1 < p < .

p
p
Since L is a linear space, f + g L .
1 1
Let q be conjugate to p, then 1 .
p q
p
Now (f + g) L
(f + g) p/q L q

1 1 1 1 1 p 1
Since 1 1
p q q p q p

(p 1) q p, |f g| p 1 q |f g| p

p
p
p–1
p
and therefore |f + g| L (f + g) p/q L because p – 1 = .
q
On applying Hölder’s inequality for f and (f + g) p/q , we get
p 1 p p 1 p
q
|f||f g| dx |f| ) dx |f g| q q dx
p
1 1
p p q
q
p
p
or |f||f g| dx |f| ) dx |f g| dx … (1)
p
Since g L , therefore interchanging f and g in (1), we get
p 1 p 1 q
p
p
q
|g||f g| dx |g| ) dx |f g| dx … (2)
Adding, we get
1 1 1
p p p p q
q
q
p
p
p
|f||f g| dx |g||f g| dx |f| dx |g| dx |f g| dx … (3)
p 1
Now |f + g| = f g f g
p
1 1 p p
But 1 1 p p 1
p q q q
p
|f + g| = f g f g q
p
p
|f| |g| |f g| q

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