September19,2013 PHDthesisinprogress,directedbyXavierTolsaMart´ıPrats AT(1)theoremforSobolevspacesondomains

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Introduction Proof of the T(P) Theorem A geometric condition for the Beurling transform

A T(1) theorem for Sobolev spaces on domains

PHD thesis in progress, directed by Xavier Tolsa

Mart´ı Prats

Universitat Aut`onoma de Barcelona

September 19, 2013

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Introduction

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The Beurling transform

The Beurling transform of a function f ∈ L^p(C) is:

Bf (z) = c₀lim

ε→0

ˆ

|w −z|>ε

f (w )

(z − w )²dm(z).

It is essential to quasiconformal mappings because B( ¯∂f ) = ∂f ∀f ∈ W^1,p. Recall that B : L^p(C) → L^p(C) is bounded for 1 < p < ∞.

Also B : W^s,p(C) → W^s,p(C) is bounded for 1 0. In particular, if z /∈ supp(f ) then Bf is analytic in an ε-neighborhood of z and

∂ⁿBf (z) = cn

ˆ

|w −z|>ε

f (w )

(z − w )ⁿ⁺²dm(z).

back to T(P)

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The Beurling transform

Bf (z) = c₀lim

ε→0

ˆ

|w −z|>ε

f (w )

(z − w )²dm(z).

It is essential to quasiconformal mappings because B( ¯∂f ) = ∂f ∀f ∈ W^1,p.

Recall that B : L^p(C) → L^p(C) is bounded for 1 < p < ∞.

Also B : W^s,p(C) → W^s,p(C) is bounded for 1 0. In particular, if z /∈ supp(f ) then Bf is analytic in an ε-neighborhood of z and

∂ⁿBf (z) = cn

ˆ

|w −z|>ε

f (w )

(z − w )ⁿ⁺²dm(z).

back to T(P)

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The Beurling transform

Bf (z) = c₀lim

ε→0

ˆ

|w −z|>ε

f (w )

(z − w )²dm(z).

Also B : W^s,p(C) → W^s,p(C) is bounded for 1 0.

In particular, if z /∈ supp(f ) then Bf is analytic in an ε-neighborhood of z and

∂ⁿBf (z) = cn

ˆ

|w −z|>ε

f (w )

(z − w )ⁿ⁺²dm(z).

back to T(P)

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The Beurling transform

Bf (z) = c₀lim

ε→0

ˆ

|w −z|>ε

f (w )

(z − w )²dm(z).

Also B : W^s,p(C) → W^s,p(C) is bounded for 1 0.

In particular, if z /∈ supp(f ) then Bf is analytic in an ε-neighborhood of z and

∂ⁿBf (z) = cn

ˆ

|w −z|>ε

f (w )

(z − w )ⁿ⁺²dm(z).

back to T(P)

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The problem we face

Let Ω be a Lipschitz domain.

When is B : W^s,p(Ω) → W^s,p(Ω) bounded?

We want an answer in terms of the geometry of the boundary.

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Known facts, part 1

In a recent paper, Cruz, Mateu and Orobitg proved that for 0 < s ≤ 1, 2 2, and ∂Ω smooth enough,

Theorem

B : W^s,p(Ω) → W^s,p(Ω) is bounded if and only if

BχΩ∈ W^s,p(Ω).

One can deduce regularity of a quasiregular mapping in terms of the regularity of its Beltrami coefficient.

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Known facts, part 1

In a recent paper, Cruz, Mateu and Orobitg proved that for 0 < s ≤ 1, 2 2, and ∂Ω smooth enough,

Theorem

B : W^s,p(Ω) → W^s,p(Ω) is bounded if and only if

BχΩ∈ W^s,p(Ω).

One can deduce regularity of a quasiregular mapping in terms of the regularity of its Beltrami coefficient.

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Introducing the Besov spaces B

_p,p^s

The geometric answer will be given in terms of Besov spaces B_p,p^s . B_p,p^s form a family closely related to W^s,p. They coincide for p = 2.

For p < 2, B_p,p^s ⊂ W^s,p. Otherwise W^s,p ⊂ B_p,p^s .

Definition

For 0 < s < ∞, 1 ≤ p < ∞, f ∈ ˙B_p,p^s (R) if

kf kB˙^s_p,p= ˆ

R

ˆ

R

∆^[s]+1_h f (x ) h^s

p

dm(h)

|h| dm(x )

!^1/p

< ∞.

Furthermore, f ∈ B_p,p^s (R) if kf k_Bs

p,p = kf k_Lp+ kf kB˙_p,p^s < ∞.

We call them homogeneous and non-homogeneous Besov spaces respectively.

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Introducing the Besov spaces B

_p,p^s

For p < 2, B_p,p^s ⊂ W^s,p. Otherwise W^s,p ⊂ B_p,p^s . Definition

For 0 < s < ∞, 1 ≤ p < ∞, f ∈ ˙B_p,p^s (R) if

kf kB˙_p,p^s = ˆ

R

ˆ

R

∆^[s]+1_h f (x ) h^s

p

dm(h)

|h| dm(x )

!^1/p

< ∞.

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Introducing the Besov spaces B

_p,p^s

For p < 2, B_p,p^s ⊂ W^s,p. Otherwise W^s,p ⊂ B_p,p^s . Definition

For 0 < s < ∞, 1 ≤ p < ∞, f ∈ ˙B_p,p^s (R) if

kf kB˙_p,p^s = ˆ

R

ˆ

R

∆^[s]+1_h f (x ) h^s

p

dm(h)

|h| dm(x )

!^1/p

< ∞.

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Known facts, part 2

In another recent paper, Cruz and Tolsa proved that for any 1 < p < ∞, and Ω a Lipschitz domain,

Theorem

If the normal vector N belongs to Bp,p^1−1/p(∂Ω), then B(χΩ) ∈ W^1,p(Ω) with

k∇B(χΩ)k_Lp(Ω)≤ ckNk_B_˙1−1/p p,p (∂Ω).

They proved also an analogous result for smoothness 0 < s < 1. This implies

Theorem

Let 0 < s ≤ 1, 2 2. If the normal vector is in the Besov space Bp,p^s−1/p(∂Ω), then the Beurling transform is bounded in W^s,p(Ω).

Tolsa proved a converse for Ω flat enough.

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Known facts, part 2

Theorem

They proved also an analogous result for smoothness 0 < s < 1.

This implies Theorem

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Known facts, part 2

Theorem

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Known facts, part 2

Theorem

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Main results

T(P) Theorem

Let 2 < p < ∞ and 1 ≤ n < ∞. Let Ω be a Lipschitz domain.

Then the Beurling transform is bounded in W^n,p(Ω) if and only if for any polynomial of degree less than n restricted to the domain, P = PχΩ, B(P) ∈ W^n,p(Ω).

This theorem is valid for any Calderon-Zygmund convolution operator with enough smoothness and for any space R^d.

Theorem (Geometric condition on the boundary) Let Ω be smooth enough. Then we can write

k∂ⁿBχΩk^p_Lp(Ω). kNk^p_Bn−1/p

p,p (∂Ω)+ H¹(∂Ω)^2−np.

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Main results

T(P) Theorem

p,p (∂Ω)+ H¹(∂Ω)^2−np.

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Main results

T(P) Theorem

p,p (∂Ω)+ H¹(∂Ω)^2−np.

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Proof of the T(P) Theorem

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Local charts

Beurling transform

We have a Lipschitz domain.

In particular, at every boundary point we can center a cube

with fixed side-length R

inducing a parametrization C^0,1. We make a covering of the boundary by N of such cubes Qk

with some controlled overlapping and find a partition of unity {ψj}^N_{j =0}. kBf k^p_Wn,p(Ω)≈ kBf k^p_Lp(Ω)+ k∇ⁿBf k^p_Lp(Ω).

k∇ⁿBf k^p_Lp(Ω)≈PN

k=0k∇ⁿB(f ψk)k^p_Lp(Q_k)+ k∇ⁿB(f ψk)k^p_Lp(Ω\Q_k)

Away from Qk we have good bounds:

|∇ⁿB(f ψ_k)(z)| . _Rn+2¹

´

Q_k|f (w )|dw

The restriction to the inner region is always bounded: f ψ0∈ W^n,p(C).

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Local charts

Beurling transform

inducing a parametrization C^0,1.

We make a covering of the boundary by N of such cubes Qk

|∇ⁿB(f ψ_k)(z)| . _Rn+2¹

´

Q_k|f (w )|dw

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Local charts

Beurling transform

with some controlled overlapping and find a partition of unity {ψj}^N_{j =0}.

kBf k^p_Wn,p(Ω)≈ kBf k^p_Lp(Ω)+ k∇ⁿBf k^p_Lp(Ω). k∇ⁿBf k^p_Lp(Ω)≈PN

|∇ⁿB(f ψ_k)(z)| . _Rn+2¹

´

Q_k|f (w )|dw

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Local charts

Beurling transform

|∇ⁿB(f ψ_k)(z)| . _Rn+2¹

´

Q_k|f (w )|dw

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Local charts

Beurling transform

|∇ⁿB(f ψ_k)(z)| . _Rn+2¹

´

Q_k|f (w )|dw

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Local charts

Beurling transform

|∇ⁿB(f ψ_k)(z)| . _Rn+2¹

´

Q_k|f (w )|dw

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Local charts

Beurling transform

|∇ⁿB(f ψ_k)(z)| . _Rn+2¹

´

Q_k|f (w )|dw

The restriction to the inner region is always bounded:

f ψ0∈ W^n,p(C).

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Local charts: Whitney decomposition

We perform an oriented Whitney covering W such that

dist(Q, ∂Ω ∩ Q) ≈ `(Q) for every Q ∈ W. The family {5Q}_Q∈W has finite superposition. ...

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Local charts: Whitney decomposition

We perform an oriented Whitney covering W such that dist(Q, ∂Ω ∩ Q) ≈ `(Q) for every Q ∈ W.

The family {5Q}_Q∈W has finite superposition. ...

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Local charts: Whitney decomposition

We perform an oriented Whitney covering W such that dist(Q, ∂Ω ∩ Q) ≈ `(Q) for every Q ∈ W.

The family {5Q}_Q∈W has finite superposition.

...

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A necessity arises: approximating polynomials

We will use the Poincar´e inequality, that is, given f ∈ W^1,p(Q), 1 ≤ p ≤ ∞,

kf − mQf k_Lp(Q). `(Q)k∇f kL^p(Q).

Equivalently, for any Sobolev function f with 0 mean on Q, kf k_Lp(Q). `(Q)k∇f kL^p(Q).

If we want to apply recursively the Poincar´e inequality we need Df to have mean 0 in 3Q for any partial derivative D.

Definition

Given f ∈ W^n,p(Ω) and a cube Q, we call Pⁿ_Qf to the polynomial of degree smaller than n restricted to Ω such that for any multiindex β with

|β| < n,

3Q

D^βPⁿ_Qf =

3Q

D^βf .

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A necessity arises: approximating polynomials

kf − mQf k_Lp(Q). `(Q)k∇f kL^p(Q). Equivalently, for any Sobolev function f with 0 mean on Q,

kf k_Lp(Q). `(Q)k∇f kL^p(Q).

Definition

|β| < n,

3Q

D^βPⁿ_Qf =

3Q

D^βf .

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A necessity arises: approximating polynomials

kf − mQf k_Lp(Q). `(Q)k∇f kL^p(Q). Equivalently, for any Sobolev function f with 0 mean on Q,

kf k_Lp(Q). `(Q)k∇f kL^p(Q).

Definition

|β| < n,

3Q

D^βPⁿ_Qf =

3Q

D^βf .

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Properties of approximating polynomials

P1.

f − Pⁿ_Qf

_L_p_(3Q). `(Q)ⁿk∇ⁿf k_Lp(3Q).

P2. Given two neighbor Whitney cubes Q1 and Q2, Pⁿ_Q₁f − Pⁿ_Q₂f

_L_∞_(3Q

1∩3Q₂). `(Q¹)ⁿ⁻²^pk∇ⁿf k_Lp(3Q1∪3Q2). ...

P5. We can bound the coefficients of the polynomial Pⁿ_Qf (w ) =P

|γ|<nmQ,γ(w − xQ)^γ:

|mQ,γ| .Pn−1 j =|γ|

∇^jf

_L∞(3Q)`(Q)^{j −|γ|}.

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Properties of approximating polynomials

P1.

f − Pⁿ_Qf

_L_∞_(3Q

1∩3Q₂). `(Q¹)ⁿ⁻²^pk∇ⁿf k_Lp(3Q1∪3Q2). ...

|mQ,γ| .Pn−1 j =|γ|

∇^jf

_L∞(3Q)`(Q)^{j −|γ|}.

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Properties of approximating polynomials

P1.

f − Pⁿ_Qf

_L_∞_(3Q

1∩3Q₂). `(Q¹)ⁿ⁻²^pk∇ⁿf k_Lp(3Q1∪3Q2). ...

|mQ,γ| .Pn−1 j =|γ|

∇^jf

_L∞(3Q)`(Q)^{j −|γ|}.

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Properties of approximating polynomials

P1.

f − Pⁿ_Qf

_L_∞_(3Q

1∩3Q₂). `(Q¹)ⁿ⁻²^pk∇ⁿf k_Lp(3Q1∪3Q2). ...

|mQ,γ| .Pn−1 j =|γ|

∇^jf

_L∞(3Q)`(Q)^{j −|γ|}.

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The proof: BP ∈ W

^n,p

(Ω) ⇒ kBf k

^p_Wn,p(Ω)

. kf k

^p_W^n,p_(Ω)

Assume that, we have a bound for the polynomials. Fix a point x0∈ Ω and call Pλ(z) = (z − x0)^λχΩ(z).

Given a cube Q, we can write

, using Newton’s binomial

Pⁿ_Qf (w ) = χΩ(w ) X

|γ|<n

mQ,γ(w − xQ)^γ

= χΩ(w ) X

|γ|<n

mQ,γ

X

(0,0)≤λ≤γ

γ λ

(w − x0)^λ(x0− xQ)^γ−λ

so

D^αB(Pⁿ_Qf )(z) = X

|γ|<n

mQ,γ

X

(0,0)≤λ≤γ

γ λ

(x0− xQ)^γ−λD^α(BPλ)(z)

where, by P5,

|m_Q,γ| .

n−1

X

j =|γ|

∇^jf

_L_∞_(3Q)`(Q)^{j −|γ|}.

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The proof: BP ∈ W

^n,p

(Ω) ⇒ kBf k

^p_Wn,p(Ω)

. kf k

^p_W^n,p_(Ω)

Given a cube Q, we can write

, using Newton’s binomial

Pⁿ_Qf (w ) = χΩ(w ) X

|γ|<n

mQ,γ(w − xQ)^γ

= χ_Ω(w ) X

|γ|<n

m_Q,γ X

(0,0)≤λ≤γ

γ λ

(w − x₀)^λ(x₀− x_Q)^γ−λ

so

|γ|<n

mQ,γ

X

(0,0)≤λ≤γ

γ λ

where, by P5,

|m_Q,γ| .

n−1

X

j =|γ|

∇^jf

_L_∞_(3Q)`(Q)^{j −|γ|}.

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The proof: BP ∈ W

^n,p

(Ω) ⇒ kBf k

^p_Wn,p(Ω)

. kf k

^p_W^n,p_(Ω)

Given a cube Q, we can write, using Newton’s binomial Pⁿ_Qf (w ) = χΩ(w ) X

|γ|<n

mQ,γ(w − xQ)^γ

= χ_Ω(w ) X

|γ|<n

m_Q,γ X

(0,0)≤λ≤γ

γ λ

(w − x₀)^λ(x₀− x_Q)^γ−λ

so

|γ|<n

mQ,γ

X

(0,0)≤λ≤γ

γ λ

where, by P5,

|m_Q,γ| .

n−1

X

j =|γ|

∇^jf

_L_∞_(3Q)`(Q)^{j −|γ|}.

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The proof: BP ∈ W

^n,p

(Ω) ⇒ kBf k

^p_Wn,p(Ω)

. kf k

^p_W^n,p_(Ω)

|γ|<n

mQ,γ(w − xQ)^γ

= χ_Ω(w ) X

|γ|<n

m_Q,γ X

(0,0)≤λ≤γ

γ λ

(w − x₀)^λ(x₀− x_Q)^γ−λ

so

|γ|<n

mQ,γ

X

(0,0)≤λ≤γ

γ λ

where, by P5,

|m_Q,γ| .

n−1

X

j =|γ|

∇^jf

_L_∞_(3Q)`(Q)^{j −|γ|}.

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The proof: BP ∈ W

^n,p

(Ω) ⇒ kBf k

^p_Wn,p(Ω)

. kf k

^p_W^n,p_(Ω)

|γ|<n

mQ,γ(w − xQ)^γ

= χ_Ω(w ) X

|γ|<n

m_Q,γ X

(0,0)≤λ≤γ

γ λ

(w − x₀)^λ(x₀− x_Q)^γ−λ

so

|γ|<n

mQ,γ

X

(0,0)≤λ≤γ

γ λ

where, by P5,

|mQ,γ| .

n−1

X

j =|γ|

∇^jf

_L_∞_(3Q)`(Q)^{j −|γ|}.

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The Sobolev Embedding Theorem appears

Thus

D^αB(Pⁿ_Qf )

p

L^p(Q).X

j <n

∇^jf

p L^∞

X

|γ|≤j 0≤λ≤γ

kD^αBPλk^p_Lp(Q)H¹(∂Ω)^{(j −|λ|)p}.

Adding with respect to Q ∈ W, by the Sobolev Embedding Theorem (

∇^jf

_L_∞_(Q∩Ω)≤ C ∇^jf

_W_1,p_(Q∩Ω)when p > 2), we get X

Q∈W

D^αB(Pⁿ_Qf )

p

L^p(Q).X

j <n

∇^jf

p

W^1,p(Q∩Ω)

X

0≤λ≤γ

kBPλk^p_Wn,p(Ω)

. kf k^pW^n,p(Q∩Ω).

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The Sobolev Embedding Theorem appears

Thus

D^αB(Pⁿ_Qf )

p

L^p(Q).X

j <n

∇^jf

p L^∞

X

|γ|≤j 0≤λ≤γ

kD^αBPλk^p_Lp(Q)H¹(∂Ω)^{(j −|λ|)p}.

Adding with respect to Q ∈ W, by the Sobolev Embedding Theorem (

∇^jf

_L_∞_(Q∩Ω)≤ C ∇^jf

_W_1,p_(Q∩Ω)when p > 2), we get X

Q∈W

D^αB(Pⁿ_Qf )

p

L^p(Q).X

j <n

∇^jf

p

W^1,p(Q∩Ω)

X

0≤λ≤γ

kBPλk^p_Wn,p(Ω)

. kf k^pW^n,p(Q∩Ω).

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Key Lemma: sticking to the essential

Lemma

Let Ω be a Lipschitz domain, Q a window, ψ ∈ C^∞(₁₀₀⁹⁹Q) with ∇^jψ

_L∞. R¹^j for j ≥ 0. Then, for any |α| = n and f = ψ · ef with ef ∈ W^n,p(Ω), TFAE:

kD^αBf k^p_Lp(Q). kf k^pW^n,p(Q∩Ω). P

Q∈W

D^αB(Pⁿ_Qf )

p

L^p(Q). kf k^pW^n,p(Q∩Ω).

Idea of the proof: separate local and non-local parts of the error term, D^αBf (z) − D^αB(Pⁿ_Qf )(z)

= D^αB(χ_2Q(f − Pⁿ_Qf ))(z) + D^αB((1 − χ_2Q)(f − Pⁿ_Qf ))(z).

Sketch of the proof

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Key Lemma: sticking to the essential

Lemma

Let Ω be a Lipschitz domain, Q a window, ψ ∈ C^∞(₁₀₀⁹⁹Q) with ∇^jψ

_L∞. R¹^j for j ≥ 0. Then, for any |α| = n and f = ψ · ef with ef ∈ W^n,p(Ω), TFAE:

kD^αBf k^p_Lp(Q). kf k^pW^n,p(Q∩Ω). P

Q∈W

D^αB(Pⁿ_Qf )

p

L^p(Q). kf k^pW^n,p(Q∩Ω).

Idea of the proof: separate local and non-local parts of the error term, D^αBf (z) − D^αB(Pⁿ_Qf )(z)

= D^αB(χ_2Q(f − Pⁿ_Qf ))(z) + D^αB((1 − χ_2Q)(f − Pⁿ_Qf ))(z).

Sketch of the proof

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A geometric condition for the Beurling transform

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k∂ⁿBχΩk^p_Lp(Ω). kNk^p_B^n−1/p

p,p (∂Ω)+ H¹(∂Ω)^2−np.

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k∂ⁿBχΩk^p_Lp(Ω). kNk^p_B^n−1/p

p,p (∂Ω)+ H¹(∂Ω)^2−np.

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Defining some generalized betas of David-Semmes

A measure of the flatness of a set Γ:

Definition (P. Jones) βΓ(Q) = infV w (V )

`(Q)

If there is no risk of confusion, we will write just β_(n)(I ).

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Defining some generalized betas of David-Semmes

A measure of the flatness of a set Γ:

Definition (P. Jones) βΓ(Q) = infV w (V )

`(Q)

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Defining some generalized betas of David-Semmes

The graph of a function y = A(x ):

Consider I ⊂ R, and define

Definition

β_∞(I , A) = inf_P∈P1

A−P

`(I )

∞

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Defining some generalized betas of David-Semmes

Consider I ⊂ R, and define Definition

β_∞(I , A) = inf_P∈P1

A−P

`(I )

∞

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Defining some generalized betas of David-Semmes

βp(I , A) = inf_P∈P1 1

`(I )

1 p

A−P

`(I )

_p

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Defining some generalized betas of David-Semmes

β_(n)(I , A) = infP∈Pⁿ 1

`(I )

A−P

`(I )

₁ If there is no risk of confusion, we will write just β_(n)(I ).

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Relation between β

_(n)

and B

_p,pⁿ

Theorem (Dorronsoro)

Let f : R → R be a function in the homogeneous Besov space ˙Bp,p^s . Then, for any n ≥ [s],

kf k^p_˙

B_p,p^s ≈X

I ∈D

β_(n)(I )

`(I )^s−1

^p

`(I ).

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Local charts: Whitney decomposition

First order derivative Second order derivative Higher order derivatives Skip higher order derivatives

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Local charts: Whitney decomposition

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Local charts: Whitney decomposition

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Local charts: Bounds for the first derivative

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Local charts: Bounds for the first derivative

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Local charts: Bounds for the first derivative

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Local charts: Second order derivative

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Local charts: Second order derivative

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Local charts: Higher order derivatives

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Local charts: Higher order derivatives

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Bounding the polynomial region

We can choose the window length R small enough so that

Proposition

If we denote by Ω_Q the region with boundary a minimizing polynomial for β_(n)(Φ(Q)), we get

∂ⁿBχ_Ω_Q ≤ C

Rⁿ.

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Bounding the polynomial region

We can choose the window length R small enough so that Proposition

If we denote by Ω_Q the region with boundary a minimizing polynomial for β_(n)(Φ(Q)), we get

∂ⁿBχ_Ω_Q ≤ C

Rⁿ.

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Bounding the interstitial region

Proposition

Choosing a minimizing polynomial for β_(n)(Φ(Q)), we get ˆ

Ω∆ΩQ

dm(w )

|z − w |ⁿ⁺² . X

I ∈D Φ(Q)⊂I ⊂Φ(Q_k)

β(n)(I )

`(I )ⁿ + 1 Rⁿ.