BONUS HOMEWORK ASSIGNMENT: LITTLEWOOD-PALEY THEORY
Exercise 1. (Khinchine’s Inequality)
Let N ∈ N be given. Let x1 , x2 , . . . , xN ∈ C. Suppose that 1 , . . . , N ∈ {−1, 1} are independent
random signs each chosen with probability 12 . Then, for all 1 ≤ p < ∞:
v
uN
N
p1
X
uX
∼p t
|xj |2 .
(1)
E |
j x j | p
j=1
j=1
In particular, letting N → ∞:
(2)
E |
∞
X
j x j |
p1
p
v
uX
u∞
∼p t
|xj |2 .
j=1
j=1
We recall that this result was used in the proof of the Littlewood-Paley inequality.
i) Show that it suffices to consider x1 , x2 , . . . , xN ∈ R.
ii) Explain why the claim holds for p = 2.
p1
PN
as an Lp norm. Explain why it
iii) From now on, it is helpful to think of E | j=1 j xj |p
suffices to show that the .p inequality in (??) holds for p ≥ 2. (Exercise continued on second page.)
[HINT: Using the fact that a probability space has measure one, note that that the .p inequality
holds for 1 ≤ p ≤ 2 and that the &p inequality holds for 2 ≤ p < ∞. Recall the logarithmic convexity
of Lp norms.]
qP
N
2
iv) Observe that it suffices to prove the claim in the case when
j=1 |xj | = 1.
PN
2
Compute the expectation E exp
. Recalling the fact that exp x2 ≥ cosh(x), show that
j=1 j xj
this expectation is . 1.
P
N
v) Use part iv) to obtain a bound on P j=1 j xj > λ and hence deduce the claim.
Exercise 2. (Bernstein’s Inequality)
Let ψ ∈ S(Rn ) be given. For j ∈ Z, we define ψj (ξ) := ψ( 2ξj ). Let 1 ≤ q ≤ p ≤ ∞ be given.
a) Show that, for all j ∈ Z:
1
1
kψj (D)f kLp (Rn ) ≤ C · 2nj ( q − p ) · kf kLq (Rn )
for some constant C > 0 which depends on ψ, n, p, q, but which is independent of j. [HINT: Write
ψj (D) as a convolution.]
b) Suppose that f ∈ Lp (Rn ) has the property that fb is supported on the set B(0, c · 2j ) for some
c > 0. Deduce that:
1
1
kf kLp (Rn ) ≤ C · 2nj ( q − p ) · kf kLq (Rn ) .
The results in parts a) and b) are called Bernstein’s Inequality.
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