Real-Anlaysis Lp Spaces, guaranteed and verified 100% Pass

1


𝐿𝑝 Spaces


Def: For 𝐸 a measurable set, 1 < 𝑝 < ∞, and a function 𝑓 ∈ 𝐿𝑝 (𝐸),
define:
1
‖𝒇‖𝒑 = (∫𝐸 |𝑓|𝑝 )𝑝 .


The functional ‖∙‖𝑝 is a norm on 𝐿𝑝 (𝐸).

It’s clear that ‖𝜆𝑓 ‖𝑝 = |𝜆|‖𝑓 ‖𝑝 , and ‖𝑓 ‖𝑝 ≥ 0 with ‖𝑓 ‖𝑝 = 0 if,
and only if, 𝑓 = 0 a.e. on 𝐸.
What is less obvious is the triangle inequality:
‖𝑓 + 𝑔‖𝑝 ≤ ‖𝑓‖𝑝 + ‖𝑔‖𝑝 .
This is called the Minkowski inequality.


𝑝
Def. The conjugate of a number 𝑝 ∈ (1, ∞) is the number 𝑞 = , which is
𝑝−1
the unique 𝑞 ∈ (1, ∞) for which:
1 1
+ = 1.
𝑝 𝑞
The conjugate of 1 is defined to be ∞, and the conjugate of ∞ is
defined to be 1.



Young’s inequality: for 1 < 𝑝 < ∞, 𝑞 the conjugate of 𝑝, and any two
positive numbers 𝑎, 𝑏,
𝑎𝑝 𝑏𝑝
𝑎𝑏 ≤
𝑝
+𝑞 .

, 2


Proof: 𝑓 (𝑥 ) = 𝑒 𝑥 has a positive second derivative and therefore is
convex, i.e. for any 𝜆 ∈ [0, 1], and any numbers 𝑢, 𝑣.

𝑒 𝜆𝑢+(1−𝜆)𝑣 ≤ 𝜆𝑒𝑢 + (1 − 𝜆)𝑒𝑣
i.e. 𝑓(𝜆𝑢 + (1 − 𝜆)𝑣) ≤ 𝜆𝑓(𝑢) + (1 − 𝜆)𝑓(𝑣).




𝑢 𝜆𝑢 + (1 − 𝜆)𝑣 𝑣



1 1
In particular, setting 𝜆 =
𝑝
, 1 − 𝜆 = 𝑞 , 𝑢 = 𝑙𝑛𝑎𝑝 , 𝑣 = 𝑙𝑛𝑏 𝑞
1 1
( ln 𝑎𝑝 + ln 𝑏𝑞 ) 1 (ln 𝑎𝑝 ) 1 𝑞
𝑒 𝑝 𝑞 ≤ 𝑒 + 𝑒(ln 𝑏 )
𝑝 𝑞
1 𝑝 1 𝑞
𝑎𝑏 ≤ 𝑎 + 𝑏 .
𝑝 𝑞




Def. 𝒔𝒈𝒏(𝒇) = 1 if 𝑓(𝑥) ≥ 0 and −1 if 𝑓 (𝑥 ) < 0.