In order to recover full Lp bound for the cluster parts while exploiting the orthogonality structure of BL(L2, L2), Zorin-Kranich adopted an extrapola-tion argument used in [BT13] by refining/localizing the L2 estimate. Yet, his argument requires a reorganization of the full collection of the tiles in-cluding the sparse parts. We come up with a similar idea without altering the configuration of the sparse parts. For starters, we state the extrapolation method matching our L2 settings:
Lemma 8.7.1 (L2Extrapolation).
Fix p > 2 and an operator T mapping Lp,1 qualitatively to Lp,∞. Suppose for any G, H ⊂ RD measurable we can find measurable subset G0⊂ G and H0 ⊂ H such that:
• Error loss control:
|G \ G0|
|G|
1/p
+ H \ H0 H
1/p0
≤ < 1,
• Testing condition:
kχH0T (χG0f )kL2≤ Λ |H|
|G|
1/2−1/p
kχG0f kL2,
we then have the following quantitative control:
kT f kLp,∞. Λ
1 − kf kLp,1.
Our goal now is to extrapolate a Lp,1→ Lp,∞bound that does not neces-sary have a exponential decay for a cluster tower in Pn. This is still okay since we can first extrapolate a bit further and interpolate with the L2
bound to spread the exponential decay. Also, for p ∈ (1, 2), we just switch to control the adjoint. With that been said, we still need to find a system-atic way to choose the subset G0, H0 for given G, H. Zorin-Kranich made the following observation:
Observation. Given measurable set A ⊂ RD and ρ ∈ (0, 1), we have:
I 6⊂ M χ−1A (ρ, ∞] =⇒ |I ∩ A|
|I| ≤ ρ.
This is equivalent to say:
I ⊂ M χ−1A (ρ, ∞] ⇐=
I
|χA| dµ = |I ∩ A|
|I| > ρ.
That reminds us the support restriction control. As we explore the idea, we would naturally come up with the following settings:
Definition 8.7.2.
Given measurable A ⊂ RD, we set:
Aρ:= M χ−1A (ρ, ∞] , where ρ ∈ (0, 1).
For a collection of tiles P ⊂ ˜D, we set:
(PA,ρ:= {P ∈ P | IP 6⊂ Aρ} P∗A,ρ:=n
P ∈ P | ˜IP 6⊂ Aρ
o . Due to our construction, we have the following:
Lemma 8.7.3 (Density Manipulation).
Given P ⊂ ˜D, a measurable set A ⊂ RD, and ρ ∈ (0, 1), we have:
I ∈ JPA,ρ∪ LPA,ρ∪ JP∗A,ρ∪ LP∗A,ρ =⇒ |I ∩ A|
|I| . ρ.
Proof. By construction, we have:
I ∈ JPA,ρ∪ LPA,ρ∪ JP∗A,ρ∪ LP∗A,ρ
=⇒ ∃P ∈ PA,ρ∪ P∗A,ρ s.t. ˜IP ⊂
κ,D∼ I
=⇒ ∃Λ .
κ,D
1 s.t. ˜IP ⊂ ΛI 6⊂ Aρ= M χ−1A (ρ, ∞]
=⇒ |I ∩ A|
|I| ≤ Λ|ΛI ∩ A|
|ΛI| ≤ Λρ.
From this, we derive:
Corollary 8.7.3.1 (In-level localized estimate).
Proof. We observe that:
(χALPχAc
Since both PA,ρand P∗A,ρare open cluster at p, applying support restriction control on χIp∩ALP∗A,ρ and χIp∩AL∗P
A,ρ gives the desired control. As a imme-diate result, natural decomposition yield the estimate for row configuration.
To control an open 2nκ-apart 2n-stack, we discard irrelevant tiles:
(χALPχAcρf = χALP∗ and proceed in the following two ways:
• To control χALP∗
A,ρ, we exploit the density manipulation to improve the extraction of separation factor. That is, given an open cluster P⊂ Pn,α at p ∈ Pn,α, we have: that are Λ-apart and E-incomparable, we have:
Therefore, for Λ-apart rows R, R0⊂ Pn,α, we also have:
This gives us the desired control to apply the Cotlar-Stein Lemma(T T∗ -T∗T argument): We first decompose P∗A,ρ into rows {Rj}2j=1n and verify
For the dual estimate, E-incomparability implies:
2n Combining the two, we have:
PA,ρ, we use orthogonality directly. After decomposing PA,ρ into rows {Rj}2j=1n , we can control its adjoint:
We now present the analogue for a cluster tower:
Lemma 8.7.4 (Cluster tower localized L2 estimate).
Given P ⊂ Pn a cluster tower, as long as κ ≥ 2/2, we have:
We again verify the condition for Cotlar-Stein Lemma but, this time, view a stack as a whole. We start with estimating χALPχAc
For the dual condition, we have:
To estimate χAL∗PχAcρ, we follow similar arguments:
For the dual condition, we have:
X
Therefore, we have:
This completes the proof.
We now use such localized estimate to extrapolate our estimate:
Theorem 8.7.5 (Cluster tower weak estimate).
Given P ⊂ Pn a cluster tower, as long as κ ≥ 2/2, we have:
kLPf kLp,∞, kL∗Pf kLp,∞ .
p
n kf kLp,1, ∀p ∈ (2, ∞).
Proof. Let T denote either LP or L∗P. We intend to use L2 Extrapolation.
• For measurable sets G, H ⊂ RD. We want to find suitable measurable subsets G0 ⊂ G and H0 ⊂ H satisfying both error loss control and testing condition.
• To verify the testing condition, we see that:
– If ρ & 1, we may just apply cluster tower L2 estimate:
kχH0T χG0f kL2 ≤ kT χG0f kL2. n2−n/2kχG0f kL2 – If ρ 1, we use cluster tower localized L2 estimate:
kχH0T χG0f kL2 =
• L2 Extrapolation yields:
kT f kLp,∞ .
p
n kf kLp,1, which completes the proof.
As a direct corollary, through interpolation, we have:
Corollary 8.7.5.1 (Cluster tower strong estimate).
Given P ⊂ Pn a cluster tower, as long as κ ≥ 2/2, we have:
kLPf kLp .
p
n2−nηpkf kLp, where ηp> 0, ∀p ∈ (1, ∞).
Corollary 8.7.5.2 (Lp bound on cluster parts).
Given the full collection of the cluster parts P ⊂ ˜D, we have:
kLPf kLp.
p
kf kLp, ∀p ∈ (1, ∞).
Remark. Through our method, instead of rearranging the whole collection as in [Zor19], we recover the result in [Lie20]. That is, the decomposition itself is effective enough for the Lp → Lp bound. Still, the formulation in [Lie20]
is similar to a decoupling inequality, which contains more information about the structure of the Lp estimate.
As we combine the estimation of sparse tower and cluster tower, we prove the main result in the following reduced form:
Theorem 8.7.6 (Main theorem for the linearized operator).
kLPnf kLp.
p
p(n)2−nηpkf kLp, where ηp> 0, ∀p ∈ (1, ∞).
Summing over n ∈ N yields:
kLf kLp.
p
kf kLp, ∀p ∈ (1, ∞).
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