Regularity of planar quasiconformal mappings

Introduction QC mappings of the whole plane QC mappings on domains The end

Regularity of planar quasiconformal mappings

Mart´ı Prats

September 8th, 2016

Introduction QC mappings of the whole plane QC mappings on domains The end

Introduction

Introduction QC mappings of the whole plane QC mappings on domains The end

Measuring smoothness and integrability in R

1 p

p“ 1 p“ 8

L⁴

Lebesgue spacesÑintegrability.

Di↵erentiablility classes Ñ smoothness. Sobolev spacesÑ both together. H¨older continuous spacesÑ fill gaps. Interpolation to generalize.

}f }L^p “`´

|f |^p˘1{p, }f }L⁸ “ ess sup|f |

}f }_C^s “ }f }_L⁸` ¨ ¨ ¨ ` }r^sf}_L⁸ }f }W^s,p “ }f }L^p` ¨ ¨ ¨ ` }r^sf}L^p

}f }C^s “

}f }L⁸` ¨ ¨ ¨ ` sup^|rtsuf_|x´y|^pxq´rtsu^tsufpyq|

}f }W^s,p,}f }B_p,q^s ,}f }F_p,q^s

By means of Sobolev embeddings, we have either continuity or extra integrability.

Introduction QC mappings of the whole plane QC mappings on domains The end

Measuring smoothness and integrability in R

1 p

s

0 2 4

C³

L⁴

Lebesgue spacesÑ integrability.

Di↵erentiablility classes Ñsmoothness.

Sobolev spacesÑ both together. H¨older continuous spacesÑ fill gaps. Interpolation to generalize.

}f }L^p “`´

|f |^p˘1{p, }f }L⁸ “ ess sup|f |

}f }C^s “ }f }L⁸` ¨ ¨ ¨ ` }r^sf}L⁸

}f }W^s,p “ }f }L^p` ¨ ¨ ¨ ` }r^sf}L^p

}f }C^s “

}f }L⁸` ¨ ¨ ¨ ` sup^|rtsuf_|x´y|^pxq´rtsu^tsufpyq|

}f }W^s,p,}f }B_p,q^s ,}f }F_p,q^s

By means of Sobolev embeddings, we have either continuity or extra integrability.

Introduction QC mappings of the whole plane QC mappings on domains The end

Measuring smoothness and integrability in R

1 p

s

0 2 4

W^3,4 C³

L⁴

Lebesgue spacesÑintegrability.

Di↵erentiablility classes Ñsmoothness.

Sobolev spacesÑbothtogether.

H¨older continuous spacesÑ fill gaps. Interpolation to generalize.

}f }L^p “`´

|f |^p˘1{p, }f }L⁸ “ ess sup|f |

}f }C^s “ }f }L⁸` ¨ ¨ ¨ ` }r^sf}L⁸

}f }W^s,p “ }f }L^p` ¨ ¨ ¨ ` }r^sf}L^p

}f }C^s “

}f }L⁸` ¨ ¨ ¨ ` sup^|rtsuf_|x´y|^pxq´rtsu^tsufpyq|

}f }W^s,p,}f }B_p,q^s ,}f }F_p,q^s

By means of Sobolev embeddings, we have either continuity or extra integrability.

Introduction QC mappings of the whole plane QC mappings on domains The end

Measuring smoothness and integrability in R

1 p

s

0 2 4

W^3,4

C^1.75

L⁴

Lebesgue spacesÑ integrability.

Di↵erentiablility classes Ñ smoothness.

Sobolev spacesÑ both together.

H¨older continuous spacesÑ fill gaps.

Interpolation to generalize.

}f }L^p “`´

|f |^p˘1{p, }f }L⁸ “ ess sup|f |

}f }C^s “ }f }L⁸` ¨ ¨ ¨ ` }r^sf}L⁸

}f }W^s,p “ }f }L^p` ¨ ¨ ¨ ` }r^sf}L^p

}f }_C^s “

}f }L⁸` ¨ ¨ ¨ ` sup^|rtsuf_|x´y|^pxq´rtsu^tsufpyq|

}f }W^s,p,}f }B_p,q^s ,}f }F_p,q^s

By means of Sobolev embeddings, we have either continuity or extra integrability.

Introduction QC mappings of the whole plane QC mappings on domains The end

Measuring smoothness and integrability in R

1 p

s

0 2 4

W^3,4

C^1.75

L⁴

W^3{2,8{5 W^1,4{3

Lebesgue spacesÑ integrability.

Di↵erentiablility classes Ñ smoothness.

Sobolev spacesÑ both together.

H¨older continuous spacesÑ fill gaps.

Interpolation to generalize.

}f }L^p “`´

|f |^p˘1{p, }f }L⁸ “ ess sup|f |

}f }C^s “ }f }L⁸` ¨ ¨ ¨ ` }r^sf}L⁸

}f }W^s,p “ }f }L^p` ¨ ¨ ¨ ` }r^sf}L^p

}f }_C^s “

}f }L⁸` ¨ ¨ ¨ ` sup^|rtsuf_|x´y|^pxq´rtsu^tsufpyq|

}f }W^s,p,}f }B_p,q^s ,}f }F_p,q^s

By means of Sobolev embeddings, we have either continuity or extra integrability.

Introduction QC mappings of the whole plane QC mappings on domains The end

Measuring smoothness and integrability in R

1 p

s

sp“ d Supercritical

Subcritical 0

2 4 3

W^s1,p1 C^s1´p1d

L^p1

W^s2,p2

L^p2˚ L^p2 d“ 3

Lebesgue spacesÑ integrability.

Di↵erentiablility classes Ñ smoothness.

Sobolev spacesÑ both together.

H¨older continuous spacesÑ fill gaps.

Interpolation to generalize.

}f }L^p “`´

|f |^p˘1{p, }f }L⁸ “ ess sup|f |

}f }C^s “ }f }L⁸` ¨ ¨ ¨ ` }r^sf}L⁸

}f }W^s,p “ }f }L^p` ¨ ¨ ¨ ` }r^sf}L^p

}f }_C^s “

}f }L⁸` ¨ ¨ ¨ ` sup^|rtsuf_|x´y|^pxq´rtsu^tsufpyq|

}f }W^s,p,}f }B_p,q^s ,}f }F_p,q^s

By means of Sobolev embeddings, we have either continuity or extra integrability.

Introduction QC mappings of the whole plane QC mappings on domains The end

Quasiconformal mappings

Conformal mappings Preserves angles

“Circles to circles” Cauchy-Riemann:

1

2pBxf ` iByfq “ 0

Quasiconformal mappings Angle distortion bounded.

“Circles to ellipses”.

|Bf | § k|Bf |

Introduction QC mappings of the whole plane QC mappings on domains The end

Quasiconformal mappings

Conformal mappings Preserves angles

“Circles to circles”

Cauchy-Riemann:

1

2pBxf ` iByfq “ 0

Quasiconformal mappings Angle distortion bounded.

“Circles to ellipses”.

|Bf | § k|Bf |

Introduction QC mappings of the whole plane QC mappings on domains The end

Quasiconformal mappings

Conformal mappings Preserves angles

“Circles to circles”

Cauchy-Riemann:

1

2pBxf ` iByfq “ 0 Bf “ 0

Quasiconformal mappings Angle distortion bounded.

“Circles to ellipses”.

|Bf | § k|Bf |

Introduction QC mappings of the whole plane QC mappings on domains The end

Quasiconformal mappings

Conformal mappings Preserves angles

“Circles to circles”

Cauchy-Riemann:

1

2pBxf ` iByfq “ 0 Bf “ 0

Quasiconformal mappings Angle distortion bounded.

“Circles to ellipses”.

|Bf | § k|Bf |

Introduction QC mappings of the whole plane QC mappings on domains The end

The Beurling transform

The Beurling transform of a function f P L^ppCq is:

Bf pzq “ 1

´⇡ lim

"Ñ0

ˆ

|w´z|°"

fpwq

pz ´ wq²dmpwq.

It is essential to quasiconformal mappings because Bp¯Bf q “ Bf @f P W^1,p. Recall thatB : L^ppCq Ñ L^ppCq is bounded for 1 † p † 8.

AlsoB : W^s,ppCq Ñ W^s,ppCq is bounded for 1 † p † 8 and s ° 0.

Introduction QC mappings of the whole plane QC mappings on domains The end

The Beurling transform

The Beurling transform of a function f P L^ppCq is:

Bf pzq “ 1

´⇡ lim

"Ñ0

ˆ

|w´z|°"

fpwq

pz ´ wq²dmpwq.

It is essential to quasiconformal mappings because Bp¯Bf q “ Bf @f P W^1,p.

Recall thatB : L^ppCq Ñ L^ppCq is bounded for 1 † p † 8.

AlsoB : W^s,ppCq Ñ W^s,ppCq is bounded for 1 † p † 8 and s ° 0.

Introduction QC mappings of the whole plane QC mappings on domains The end

The Beurling transform

The Beurling transform of a function f P L^ppCq is:

Bf pzq “ 1

´⇡ lim

"Ñ0

ˆ

|w´z|°"

fpwq

pz ´ wq²dmpwq.

It is essential to quasiconformal mappings because Bp¯Bf q “ Bf @f P W^1,p. Recall thatB : L^ppCq Ñ L^ppCq is bounded for 1 † p † 8.

AlsoB : W^s,ppCq Ñ W^s,ppCq is bounded for 1 † p † 8 and s ° 0.

Introduction QC mappings of the whole plane QC mappings on domains The end

QC mappings of the whole plane

Introduction QC mappings of the whole plane QC mappings on domains The end

The Beltrami equation

1 p

p“ 1 p“ 8

µ

Let µP L⁸c pCq with  :“ }µ}₈† 1.

The Beltrami equation

¯Bf pzq “ µpzqBf pzq has a unique solution f P Wloc^1,2such that fpzq “ z ` Op1{zq as z Ñ 8. Consider

h:“ µ ` µBpµq ` µBpµBpµqq ` ¨ ¨ ¨

“pI ´ µBq^´1pµq,

since}µ ¨ B}_p2,2q§ }B}_p2,2q“  † 1. Then, hP L² and f “ ⇡z¹ ˚ h ` z. This remains true if}B}_pp,pq† 1{.

Introduction QC mappings of the whole plane QC mappings on domains The end

The Beltrami equation

1 p

µ

W_loc^s,p

0 2 s

f

Let µP L⁸c pCq with  :“ }µ}₈† 1.

The Beltrami equation

¯Bf pzq “ µpzqBf pzq has a unique solution f P Wloc^1,2such that fpzq “ z ` Op1{zq as z Ñ 8.

Consider

h:“ µ ` µBpµq ` µBpµBpµqq ` ¨ ¨ ¨

“pI ´ µBq^´1pµq,

since}µ ¨ B}_p2,2q§ }B}_p2,2q“  † 1. Then, hP L² and f “ ⇡z¹ ˚ h ` z. This remains true if}B}_pp,pq† 1{.

Introduction QC mappings of the whole plane QC mappings on domains The end

The Beltrami equation

1 p

µ

W_loc^s,p

0 2 s

f

Let µP L⁸c pCq with  :“ }µ}₈† 1.

The Beltrami equation

¯Bf pzq “ µpzqBf pzq has a unique solution f P Wloc^1,2such that fpzq “ z ` Op1{zq as z Ñ 8.

Consider

h:“ µ ` µBpµq ` µBpµBpµqq ` ¨ ¨ ¨

“pI ´ µBq^´1pµq,

since}µ ¨ B}_p2,2q§ }B}_p2,2q“  † 1. Then, hP L² and f “ ⇡z¹ ˚ h ` z. This remains true if}B}_pp,pq† 1{.

Introduction QC mappings of the whole plane QC mappings on domains The end

The Beltrami equation

1 p

µ

W_loc^s,p

0 2 s

f

Let µP L⁸c pCq with  :“ }µ}₈† 1.

The Beltrami equation

¯Bf pzq “ µpzqBf pzq has a unique solution f P Wloc^1,2such that fpzq “ z ` Op1{zq as z Ñ 8.

Consider

h:“ µ ` µBpµq ` µBpµBpµqq ` ¨ ¨ ¨

“pI ´ µBq^´1pµq,

since}µ ¨ B}_p2,2q§ }B}_p2,2q“  † 1. Then, hP L² and f “ ⇡z¹ ˚ h ` z. This remains true if}B}_pp,pq† 1{.

Introduction QC mappings of the whole plane QC mappings on domains The end

The Beltrami equation

1 p

µ

W_loc^s,p

0 2 s

f

Let µP L⁸c pCq with  :“ }µ}₈† 1.

The Beltrami equation

¯Bf pzq “ µpzqBf pzq has a unique solution f P Wloc^1,2such that fpzq “ z ` Op1{zq as z Ñ 8.

Consider

h:“ µ ` µBpµq ` µBpµBpµqq ` ¨ ¨ ¨

“pI ´ µBq^´1pµq,

since}µ ¨ B}_p2,2q§ }B}_p2,2q“  † 1.

Then, hP L² and f “ ⇡z¹ ˚ h ` z. This remains true if}B}_pp,pq† 1{.

Introduction QC mappings of the whole plane QC mappings on domains The end

The Beltrami equation

1 p

µ

W_loc^s,p

0 2 s

f

h

Let µP L⁸c pCq with  :“ }µ}₈† 1.

The Beltrami equation

¯Bf pzq “ µpzqBf pzq has a unique solution f P Wloc^1,2such that fpzq “ z ` Op1{zq as z Ñ 8.

Consider

h:“ µ ` µBpµq ` µBpµBpµqq ` ¨ ¨ ¨

“pI ´ µBq^´1pµq,

since}µ ¨ B}_p2,2q§ }B}_p2,2q“  † 1.

Then, hP L²

and f “ ⇡z¹ ˚ h ` z. This remains true if}B}_pp,pq† 1{.

Introduction QC mappings of the whole plane QC mappings on domains The end

The Beltrami equation

1 p

µ

W_loc^s,p

0 2 s

f

h

Let µP L⁸c pCq with  :“ }µ}₈† 1.

The Beltrami equation

¯Bf pzq “ µpzqBf pzq has a unique solution f P Wloc^1,2such that fpzq “ z ` Op1{zq as z Ñ 8.

Consider

h:“ µ ` µBpµq ` µBpµBpµqq ` ¨ ¨ ¨

“pI ´ µBq^´1pµq,

since}µ ¨ B}_p2,2q§ }B}_p2,2q“  † 1.

Then, hP L² and f “ ⇡z¹ ˚ h ` z.

This remains true if}B}_pp,pq† 1{.

Introduction QC mappings of the whole plane QC mappings on domains The end

The Beltrami equation

1 p

µ

W_loc^s,p

0 2 s

f

h

Let µP L⁸c pCq with  :“ }µ}₈† 1.

The Beltrami equation

¯Bf pzq “ µpzqBf pzq has a unique solution f P Wloc^1,2such that fpzq “ z ` Op1{zq as z Ñ 8.

Consider

h:“ µ ` µBpµq ` µBpµBpµqq ` ¨ ¨ ¨

“pI ´ µBq^´1pµq,

since}µ ¨ B}_p2,2q§ }B}_p2,2q“  † 1.

Then, hP L² and f “ ⇡z¹ ˚ h ` z.

This remains true if}B}_pp,pq † 1{.

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

µ 

`1 1

`1

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01]. µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM]. µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

µ 1 h

p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM]. µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

f

µ 1 h

p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM]. µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

µ ₁

p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq

ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM]. µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

µ h

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM]. µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

f

µ h

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM]. µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

µ

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^n`"

ùñ h P C_loc^n`"[AIM]. µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

µ h

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM].

µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

f

µ h

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM].

µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

µ

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM].

µP A^sp,q

ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

µ h

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM].

µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

f

µ h

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM].

µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

µ

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM].

µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2

ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

µ h

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM].

µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0

2 f

µ h

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM].

µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

µ

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM].

µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p

ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0 2

µ h

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM].

µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Results without boundaries

W_loc^s,p

1 p

s

0

2 f

µ h

1 p

1 p¹

Let µP L⁸c pCq with  :“ }µ}₈† 1.

hP L^p for _p¹

 † ¹_p [A92, AIS01].

µP VMOpˆCq ùñ h P L^p for 1† p † 8. [I]

µP C_loc^{n`"</sup> ùñ h P C<sub>loc</sub><sup>n`"}[AIM].

µP A^sp,q ùñ h P A^sp,q for sp° 2 [CMO].

µP W^1,2 ùñ h P W^1,2´"for p“ 2 [CFMOZ].

µP W^1,p ùñ h P W^1,q for p† 2,

1

q °¹_p`_p¹_ [CFMOZ].

Introduction QC mappings of the whole plane QC mappings on domains The end

Recent progress

Theorem (P.)

Let 0† s † 2, 1 † p † 8, let µ P W^s,pX L⁸, with µ§  D and let f be the principal solution to the Beltrami equation ¯Bf “ µBf .

If s“ ²_p, then

Bf P W¯ ^s,q for every 1 q ° 1

p. If s† ²_p and ¹_p † _p¹¹_ ´_p¹_ “ ^1´_1`, then

¯Bf P W^s,q for every 1 q ° 1

p` 1 p

.

See [Clop, Faraco, Ruiz] for previous weaker results and Bais´on’s thesis for a stronger result in the critical setting with s° 1{2.

It remains unclear if the condition ¹_p † _p¹1

 ´_p¹_ can be replaced by ¹_p †_p¹¹_, which is more natural and is achieved for s“ 1.

Introduction QC mappings of the whole plane QC mappings on domains The end

Recent progress

Theorem (P.)

Let 0† s † 2, 1 † p † 8, let µ P W^s,pX L⁸, with µ§  D and let f be the principal solution to the Beltrami equation ¯Bf “ µBf .

If s“ ²_p, then

Bf P W¯ ^s,q for every 1 q ° 1

p. If s† ²_p and ¹_p † _p¹¹_ ´_p¹_ “ ^1´_1`, then

¯Bf P W^s,q for every 1 q ° 1

p` 1 p

.

See [Clop, Faraco, Ruiz] for previous weaker results and Bais´on’s thesis for a stronger result in the critical setting with s° 1{2.

It remains unclear if the condition ¹_p † _p¹1

 ´_p¹_ can be replaced by ¹_p †_p¹¹_, which is more natural and is achieved for s“ 1.

Introduction QC mappings of the whole plane QC mappings on domains The end

Recent progress

Theorem (P.)

Let 0† s † 2, 1 † p † 8, let µ P W^s,pX L⁸, with µ§  D and let f be the principal solution to the Beltrami equation ¯Bf “ µBf .

If s“ ²_p, then

Bf P W¯ ^s,q for every 1 q ° 1

p. If s† ²_p and ¹_p † _p¹¹_ ´_p¹_ “ ^1´_1`, then

¯Bf P W^s,q for every 1 q ° 1

p` 1 p

.

See [Clop, Faraco, Ruiz] for previous weaker results and Bais´on’s thesis for a stronger result in the critical setting with s° 1{2.

It remains unclear if the condition ¹_p † p¹_¹ ´p¹ can be replaced by _p¹ †_p¹¹_, which is more natural and is achieved for s“ 1.

Introduction QC mappings of the whole plane QC mappings on domains The end

Idea

W^s,p

1 p

s

0 2

µ

µ

1 p

1 p¹_

µP L⁸X W^s,p

h :“ pI ´ µBq^´1pµq h“ µBh ` µ.

BµBh P L^q, qP pp¹, pq: H¨older rD^s, µsBh P L^q: Kato-Ponce

s“ 1

Bh “ BpµBhq ` Bµ Bh “ µBBh ` BµBh ` Bµ Bh “ µBBh ` BµBh ` Bµ pI ´ µBqBh “ BµBh ` Bµ Bh “ pI ´ µBq^´1pBµBh ` Bµq

s† 1

D^sh“ D^spµBhq ` D^sµ

D^sh“ µD^sBh ´ rD^s, µspBhq ` D<sup>s</sup>µ D<sup>s</sup>h“ µBD<sup>s</sup>h´ rD<sup>s</sup>, µspBhq ` D^sµ pI ´ µBqD^sh“ ´rD^s, µspBhq ` D^sµ D^sh“ pI ´µBq^´1pD^sµ´rD^s, µspBhqq

Introduction QC mappings of the whole plane QC mappings on domains The end

Idea

W^s,p

1 p

s

0 2

h µ

1 p

1 p_¹

µP L⁸X W^s,p h :“ pI ´ µBq^´1pµq

h“ µBh ` µ.

BµBh P L^q, qP pp¹, pq: H¨older rD^s, µsBh P L^q: Kato-Ponce

s“ 1

Bh “ BpµBhq ` Bµ Bh “ µBBh ` BµBh ` Bµ Bh “ µBBh ` BµBh ` Bµ pI ´ µBqBh “ BµBh ` Bµ Bh “ pI ´ µBq^´1pBµBh ` Bµq

s† 1

D^sh“ D^spµBhq ` D^sµ

D^sh“ µD^sBh ´ rD^s, µspBhq ` D<sup>s</sup>µ D<sup>s</sup>h“ µBD<sup>s</sup>h´ rD<sup>s</sup>, µspBhq ` D^sµ pI ´ µBqD^sh“ ´rD^s, µspBhq ` D^sµ D^sh“ pI ´µBq^´1pD^sµ´rD^s, µspBhqq

Introduction QC mappings of the whole plane QC mappings on domains The end

Idea

W^s,p

1 p

s

0 2

h µ

1 p

1 p_¹

µP L⁸X W^s,p h :“ pI ´ µBq^´1pµq h“ µBh ` µ.

BµBh P L^q, qP pp¹, pq: H¨older rD^s, µsBh P L^q: Kato-Ponce

s“ 1

Bh “ BpµBhq ` Bµ Bh “ µBBh ` BµBh ` Bµ Bh “ µBBh ` BµBh ` Bµ pI ´ µBqBh “ BµBh ` Bµ Bh “ pI ´ µBq^´1pBµBh ` Bµq

s† 1

D^sh“ D^spµBhq ` D^sµ

D^sh“ µD^sBh ´ rD^s, µspBhq ` D<sup>s</sup>µ D<sup>s</sup>h“ µBD<sup>s</sup>h´ rD<sup>s</sup>, µspBhq ` D^sµ pI ´ µBqD^sh“ ´rD^s, µspBhq ` D^sµ D^sh“ pI ´µBq^´1pD^sµ´rD^s, µspBhqq

Introduction QC mappings of the whole plane QC mappings on domains The end

Idea

W^s,p

1 p

s

0 2

h µ

1 p

1 p_¹

µP L⁸X W^s,p h :“ pI ´ µBq^´1pµq h“ µBh ` µ.

BµBh P L^q, qP pp¹, pq: H¨older rD^s, µsBh P L^q: Kato-Ponce

s“ 1

Bh “ BpµBhq ` Bµ Bh “ µBBh ` BµBh ` Bµ Bh “ µBBh ` BµBh ` Bµ pI ´ µBqBh “ BµBh ` Bµ Bh “ pI ´ µBq^´1pBµBh ` Bµq

s† 1

D^sh“ D^spµBhq ` D^sµ

D^sh“ µD^sBh ´ rD^s, µspBhq ` D<sup>s</sup>µ D<sup>s</sup>h“ µBD<sup>s</sup>h´ rD<sup>s</sup>, µspBhq ` D^sµ pI ´ µBqD^sh“ ´rD^s, µspBhq ` D^sµ D^sh“ pI ´µBq^´1pD^sµ´rD^s, µspBhqq

Introduction QC mappings of the whole plane QC mappings on domains The end

Idea

W^s,p

1 p

s

0 2

h µ

1 p

1 p_¹

µP L⁸X W^s,p h :“ pI ´ µBq^´1pµq h“ µBh ` µ.

BµBh P L^q, qP pp¹, pq: H¨older rD^s, µsBh P L^q: Kato-Ponce

s“ 1

Bh “ BpµBhq ` Bµ

Bh “ µBBh ` BµBh ` Bµ Bh “ µBBh ` BµBh ` Bµ pI ´ µBqBh “ BµBh ` Bµ Bh “ pI ´ µBq<sup>´1</sup>pBµBh ` Bµq

s† 1

D^sh“ D^spµBhq ` D^sµ

D^sh“ µD^sBh ´ rD^s, µspBhq ` D<sup>s</sup>µ D<sup>s</sup>h“ µBD<sup>s</sup>h´ rD<sup>s</sup>, µspBhq ` D^sµ pI ´ µBqD^sh“ ´rD^s, µspBhq ` D^sµ D^sh“ pI ´µBq^´1pD^sµ´rD^s, µspBhqq

Introduction QC mappings of the whole plane QC mappings on domains The end

Idea

W^s,p

1 p

s

0 2

h µ

1 p

1 p_¹

µP L⁸X W^s,p h :“ pI ´ µBq^´1pµq h“ µBh ` µ.

BµBh P L^q, qP pp¹, pq: H¨older rD^s, µsBh P L^q: Kato-Ponce

s“ 1

Bh “ BpµBhq ` Bµ

Bh “ µBBh ` BµBh ` Bµ Bh “ µBBh ` BµBh ` Bµ pI ´ µBqBh “ BµBh ` Bµ Bh “ pI ´ µBq<sup>´1</sup>pBµBh ` Bµq

s† 1

D^sh“ D^spµBhq ` D^sµ

D^sh“ µD^sBh ´ rD^s, µspBhq ` D<sup>s</sup>µ D<sup>s</sup>h“ µBD<sup>s</sup>h´ rD<sup>s</sup>, µspBhq ` D^sµ pI ´ µBqD^sh“ ´rD^s, µspBhq ` D^sµ D^sh“ pI ´µBq^´1pD^sµ´rD^s, µspBhqq

Introduction QC mappings of the whole plane QC mappings on domains The end

Idea

W^s,p

1 p

s

0 2

h µ

1 p

1 p_¹

µP L⁸X W^s,p h :“ pI ´ µBq^´1pµq h“ µBh ` µ.

BµBh P L^q, qP pp¹, pq: H¨older rD^s, µsBh P L^q: Kato-Ponce

s“ 1

Bh “ BpµBhq ` Bµ

Bh “ µBBh ` BµBh ` Bµ Bh “ µBBh ` BµBh ` Bµ pI ´ µBqBh “ BµBh ` Bµ Bh “ pI ´ µBq<sup>´1</sup>pBµBh ` Bµq

s† 1

D^sh“ D^spµBhq ` D^sµ

D^sh“ µD^sBh ´ rD^s, µspBhq ` D<sup>s</sup>µ D<sup>s</sup>h“ µBD<sup>s</sup>h´ rD<sup>s</sup>, µspBhq ` D^sµ pI ´ µBqD^sh“ ´rD^s, µspBhq ` D^sµ D^sh“ pI ´µBq^´1pD^sµ´rD^s, µspBhqq

Introduction QC mappings of the whole plane QC mappings on domains The end

Idea

W^s,p

1 p

s

0 2

h µ

1 p

1 p_¹

µP L⁸X W^s,p h :“ pI ´ µBq^´1pµq h“ µBh ` µ.

BµBh P L^q, qP pp¹, pq: H¨older rD^s, µsBh P L^q: Kato-Ponce

s“ 1

Bh “ BpµBhq ` Bµ Bh “µBBh ` BµBh` Bµ

Bh “ µBBh ` BµBh ` Bµ pI ´ µBqBh “ BµBh ` Bµ Bh “ pI ´ µBq<sup>´1</sup>pBµBh ` Bµq

s† 1

D^sh“ D^spµBhq ` D^sµ

D^sh“ µD^sBh ´ rD^s, µspBhq ` D<sup>s</sup>µ D<sup>s</sup>h“ µBD<sup>s</sup>h´ rD<sup>s</sup>, µspBhq ` D^sµ pI ´ µBqD^sh“ ´rD^s, µspBhq ` D^sµ D^sh“ pI ´µBq^´1pD^sµ´rD^s, µspBhqq

Introduction QC mappings of the whole plane QC mappings on domains The end

Idea

W^s,p

1 p

s

0 2

h µ

1 p

1 p_¹

µP L⁸X W^s,p h :“ pI ´ µBq^´1pµq h“ µBh ` µ.

BµBh P L^q, qP pp¹, pq: H¨older rD^s, µsBh P L^q: Kato-Ponce

s“ 1

Bh “ BpµBhq ` Bµ Bh “ µBBh ` BµBh ` Bµ Bh “ µBBh` BµBh ` Bµ

pI ´ µBqBh “ BµBh ` Bµ Bh “ pI ´ µBq<sup>´1</sup>pBµBh ` Bµq

s† 1

D^sh“ D^spµBhq ` D^sµ

D^sh“ µD^sBh ´ rD^s, µspBhq ` D<sup>s</sup>µ D<sup>s</sup>h“ µBD<sup>s</sup>h´ rD<sup>s</sup>, µspBhq ` D^sµ pI ´ µBqD^sh“ ´rD^s, µspBhq ` D^sµ D^sh“ pI ´µBq^´1pD^sµ´rD^s, µspBhqq