December13and15,2021 JulianWeigt HigherDimensionalTechniquesfortheRegularityofMaximalFunctions

History Core Techniques Covering Techniques Summary References

Higher Dimensional Techniques for the Regularity of Maximal Functions

Julian Weigt

Aalto University

December 13 and 15, 2021

History Core Techniques Covering Techniques Summary References

Outline

1 History Background

Onedimensional case

2 Core Techniques

Reduction and decomposition High density case

Low density case

3 Covering Techniques Boundary of large balls High density, general version Dyadic cubes to general cubes

4 Summary

History Core Techniques Covering Techniques Summary References

History

1 History Background

Onedimensional case

2 Core Techniques

Reduction and decomposition High density case

Low density case

3 Covering Techniques Boundary of large balls High density, general version Dyadic cubes to general cubes

4 Summary

History Core Techniques Covering Techniques Summary References

Background

1 History Background

Onedimensional case

2 Core Techniques

Reduction and decomposition High density case

Low density case

3 Covering Techniques Boundary of large balls High density, general version Dyadic cubes to general cubes

4 Summary

History Core Techniques Covering Techniques Summary References

Background

Forf :R^d →Rthe centered Hardy-Littlewood maximal function is defined by

M^cf(x) = sup

r>0

f_B(x,r) with f_B(x,r) = 1 L(B(x,r))

ˆ

B(x,r)|f|.

The Hardy-Littlewood maximal function theorem:

kM^cfkL^p(R^d)≤C_d,pkfkL^p(R^d) if and only ifp >1

Juha Kinnunen (1997):

k∇M^cfkL^p(R^d)≤C_d,pk∇fkL^p(R^d) ifp >1

Question (Haj lasz and Onninen 2004) Is it true that

k∇M^cfkL¹(R^d)≤C_dk∇fkL¹(R^d)?

History Core Techniques Covering Techniques Summary References

Background

Forf :R^d →Rthe centered Hardy-Littlewood maximal function is defined by

M^cf(x) = sup

r>0

f_B(x,r) with f_B(x,r) = 1 L(B(x,r))

ˆ

B(x,r)|f|. The Hardy-Littlewood maximal function theorem:

kM^cfkL^p(R^d)≤C_d,pkfkL^p(R^d) if and only ifp >1

Juha Kinnunen (1997):

k∇M^cfkL^p(R^d)≤C_d,pk∇fkL^p(R^d) ifp >1

Question (Haj lasz and Onninen 2004) Is it true that

k∇M^cfkL¹(R^d)≤C_dk∇fkL¹(R^d)?

History Core Techniques Covering Techniques Summary References

Background

Forf :R^d →Rthe centered Hardy-Littlewood maximal function is defined by

M^cf(x) = sup

r>0

f_B(x,r) with f_B(x,r) = 1 L(B(x,r))

ˆ

B(x,r)|f|. The Hardy-Littlewood maximal function theorem:

kM^cfkL^p(R^d)≤C_d,pkfkL^p(R^d) if and only ifp >1 Juha Kinnunen (1997):

k∇M^cfkL^p(R^d)≤C_d,pk∇fkL^p(R^d) ifp >1

Question (Haj lasz and Onninen 2004) Is it true that

k∇M^cfkL¹(R^d)≤C_dk∇fkL¹(R^d)?

History Core Techniques Covering Techniques Summary References

Background

Forf :R^d →Rthe centered Hardy-Littlewood maximal function is defined by

M^cf(x) = sup

r>0

f_B(x,r) with f_B(x,r) = 1 L(B(x,r))

ˆ

B(x,r)|f|. The Hardy-Littlewood maximal function theorem:

kM^cfkL^p(R^d)≤C_d,pkfkL^p(R^d) if and only ifp >1 Juha Kinnunen (1997):

k∇M^cfkL^p(R^d)≤C_d,pk∇fkL^p(R^d) ifp >1

Question (Haj lasz and Onninen 2004) Is it true that

k∇M^cfkL¹(R^d)≤C_dk∇fkL¹(R^d)?

History Core Techniques Covering Techniques Summary References

Proof

Fore ∈R^d by the sublinearity of M^c

∂_eM^cf(x)∼ M^cf(x+he)−M^cf(x) h

≤ M^c(f(·+he)−f)(x) h

=M^cf(·+he)−f) h

(x)∼M^c(∂_ef)(x) By the Hardy-Littlewood maximal function theorem forp >1

k∇M^cfkL^p(R^d).kM^c(|∇f|)kL^p(R^d) .k∇fkL^p(R^d)

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Proof

Fore ∈R^d by the sublinearity of M^c

∂_eM^cf(x)∼ M^cf(x+he)−M^cf(x) h

≤ M^c(f(·+he)−f)(x) h

=M^cf(·+he)−f) h

(x)∼M^c(∂_ef)(x) By the Hardy-Littlewood maximal function theorem forp >1

k∇M^cfkL^p(R^d).kM^c(|∇f|)kL^p(R^d) .k∇fkL^p(R^d)

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Proof

Fore ∈R^d by the sublinearity of M^c

∂_eM^cf(x)∼ M^cf(x+he)−M^cf(x) h

≤ M^c(f(·+he)−f)(x) h

=M^cf(·+he)−f) h

(x)

∼M^c(∂_ef)(x) By the Hardy-Littlewood maximal function theorem forp >1

k∇M^cfkL^p(R^d).kM^c(|∇f|)kL^p(R^d) .k∇fkL^p(R^d)

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Proof

Fore ∈R^d by the sublinearity of M^c

∂_eM^cf(x)∼ M^cf(x+he)−M^cf(x) h

≤ M^c(f(·+he)−f)(x) h

=M^cf(·+he)−f) h

(x)∼M^c(∂_ef)(x)

By the Hardy-Littlewood maximal function theorem forp >1 k∇M^cfkL^p(R^d).kM^c(|∇f|)kL^p(R^d) .k∇fkL^p(R^d)

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Proof

Fore ∈R^d by the sublinearity of M^c

∂_eM^cf(x)∼ M^cf(x+he)−M^cf(x) h

≤ M^c(f(·+he)−f)(x) h

=M^cf(·+he)−f) h

(x)∼M^c(∂_ef)(x) By the Hardy-Littlewood maximal function theorem forp >1

k∇M^cfkL^p(R^d).kM^c(|∇f|)kL^p(R^d) .k∇fkL^p(R^d)

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Uncentered maximal operator

Forf :R^d →Rthe uncentered Hardy-Littlewood maximal function is defined by

Mfe (x) = sup

B3x

f_B.

The result by Kinnunen also holds forMe and various other maximal operators, and the question by Ha ljasz and Onninen is being investigated.

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Uncentered maximal operator

Forf :R^d →Rthe uncentered Hardy-Littlewood maximal function is defined by

Mfe (x) = sup

B3x

f_B.

The result by Kinnunen also holds forMe and various other maximal operators, and the question by Ha ljasz and Onninen is being investigated.

History Core Techniques Covering Techniques Summary References

Onedimensional case

1 History Background

Onedimensional case

2 Core Techniques

Reduction and decomposition High density case

Low density case

3 Covering Techniques Boundary of large balls High density, general version Dyadic cubes to general cubes

4 Summary

History Core Techniques Covering Techniques Summary References

Onedimensional case

In 2002 Tanaka proved

varMfe ≤varf

forf :R→R, but with a factor 2 on the right hand side. In 2007 Aldaz and P´erez L´azaro reduced that factor to the optimal value 1.

They use that in one dimension we have varf = sup

n∈N,x1<...<xn

n−1X

i=1

|f(xn+1)−f(xn)|.

Main ingredient: Mfe is convex on connected components of {x∈R:Mfe (x)>f(x)}.

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Onedimensional case

In 2002 Tanaka proved

varMfe ≤varf

forf :R→R, but with a factor 2 on the right hand side. In 2007 Aldaz and P´erez L´azaro reduced that factor to the optimal value 1.

They use that in one dimension we have varf = sup

n∈N,x1<...<xn

n−1X

i=1

|f(xn+1)−f(xn)|.

Main ingredient: Mfe is convex on connected components of {x∈R:Mfe (x)>f(x)}.

History Core Techniques Covering Techniques Summary References

Onedimensional case

In 2002 Tanaka proved

varMfe ≤varf

forf :R→R, but with a factor 2 on the right hand side. In 2007 Aldaz and P´erez L´azaro reduced that factor to the optimal value 1.

They use that in one dimension we have varf = sup

n∈N,x1<...<xn

n−1X

i=1

|f(xn+1)−f(xn)|.

Main ingredient: Mfe is convex on connected components of {x∈R:Mfe (x)>f(x)}.

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x₀ x₁ x₂

˜f Mf

varMfe = var_[0,x0]Mfe + var_[x2,1]Mfe

+|Mfe (x₀)−Mfe (x₁)|+|Mfe (x₂)−Mfe (x₁)|

≤var_[0,x0]f + var_[x2,1]f

+|f(x₀)−f(x₁)|+|f(x₂)−f(x₁)|

≤var_[0,x0]f + var_[x2,1]f + var_[x0,x2]f = varf

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x₀ x₁ x₂

˜f Mf

varMfe = var_[0,x0]Mfe + var_[x2,1]Mfe

+|Mfe (x₀)−Mfe (x₁)|+|Mfe (x₂)−Mfe (x₁)|

≤var_[0,x0]f + var_[x2,1]f

+|f(x₀)−f(x₁)|+|f(x₂)−f(x₁)|

≤var_[0,x0]f + var_[x2,1]f + var_[x0,x2]f = varf

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x₀ x₁ x₂

˜f Mf

varMfe = var_[0,x0]Mfe + var_[x2,1]Mfe

+|Mfe (x₀)−Mfe (x₁)|+|Mfe (x₂)−Mfe (x₁)|

≤var_[0,x0]f + var_[x2,1]f

+|f(x₀)−f(x₁)|+|f(x₂)−f(x₁)|

≤var_[0,x0]f + var_[x2,1]f + var_[x0,x2]f = varf

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x₀ x₁ x₂

˜f Mf

varMfe = var_[0,x0]Mfe + var_[x2,1]Mfe

+|Mfe (x₀)−Mfe (x₁)|+|Mfe (x₂)−Mfe (x₁)|

≤var_[0,x0]f + var_[x2,1]f

+|f(x₀)−f(x₁)|+|f(x₂)−f(x₁)|

≤var_[0,x0]f + var_[x2,1]f + var_[x0,x2]f

= varf

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x₀ x₁ x₂

˜f Mf

varMfe = var_[0,x0]Mfe + var_[x2,1]Mfe

+|Mfe (x₀)−Mfe (x₁)|+|Mfe (x₂)−Mfe (x₁)|

≤var_[0,x0]f + var_[x2,1]f

+|f(x₀)−f(x₁)|+|f(x₂)−f(x₁)|

≤var_[0,x0]f + var_[x2,1]f + var_[x0,x2]f = varf

History Core Techniques Covering Techniques Summary References

Onedimensional case

For the centered maximal functionM^cf the convexity property does not hold. Nevertheless,

centered

Kurka proved varM^cf ≤Cvarf for f :R→R in a very involved paper in 2015.

He did case distinctions with respect to the shape of triples x₀ <x₁<x₂ withM^cf(x₀)<M^cf(x₁)>M^cf(x₂) and a decomposition in scales.

History Core Techniques Covering Techniques Summary References

Onedimensional case

For the centered maximal functionM^cf the convexity property does not hold. Nevertheless,

centered

Kurka proved varM^cf ≤Cvarf for f :R→R in a very involved paper in 2015.

He did case distinctions with respect to the shape of triples x₀ <x₁<x₂ withM^cf(x₀)<M^cf(x₁)>M^cf(x₂) and a decomposition in scales.

History Core Techniques Covering Techniques Summary References

Onedimensional case

For radial functionsf :R^d →R with f(x) =f(|x|) we have k∇fkL¹(R^d) =

ˆ ∞

0 |∇f(r)|r^d−1dr and alsoMfe is radial.

radial

In 2018 Luiro used this one-dimensional representation to prove k∇Mfe kL¹(R^d)≤C_dk∇fkL¹(R^d) for radial functionsf :R^d →R. block-decreasing

In 2009 Aldaz and P´erez L´azaro proved

k∇Mfe kL¹(R^d)≤C_dk∇fkL¹(R^d) for block-decreasing f :R^d →R, which are to some extent similar to radially decreasing functions.

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Onedimensional case

For radial functionsf :R^d →R with f(x) =f(|x|) we have k∇fkL¹(R^d) =

ˆ ∞

0 |∇f(r)|r^d−1dr and alsoMfe is radial.

radial

In 2018 Luiro used this one-dimensional representation to prove k∇Mfe kL¹(R^d)≤C_dk∇fkL¹(R^d) for radial functionsf :R^d →R.

block-decreasing

In 2009 Aldaz and P´erez L´azaro proved

k∇Mfe kL¹(R^d)≤C_dk∇fkL¹(R^d) for block-decreasing f :R^d →R, which are to some extent similar to radially decreasing functions.

History Core Techniques Covering Techniques Summary References

Onedimensional case

For radial functionsf :R^d →R with f(x) =f(|x|) we have k∇fkL¹(R^d) =

ˆ ∞

0 |∇f(r)|r^d−1dr and alsoMfe is radial.

radial

In 2018 Luiro used this one-dimensional representation to prove k∇Mfe kL¹(R^d)≤C_dk∇fkL¹(R^d) for radial functionsf :R^d →R. block-decreasing

In 2009 Aldaz and P´erez L´azaro proved

k∇Mfe kL¹(R^d)≤C_dk∇fkL¹(R^d) for block-decreasing f :R^d →R, which are to some extent similar to radially decreasing functions.

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Other maximal operators and related questions

fractional maximal operators convolution operators local maximal operators discrete maximal operators bilinear maximal operators

any combinations of the above

bounds on other spaces than Sobolev spaces

related: Continuity of the operator given byf 7→ ∇Mf on W^1,1(R^d)→L¹(R^d). This is a stronger property than boundedness.

History Core Techniques Covering Techniques Summary References

Other maximal operators and related questions

fractional maximal operators convolution operators local maximal operators discrete maximal operators bilinear maximal operators any combinations of the above

bounds on other spaces than Sobolev spaces

related: Continuity of the operator given byf 7→ ∇Mf on W^1,1(R^d)→L¹(R^d). This is a stronger property than boundedness.

History Core Techniques Covering Techniques Summary References

Other maximal operators and related questions

fractional maximal operators convolution operators local maximal operators discrete maximal operators bilinear maximal operators any combinations of the above

bounds on other spaces than Sobolev spaces

related: Continuity of the operator given byf 7→ ∇Mf on W^1,1(R^d)→L¹(R^d). This is a stronger property than boundedness.

History Core Techniques Covering Techniques Summary References

Other maximal operators and related questions

fractional maximal operators convolution operators local maximal operators discrete maximal operators bilinear maximal operators any combinations of the above

bounds on other spaces than Sobolev spaces

related: Continuity of the operator given byf 7→ ∇Mf on W^1,1(R^d)→L¹(R^d). This is a stronger property than boundedness.

History Core Techniques Covering Techniques Summary References

Core Techniques

1 History Background

Onedimensional case

2 Core Techniques

Reduction and decomposition High density case

Low density case

3 Covering Techniques Boundary of large balls High density, general version Dyadic cubes to general cubes

4 Summary

History Core Techniques Covering Techniques Summary References

Reduction and decomposition

1 History Background

Onedimensional case

2 Core Techniques

Reduction and decomposition High density case

Low density case

3 Covering Techniques Boundary of large balls High density, general version Dyadic cubes to general cubes

4 Summary

History Core Techniques Covering Techniques Summary References

reformulations

definition

varf = supnˆ

f divϕ:ϕ∈C_c¹(R^d;R^d), |ϕ| ≤1o

=k∇fkL¹(R^d) iff ∈W^1,1(R^d). coarea formula

varf = ˆ

R

H^d−1(∂{x∈R^d :f(x)> λ})dλ

superlevel sets

{Mf > λ}=

{x∈R^d :Mf(x)> λ}=[

{B :f_B > λ} foruncentered maximal operators.

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reformulations

definition

varf = supnˆ

f divϕ:ϕ∈C_c¹(R^d;R^d), |ϕ| ≤1o

=k∇fkL¹(R^d) iff ∈W^1,1(R^d).

coarea formula varf =

ˆ

R

H^d−1(∂{x∈R^d :f(x)> λ})dλ

superlevel sets

{Mf > λ}=

{x∈R^d :Mf(x)> λ}=[

{B :f_B > λ} foruncentered maximal operators.

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reformulations

definition

varf = supnˆ

f divϕ:ϕ∈C_c¹(R^d;R^d), |ϕ| ≤1o

=k∇fkL¹(R^d) iff ∈W^1,1(R^d).

coarea formula varf =

ˆ

R

H^d−1(∂{x∈R^d :f(x)> λ})dλ

superlevel sets

{Mf > λ}=

{x∈R^d :Mf(x)> λ}=[

{B :f_B > λ} foruncentered maximal operators.

History Core Techniques Covering Techniques Summary References

reformulations

definition

varf = supnˆ

f divϕ:ϕ∈C_c¹(R^d;R^d), |ϕ| ≤1o

=k∇fkL¹(R^d) iff ∈W^1,1(R^d).

coarea formula varf =

ˆ

R

H^d−1(∂{x∈R^d :f(x)> λ})dλ

superlevel sets

{Mf > λ}=

{x∈R^d :Mf(x)> λ}=[

{B:f_B > λ} foruncentered maximal operators.

History Core Techniques Covering Techniques Summary References

reformulations

definition

varf = supnˆ

f divϕ:ϕ∈C_c¹(R^d;R^d), |ϕ| ≤1o

=k∇fkL¹(R^d) iff ∈W^1,1(R^d).

coarea formula varf =

ˆ

R

H^d−1(∂{x∈R^d :f(x)> λ})dλ

superlevel sets

{Mf > λ}={x∈R^d :Mf(x)> λ}=[

{B:f_B > λ} foruncentered maximal operators.

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Denote

B^<λ ={B :f_B > λ, L(B∩ {f > λ})<L(B)/2} andB_λ^≥ accordingly. We split the boundary

∂[

{B :f_B > λ} ⊂∂[

B^<λ ∪∂[

B_λ^≥. (1)

SinceMf ≥f a.e. we have {f > λ} ⊂ {Mf > λ} up to measure zero, and thus

∂[

{B :f_B > λ} ⊂

∂[

{B:f_B > λ}

\ {f > λ} ∪∂{f > λ}. (2) Plug (1) into (2) and that into the coarea formula

varMf = ˆ ∞

0 H^d−1

∂[

{B :fB > λ} dλ.

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Denote

B^<λ ={B :f_B > λ, L(B∩ {f > λ})<L(B)/2} andB_λ^≥ accordingly. We split the boundary

∂[

{B :f_B > λ} ⊂∂[

B^<λ ∪∂[

B_λ^≥. (1) SinceMf ≥f a.e. we have {f > λ} ⊂ {Mf > λ} up to measure zero, and thus

∂[

{B :f_B > λ} ⊂

∂[

{B:f_B > λ}

\ {f > λ} ∪∂{f > λ}. (2)

Plug (1) into (2) and that into the coarea formula varMf =

ˆ ∞

0 H^d−1

∂[

{B :fB > λ} dλ.

History Core Techniques Covering Techniques Summary References

Denote

B^<λ ={B :f_B > λ, L(B∩ {f > λ})<L(B)/2} andB_λ^≥ accordingly. We split the boundary

∂[

{B :f_B > λ} ⊂∂[

B^<λ ∪∂[

B_λ^≥. (1) SinceMf ≥f a.e. we have {f > λ} ⊂ {Mf > λ} up to measure zero, and thus

∂[

{B :f_B > λ} ⊂

∂[

{B:f_B > λ}

\ {f > λ} ∪∂{f > λ}. (2) Plug (1) into (2) and that into the coarea formula

varMf = ˆ ∞

0

H^d−1

∂[

{B :fB > λ} dλ.

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Decomposition of the boundary

decomposition

varMf ≤ ˆ _∞

0

H^d−1

∂[

B^<_λ dλ +

ˆ ∞

0 H^d−1

∂[

B_λ^≥

\ {f > λ} dλ + varf

X

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Decomposition of the boundary

decomposition

varMf ≤ ˆ _∞

0

H^d−1

∂[

B^<_λ dλ +

ˆ ∞

0 H^d−1

∂[

B_λ^≥

\ {f > λ} dλ + varf X

History Core Techniques Covering Techniques Summary References

High density case

1 History Background

Onedimensional case

2 Core Techniques

Reduction and decomposition High density case

Low density case

3 Covering Techniques Boundary of large balls High density, general version Dyadic cubes to general cubes

4 Summary

History Core Techniques Covering Techniques Summary References

Relative isoperimetric inequality

Ais a John domainif there is a K >0 and point x∈Asuch that for anyy∈Athere is a pathγ fromx toy with

dist(γ(t),A^{)≥K⁻¹|γ(t)−y|.

Relative isoperimetric inequality

LetAbe a John domain and L(A∩E)≤ L(A)/2. Then L(A∩E)^d−1d .H^d−1(A∩∂E)

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Relative isoperimetric inequality

Ais a John domainif there is a K >0 and point x∈Asuch that for anyy∈Athere is a pathγ fromx toy with

dist(γ(t),A^{)≥K⁻¹|γ(t)−y|.

Relative isoperimetric inequality

LetAbe a John domain and L(A∩E)≤ L(A)/2. Then L(A∩E)^d−1d .H^d−1(A∩∂E)

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High density case

Corollary: For a ball or cubeB with L(B)/4≤ L(B∩E)≤ L(B)/2 we have

H^d−1(∂B).L(B)^d−1d .L(B∩E)^d−1d .H^d−1(B∩∂E).

Proposition (High density) ForL(B∩E)≥ L(B)/2 we have

H^d−1(∂B\E).H^d−1(B∩∂E).

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High density case

Corollary: For a ball or cubeB with L(B)/4≤ L(B∩E)≤ L(B)/2 we have

H^d−1(∂B).L(B)^d−1d .L(B∩E)^d−1d .H^d−1(B∩∂E).

Proposition (High density) ForL(B∩E)≥ L(B)/2 we have

H^d−1(∂B\E).H^d−1(B∩∂E).

E

Q

L(Q∩E)≥εL(Q)

=⇒ H^d−1(∂Q\E).εH^d−1(Q∩∂E)

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Proof of high density proposition

Idea: Decompose∂B\E according to distance to significant part ofE.

For everyx∈∂B\E there is an ε >0 with L(B(x, ε)∩E) = 0,

L(B∩B(x,diam(B))∩E)≥ L(B)/2 = 2^−d−1L(B(x,diam(B))) Thus∃r ∈[ε,diam(B)]

L(B(x,r)∩E) = 2^−d−1L(B(x,r))

LetB be the collection of all such ballsB(x,r) and apply the Vitali covering. LetS be the resulting disjoint subset.

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Proof of high density proposition

Idea: Decompose∂B\E according to distance to significant part ofE.

For everyx∈∂B\E there is an ε >0 with L(B(x, ε)∩E) = 0,

L(B∩B(x,diam(B))∩E)≥ L(B)/2 = 2^−d−1L(B(x,diam(B)))

Thus∃r ∈[ε,diam(B)]

L(B(x,r)∩E) = 2^−d−1L(B(x,r))

LetB be the collection of all such ballsB(x,r) and apply the Vitali covering. LetS be the resulting disjoint subset.

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Proof of high density proposition

Idea: Decompose∂B\E according to distance to significant part ofE.

For everyx∈∂B\E there is an ε >0 with L(B(x, ε)∩E) = 0,

L(B∩B(x,diam(B))∩E)≥ L(B)/2 = 2^−d−1L(B(x,diam(B))) Thus∃r ∈[ε,diam(B)]

L(B(x,r)∩E) = 2^−d−1L(B(x,r))

LetB be the collection of all such ballsB(x,r) and apply the Vitali covering. LetS be the resulting disjoint subset.

History Core Techniques Covering Techniques Summary References

Proof of high density proposition

Idea: Decompose∂B\E according to distance to significant part ofE.

For everyx∈∂B\E there is an ε >0 with L(B(x, ε)∩E) = 0,

L(B∩B(x,diam(B))∩E)≥ L(B)/2 = 2^−d−1L(B(x,diam(B))) Thus∃r ∈[ε,diam(B)]

L(B(x,r)∩E) = 2^−d−1L(B(x,r))

LetB be the collection of all such ballsB(x,r) and apply the Vitali covering. LetS be the resulting disjoint subset.

History Core Techniques Covering Techniques Summary References

Relative isoperimetric inequality

For eachB(x,r)∈ S the setA=B∩B(x,r) is a John domain and thus satisfies the

relative isoperimetric inequality

min{L(A∩E),L(A\E)}^d−1d .H^d−1(∂E ∩A)

Thus by the choice ofr

H^d−1(∂B(x,r)).L(B∩B(x,r))^d−1d

.H^d−1(∂E ∩B∩B(x,r)). (Proof of first inequality can be made precise.)

History Core Techniques Covering Techniques Summary References

Relative isoperimetric inequality

For eachB(x,r)∈ S the setA=B∩B(x,r) is a John domain and thus satisfies the

relative isoperimetric inequality

min{L(A∩E),L(A\E)}^d−1d .H^d−1(∂E ∩A) Thus by the choice ofr

H^d−1(∂B(x,r)).L(B∩B(x,r))^d−1d

.H^d−1(∂E ∩B∩B(x,r)).

(Proof of first inequality can be made precise.)

History Core Techniques Covering Techniques Summary References

Relative isoperimetric inequality

For eachB(x,r)∈ S the setA=B∩B(x,r) is a John domain and thus satisfies the

relative isoperimetric inequality

min{L(A∩E),L(A\E)}^d−1d .H^d−1(∂E ∩A) Thus by the choice ofr

H^d−1(∂B(x,r)).L(B∩B(x,r))^d−1d

.H^d−1(∂E ∩B∩B(x,r)).

(Proof of first inequality can be made precise.)

History Core Techniques Covering Techniques Summary References

S Vitali covering of∂B\E. We can conclude H^d−1

∂B\E

=H^d−1[

B ∩∂B\E

≤ H^d−1[

B ∩∂B

=H^d−1[

5S ∩∂B

≤X

S∈S

H^d−1(5S∩∂B) . X

S∈S

H^d−1(∂5S).X

S∈S

H^d−1(∂S) . X

S∈S

H^d−1(∂E ∩B∩S)≤ H^d−1(∂E∩B)

(Proof of fifth step can be made precise.)

History Core Techniques Covering Techniques Summary References

S Vitali covering of∂B\E. We can conclude H^d−1

∂B\E

=H^d−1[

B ∩∂B\E

≤ H^d−1[

B ∩∂B

=H^d−1[

5S ∩∂B

≤X

S∈S

H^d−1(5S∩∂B) . X

S∈S

H^d−1(∂5S).X

S∈S

H^d−1(∂S) . X

S∈S

H^d−1(∂E ∩B∩S)≤ H^d−1(∂E∩B) (Proof of fifth step can be made precise.)

History Core Techniques Covering Techniques Summary References

High density case

Proposition (High density, general version)

LetB be a set of ballsB with L(B∩E)≥εL(B). Then H^d−1

∂[ B \E

.εH^d−1[

B ∩∂E

.

ˆ _∞

0

H^d−1

∂[

B^≥_λ

\ {f > λ} dλ .

ˆ ∞

0 H^d−1[

B^≥_λ ∩∂{f > λ} dλ

≤varf.

Proof works almost the same as withB={B} if all balls inB have the same scale. But we need one extra covering tool from the next section. Then we prove a modified version for each scale separately and add up all scales.

History Core Techniques Covering Techniques Summary References

High density case

Proposition (High density, general version)

LetB be a set of ballsB with L(B∩E)≥εL(B). Then H^d−1

∂[ B \E

.εH^d−1[

B ∩∂E

.

ˆ _∞

0 H^d−1

∂[

B^≥_λ

\ {f > λ} dλ .

ˆ ∞

0 H^d−1[

B^≥_λ ∩∂{f > λ} dλ

≤varf.

Proof works almost the same as withB={B} if all balls inB have the same scale. But we need one extra covering tool from the next section. Then we prove a modified version for each scale separately and add up all scales.

History Core Techniques Covering Techniques Summary References

High density case

Proposition (High density, general version)

LetB be a set of ballsB with L(B∩E)≥εL(B). Then H^d−1

∂[ B \E

.εH^d−1[

B ∩∂E

.

ˆ _∞

0 H^d−1

∂[

B^≥_λ

\ {f > λ} dλ .

ˆ ∞

0 H^d−1[

B^≥_λ ∩∂{f > λ} dλ

≤varf.

Proof works almost the same as withB={B} if all balls inB have the same scale. But we need one extra covering tool from the next section.

Then we prove a modified version for each scale separately and add up all scales.

History Core Techniques Covering Techniques Summary References

High density case

Proposition (High density, general version)

LetB be a set of ballsB with L(B∩E)≥εL(B). Then H^d−1

∂[ B \E

.εH^d−1[

B ∩∂E

.

ˆ _∞

0 H^d−1

∂[

B^≥_λ

\ {f > λ} dλ .

ˆ ∞

0 H^d−1[

B^≥_λ ∩∂{f > λ} dλ

≤varf.

Proof works almost the same as withB={B} if all balls inB have the same scale. But we need one extra covering tool from the next section. Then we prove a modified version for each scale separately and add up all scales.

History Core Techniques Covering Techniques Summary References

Low density case

1 History Background

Onedimensional case

2 Core Techniques

Reduction and decomposition High density case

Low density case

3 Covering Techniques Boundary of large balls High density, general version Dyadic cubes to general cubes

4 Summary

History Core Techniques Covering Techniques Summary References

Low density case

Have to bound ˆ ∞

0 H^d−1

∂[

B_λ^<

dλ.varf, where

B^<_λ ={B :fB > λ, L(B∩ {f > λ})<L(B)/2}.

I can’t :(

dyadic maximal operator

M^df(x) = sup

Q3x,Qdyadic

f_Q.

{x:M^df(x)> λ}=[

{maximal dyadicQ :f_Q > λ}

=[

Q^<_λ∪Q^<_λ

History Core Techniques Covering Techniques Summary References

Low density case

Have to bound ˆ ∞

0 H^d−1

∂[

B_λ^<

dλ.varf, where

B^<_λ ={B :fB > λ, L(B∩ {f > λ})<L(B)/2}. I can’t :(

dyadic maximal operator

M^df(x) = sup

Q3x,Qdyadic

f_Q.

{x:M^df(x)> λ}=[

{maximal dyadicQ :f_Q > λ}

=[

Q^<_λ∪Q^<_λ

History Core Techniques Covering Techniques Summary References

Low density case

Have to bound ˆ ∞

0 H^d−1

∂[

B_λ^<

dλ.varf, where

B^<_λ ={B :fB > λ, L(B∩ {f > λ})<L(B)/2}. I can’t :(

dyadic maximal operator

M^df(x) = sup

Q3x,Qdyadic

f_Q.

{x:M^df(x)> λ}=[

{maximal dyadicQ :f_Q > λ}

=[

Q^<_λ∪Q^<_λ

History Core Techniques Covering Techniques Summary References

Low density case

Have to bound ˆ ∞

0 H^d−1

∂[

B_λ^<

dλ.varf, where

B^<_λ ={B :fB > λ, L(B∩ {f > λ})<L(B)/2}. I can’t :(

dyadic maximal operator

M^df(x) = sup

Q3x,Qdyadic

f_Q.

{x:M^df(x)> λ}=[

{maximal dyadicQ :f_Q > λ}

=[

Q^<_λ∪Q^<_λ

History Core Techniques Covering Techniques Summary References

Low density case

Have to bound ˆ ∞

0 H^d−1

∂[

B_λ^<

dλ.varf, where

B^<_λ ={B :fB > λ, L(B∩ {f > λ})<L(B)/2}. I can’t :(

dyadic maximal operator

M^df(x) = sup

Q3x,Qdyadic

f_Q.

{x:M^df(x)> λ}=[

{maximal dyadicQ :f_Q > λ}=[

Q^<_λ∪Q^<_λ

History Core Techniques Covering Techniques Summary References

Definition

Q is maximal forλ <fQ if for allP )Q we havefP ≤λ.

Given Q, letλ_Q be the smallest such λ.

ˆ

R

H^d−1(∂[

Q^<_λ)dλ

≤ ˆ

R

X

Q∈Q^<

λ

H^d−1(∂Q)dλ

= ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q)dλ

=X

Q

(f_Q −λ˜_Q)H^d−1(∂Q) where

λ˜_Q =

supn λ_Q,

sup{λ:L(Q∩ {f >λ˜_Q})≥2⁻¹· L(Q) }

o

History Core Techniques Covering Techniques Summary References

Definition

Q is maximal forλ <fQ if for allP )Q we havefP ≤λ.

Given Q, letλ_Q be the smallest such λ.

ˆ

R

H^d−1(∂[

Q^<_λ)dλ

≤ ˆ

R

X

Q∈Q^<_λ

H^d−1(∂Q)dλ

= ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q)dλ

=X

Q

(f_Q −λ˜_Q)H^d−1(∂Q) where

λ˜Q =

supn λ_Q,

sup{λ:L(Q∩ {f >λ˜Q})≥2⁻¹· L(Q) }

o

History Core Techniques Covering Techniques Summary References

Definition

Q is maximal forλ <fQ if for allP )Q we havefP ≤λ.

Given Q, letλ_Q be the smallest such λ.

ˆ

R

H^d−1(∂[

Q^<_λ)dλ≤ ˆ

R

X

Q∈Q^<_λ

H^d−1(∂Q)dλ

= ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q)dλ

=X

Q

(f_Q −λ˜_Q)H^d−1(∂Q) where

λ˜Q =

supn λ_Q,

sup{λ:L(Q∩ {f >λ˜Q})≥2⁻¹· L(Q) }

o

History Core Techniques Covering Techniques Summary References

Definition

Q is maximal forλ <fQ if for allP )Q we havefP ≤λ.

Given Q, letλ_Q be the smallest such λ.

ˆ

R

H^d−1(∂[

Q^<_λ)dλ≤ ˆ

R

X

Q∈Q^<_λ

H^d−1(∂Q)dλ

= ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q)dλ

=X

Q

(f_Q −λ˜_Q)H^d−1(∂Q) where

λ˜Q =

supn λ_Q,

sup{λ:L(Q∩ {f >λ˜Q})≥2⁻¹· L(Q) }

o

History Core Techniques Covering Techniques Summary References

Definition

Q is maximal forλ <fQ if for allP )Q we havefP ≤λ. Given Q, letλ_Q be the smallest such λ.

ˆ

R

H^d−1(∂[

Q^<_λ)dλ≤ ˆ

R

X

Q∈Q^<_λ

H^d−1(∂Q)dλ

= ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q)dλ

=X

Q

(f_Q −λ˜_Q)H^d−1(∂Q) where

λ˜Q =

supn λ_Q,

sup{λ:L(Q∩ {f >λ˜Q})≥2⁻¹· L(Q) }

o

History Core Techniques Covering Techniques Summary References

Definition

Q is maximal forλ <fQ if for allP )Q we havefP ≤λ. Given Q, letλ_Q be the smallest such λ.

ˆ

R

H^d−1(∂[

Q^<_λ)dλ≤ ˆ

R

X

Q∈Q^<_λ

H^d−1(∂Q)dλ

= ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q)dλ

=X

Q

(f_Q −λ˜_Q)H^d−1(∂Q)

where λ˜Q =

sup n

λQ,

sup{λ:L(Q∩ {f >λ˜Q})≥2⁻¹· L(Q) }

o

History Core Techniques Covering Techniques Summary References

Definition

Q is maximal forλ <fQ if for allP )Q we havefP ≤λ. Given Q, letλ_Q be the smallest such λ.

ˆ

R

H^d−1(∂[

Q^<_λ)dλ≤ ˆ

R

X

Q∈Q^<_λ

H^d−1(∂Q)dλ

= ˆ

R

X

Q:˜λQ<λ<fQ

H^d−1(∂Q)dλ

=X

Q

(f_Q −λ˜_Q)H^d−1(∂Q)

where λ˜Q = sup

n

λQ,sup{λ:L(Q∩ {f >λ˜Q})≥2⁻¹· L(Q) }o