Page 67 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
1 1
1
q p
p
1 p
q

p
p 1
q

1 p 1 1 1 1 1
If we take p and , then 1 and since 0 p 1 1 P 1 ,
P P Q p Q 0 P

1 p 1
i.e. 1< P < < and also 1 p 0 1 as 0 < p < 1 Q > 1.
Q q Q
P, Q are conjugate numbers with 1 < P < .
If we take |fg| = F and |g| = G .
Q
p
q
1 q p
1 q
p
p
Q
Then fg = |f | |g| = |f| |g| p Q q
g
p
= |f| .
f, g are non-negative measurable functions s.t.
q
p
Also f L and g L .
Applying the Hölder’s inequality for P, Q to the functions f and g, we get
|FG| F G
P Q
1 1
|f| p |F| P P |G| Q Q as |fg| = fg = |f| p

p
p q
|f| p |fg| |g| q

1 1
p q q
p
|f| |fg| |g|

1 |fg|
p q
p
|f| 1 , provided |g| 0
q
q
|g|
1 1
|fg| |f| p p |q| q q f g
p q

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