Problem set for Math 158, unit on decoupling in Fourier analysis
1. Locally constant property. Suppose that θ ⊂ R2 is a rectangle. Let θ∗ denote
the dual rectangle. If we rotate and translate so that θ = [0, θ1 ] × [0, θ2 ], then the
dual rectangle θ∗ is [0, θ1−1 ], [0, θ2 ]−1 . Suppose that the support of fˆ is contained in
θ.
Prove that |f | is roughly locally constant on translates of θ∗ in the following sense:
for any x0 ∈ R2 ,
kf kL∞ (θ∗ +x0 ) . kf kL1avg (Wθ∗ +x0 ) .
Here Wθ∗ +x0 is a weight function which is ∼ 1 on θ∗ + x0 and decays rapidly away
from θ∗ + x0 . If θ∗ is [0, θ1−1 ], [0, θ2 ]−1 , and x0 = 0, then for any N ≥ 1,
|W (y)| .N (1 + θ1 y1 + θ2 y2 )−N .
For a weight W , the averaged norm Lpavg (W ) is defined as
R
1/p
W |f |p
R
kf kLpavg (W ) :=
.
W
2. The simplest decoupling problem. Let τ = [0, A] ⊂ R and decompose τ into
unit intervals τ = ∪θ. Define the decoupling constant D[interval]p (A) to be the best
constant in the following inequality:
P
Whenever fˆθ is supported in θ and f = θ fθ , then
!1/2
kf kLp (R) ≤ D[interval]p (A)
X
kfθ k2Lp (R)
.
θ
Give the best upper and lower bounds that you can for D[interval]p (A).
I might add one more problem next week.
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