Explicit Construction of Smooth Carpet

We resume to prove the mollification lemma6.4.5. In the original literature [Zor19], Zorin-Kranich neither gave an explicit construction nor verified the δ-covering relation. For the sake of completeness, we present our arguments with explicit construction. A reasonable starting point is to first consider the following question: What is the simplest non-trivial smooth carpet? A direct guess leads us to the next definition:

Definition 6.5.1 (The Ink-bleeding).

Given A ∈ D, we define the Ink-bleeding of A:

β_A∈ m {A ∈ X^∞| {A} ≺ A}

as the ≺-minimal smooth carpet that covers the one cube carpet {A} con-structed through the following process:

1. For some s ∈ Z, A ∈ Ds. We set A0:= {A} ∈ X at our initial stage.

2. Suppose we have A^k−1∈ X at k − 1th stage, we build Ak∈ X as such:

Ak:=M



Ak−1∪ [

J ∈Ak−1

n

I ∈ D | `<sup>I</sup> ≤ 2<sup>−κ</sup>`J ∧ ˜I ∩ J 6= ∅o





=A^k−1tn

I ∈ D^s−k\ A^⊂k−1| ˜I ∩G

Ak−16= ∅o .

Essentially, we attempt to use greedy algorithm by adding the bare re-quirement for it to be smoother. Incidentally, the process adds barely smaller layer of cubes on the edge of the carpet.

We define β_A:= [

k∈N

Ak. It is easy to check that {A} ≺ β_A∈ X:

{A} ⊂ A0⊂ A1⊂ · · · ⊂ Ak⊂ · · · ⊂ β_A∈ X.

By construction, β_A∈ X^∞ since, given (I, J ) ∈ D × Ak−1, we have:



`I ≤ 2<sup>−κ</sup>`J ∧ ˜I ∩ J 6= ∅

=⇒ ∃J⁰ ∈ Ak s.t. 2^−κ`J≤ `J⁰ ∧ I ⊂ J⁰
.

Figure 1: A³(with D = 2, κ = 1 and I, ˜I: red v.s. J : blue)

A

Also, minimality is guaranteed by the greedy algorithm. Lastly, we give some quantitative description:

Gβ_I = 1 + 2 (n_D2^κ+ 1)X

k∈N

2^−κk

! A

=(2nD+ 1) 2^κ+ 1

2^κ− 1 A ⊂ CDA, where CD:= 4nD+ 3.

With building blocks constructed, we still need ways to sew things together:

Properties 6.5.2 (Sewing).

Given Y ⊂ X^∞ and B ∈ X, we have:

(∀A ∈ Y, A ≺ B) =⇒ B _

Y := M [

Y ∈ X^∞.

Proof. By construction, we only need to verify the smoothness. Given I ∈ D

and J ∈W

Y, since there is A ∈ Y such that J ∈ A, we have:



`I ≤ 2<sup>−κ</sup>`J ∧ ˜I ∩ J 6= ∅

=⇒ ∃J⁰ ∈ A s.t. 2^−κ`J≤ `J⁰ ∧ I ⊂ J⁰

=⇒ ∃J⁰⁰∈_

Y s.t. 2^−κ`J ≤ `J⁰≤ `J⁰⁰ ∧ I ⊂ J⁰⊂ J⁰⁰
. As a result, B W

Y ∈ X^∞.

Now we are ready to prove the mollification lemma:

Proof (Lemma6.4.5). Through sewing Ink-bleedings, we immediately have:

∵ ∀A ∈ A, {A} ≺ β^A≺ B ∴ A ≺ βA ≺ B, where βA := _

A∈A

βA∈ X^∞.

On the other hand, since A ≺

δ B with δ ∈ (0, 1), we must have:

∀(A, B) ∈ A × B, A ⊂ B =⇒ `A≤ 2<sup>−κ</sup>`B
.

Consequently, given (A, B) ∈ A × B, we have C^κ,D:= 1 + 2^−κCD such that:

∃I ∈ βA s.t. I ⊂ B =⇒ B ∩ CDA 6= ∅ =⇒ A ⊂ C^κ,DB.

We now verify the quantitative covering relation. Given B ∈ B, since B ∈ X^∞ (scale of cubes varies smoothly in B), previous estimate yields:

X

I∈βA I⊂B

|I| ≤ X

A⊂CA∈A_κ,DB

X

I∈βA

|I| ≤ X

B⁰∈B B⁰∩Cκ,DB6=∅

X

A∈A A⊂B⁰

µG βA



.

D

X

B⁰∈B B⁰∩Cκ,DB6=∅

X

A∈A A⊂B⁰

|A| ≤ X

B⁰∈B B⁰∩Cκ,DB6=∅

δ|B⁰| .

κ,D

δ|B|.

7 Sparse Domination of Sparse Parts

With Eiffel Tower construction, we have set up for our decomposition scheme. The rest of the work is to provide good control on both sparse parts and cluster parts. Here, we choose the the setting: l . m = n to do the decomposition and present the argument for sparse parts in the form of sparse form dominance and pointwise sparse dominance.

7.1 Reductions

Definition 7.1.1.

C_P:=X

P ∈P

χE_P, where P ⊂ ˜D.

Definition 7.1.2 (Spectral η-control).

Given P_j∈ ˜D, we define:

(P0.^<P1 ⇐⇒ sP₀ ≤ sP₁ ∧ ∆(P0, P1) < η P₀.≥P₁ ⇐⇒ s_P0 ≤ s_P1 ∧ ∆(P₀, P₁) ∈ [η, ∞) .

Notice that either relation implies ˜IP₀∩ ˜IP₁ 6= ∅ and thus, IP₀ ⊂ 2 ˜IP₁. Addi-tionally, given P ⊂ ˜D and P ∈D, we define:˜

(

PP,<:= {P⁰ ∈ P | P⁰.^<P } PP,≥:= {P⁰ ∈ P | P⁰.≥P } Lemma 7.1.3 (Tile-tile interaction).

Given Pj∈ ˜D, we have:







L^∗_P1LP₀f

= 0 ⇐= P0, P1 are E -incomparable LP₁L^∗_P0f

 .

κ,D,d

h∆ (P0, P1)i^{τ /d} |^I˜_P0∩ ˜I_P1|

|^I_P0|^·|^I_P1|kf k_L1(^EP0) χ^EP1. Proof. The first relation is trivial since:

P0, P1 E -incomparable =⇒ E^P0∩ EP₁ = ∅.

The second relation follows from estimating the kernel:

LP₁L^∗_P0f (·) = ˆ

KP₀,P₁(·, y)f (y)dy, where the explicit form of K_P0,P1 is:

K_P0,P1(x, y) = ˆ

I˜_P0∩ ˜I_P1

e^{i(qx−qy)(z)}K_sP1(x, z)K_sP0(y, z)dz · χ_EP1(x)χ_EP0(y).

Considering J := ˜IP₀∩ ˜IP₁ 6= ∅ and (x, y) ∈ EP₁× EP₀, we have:

∵ (qx, q_y) ∈ ω_P1× ωP0 ∴ kqx− qykI˜_P0∩ ˜I_P1 ≥ ∆ (P0, P₁) .

To applyVan der Corput estimate, we need a way to measures the Oscil-lation of ψP₀,P₁(·) := Ks_P1(x, ·)Ks_P0(y, ·). Using kernel’s properties: L^∞\Size Control and Locally τ -H¨older Continuity(3.4.3.1), we have:

k∆k . `J =⇒ |ψP₀,P₁− τ∆ψP₀,P₁| .

κ,D,d

(k∆k/`J)^τ|IP₀|⁻¹|IP₁|⁻¹.

Plugging everything into the estimate yields:

|KP0,P1(x, y)| .

D,d

sup

k∆k

`J <hkqx−qykJi^1/d

kψP0,P1− τ∆ψ_P0,P1k_L∞|J |

.

κ,D,d

kqx− q_yk_J^{τ /d}

|I_P0|⁻¹|I_P1|⁻¹|J |

≤ h∆ (P0, P1)i^{τ /d}   

I˜P₀∩ ˜IP₁

  

|IP₀| · |IP₁|.

Remark. Comparing to thesingle tile estimate, we successfully extract the dis-tance factor and keep all other the good estimate.

Through single tile estimate and tile-tile interaction, we aim to control the behavior of the sparse part. For starters, we first observe that: Given P ⊂ Pn be sparse parts, we have two ways to proceed with our control:

• Pointwise Dominance: Using single tile estimate, we suspect that:

|L_P| .X

P ∈P

|f |I˜_PχE_P

?

.X

I∈S

|f |ΛIχI,

for some large constant Λ . 1 and S ⊂ D p(n)-carleson with p(·) be a prescribed polynomial. By Sparse-Maximal dominance, we expect:

kL_Pf k_Lp. p(n) kM f kL^p. p(n)kf k^Lp, ∀p ∈ (1, ∞).

• L² control: Expanding the L² norm explicitly, we have:

kL^∗_Pf k²_L2 = hL^∗_Pf, L^∗_Pf i . X

P_j∈P s_P0≤s_P1

L^∗_P0f, L^∗_P

1f 

≤

* X

P_j∈P s_P0≤s_P1

L_P1L^∗_P

0f , |f |

+ .

To control the L² norm is to control the first term in the last expression.

With Tile-tile interaction, we have:

Applying H¨older’s inequality, we get:

X

the RHS is again possible to be dominated by the corresponding sparse operator with a p(n)-carlson sparse cubes S⁰ ⊂ D. This in turn can further be norm dominated by Mrf :

X

As a result, through duality, we shall expect:

kL_Pf k_L2 . p(n)2^−n/2kf k_L2.

Suppose everything works as intended, we can easily spread out the 2^−n/2 decay in L² to all L^p and sum over n ∈ N to complete the L^p control:

Theorem 7.1.4 (L^p bound on sparse parts).

Given P ⊂ ˜D be the full collection of the sparse parts, we have:

kL_Pf k_Lp.

p

kf k_Lp, ∀p ∈ (0, ∞).

For a more precise analysis, we consider the following configuration:

Definition 7.1.5 (Sparse tower or Sparse Forest in [Lie20], Anti-chain and boundary in [Zor19]).

Given P ⊂ Pⁿ, we say:

P is

(an anti-chain

a . n-decay tower

⇐⇒ P ∩ Pn,α is

(an anti-chain

a . n-decay stack ∀α ∈ N In either case, we call P a sparse tower.

Remark. In our case, using decomposition scheme on Eiffel Tower con-struction with l . m = n gives us:

Pn







. n² anti-chain towers . n . n-decay towers A lot of clusters

Therefore, to compensate the polynomial growth of the number of sparse towers, we shall extract some exponential decay from the estimate of a sparse tower:

Theorem 7.1.6 (Sparse tower estimate).

Given P ⊂ Pⁿ a sparse tower, we have:

kL_Pf k_L2 . p(n)2^−nη2kf k_L2,

and we can construct a p(n)-carleson collection S ⊂ D such that:

|L_Pf | .X

I∈S

|f |I˜χI.

As a result, we have full control:

kL_Pf k_Lp.

p

p(n)2^−nηpkf k_Lp, where ηp> 0, ∀p ∈ (1, ∞).

The theorem follows directly from the following two lemmas.

Lemma 7.1.7 (Sparse dominance).

Given P ⊂ Pn be sparse tower, we can find p(n)-carleson S ⊂ D such that:

X

P ∈P

|f |_ΛI

P,rχE_P ≤X

I∈S

|f |ΛI,rχI, ∀r ∈ [1, ∞).

Lemma 7.1.8 (Density extraction).

Given P ⊂ Pⁿ be sparse tower, P⁰ ∈ ˜D, and r ∈ (1, ∞), we have:





   C_{PP 0 ,≥}

  _{2 ˜I}

P 0,r⁰ .

r

p(n) 

 C_{PP 0 ,<}



2 ˜I_{P 0},r⁰ .

r

p(n)2^−n/r0(1 + η)^(dD+)/r0.

Remark. To apply density extraction to the proof of theorem, we fine tune η,  ∈ R+ and r ∈ (1, 2) so that:

(1 + η)^{−τ /d}+ 2^−n/r0(1 + η)^(dD+)/r0 . 2^−nη2.

Before we proceed with the proof of the lemmas, we present our plan:

1. Prove the lemmas with P ⊂ Pn,α be an anti-chain.

2. For any . n-decay stack P ⊂ P^n,α, we can construct a decomposition on P with respect to a decomposition on its temporal projection to encode the decay property. We first recall that there is s∆h n such that:

s⁰− s ≥ s_∆ =⇒ ∀J ∈ Ds⁰, X

I∈IPs

I⊂J

|I| ≤ 2^−κ|J |.

We now reorganize the collection by modding out s_∆on the scaling:

IP=

s_∆

G

j=1

I^j_P, where I^j_P:=G

t∈Z

IP,s∆t+j,

and do the following canonical decomposition into carpets:

I^j_P= G

k∈N

M^j_P,k, where M^j_P,k := M I^j_P\ G

l<k

M^j_P,l

!

∈ X, ∀k ∈ N.

By construction, if (I, J ) ∈

Ds∩ M^j_P,k+1

×

Ds⁰∩ M^j_P,k , then:

I ⊂ J =⇒ s⁰− s s∆ ∈ N.

Therefore, we can verify the following covering condition:

∀J ∈ Ds⁰∩ M^j_P,k, X

I∈M^j_P,k+1 I⊂J

|I| = X

s∈Z

s0 −s s∆ ∈N

X

I∈Ds∩M^jP,k+1

I⊂J

|I|

by decay property, ≤ X

s∈Z

s0 −s s∆ ∈N

2⁻

s0 −s s∆ κ

|J | = 1 2^κ− 1|J |.

That is, we have: As a direct consequence, I^j_P is . 1-carleson. Correspondingly, we define:

P =

Notice that P^jks are anti-chains. With the the ≺

1 2κ −1

-chain structure, estimate from individual anti-chain P^jk can be sum up to similar order.

3. For P ⊂ Pⁿbe an sparse tower, we decompose the collection with respect to the level/cell structure:

P = G

α∈N

P^(α), where P^(α):= P ∩ P^n,α.

δ-covering relation among A^αs should allow us to sum everything up.

在文檔中 多項式調變不變奇異積分算子 (頁 52-60)