INTERPOLATION

AND

THE BELTRAMI EQUATION

Prague, September 2011

István Prause

Quasiconformal mappingsin the plane

Beltrami equation Beurling transform

Beltrami equation

! ∈ "!,"#$% (C)

MorreyBojarski,

Ahlfors-Bers

existence and uniqueness

homeomorphism

analytic dependence

regularity

Astala,etc.

!(", #) > ! "#$#

i(}) = } +O(4/}) principal quasiconformal mapping

N =4+ n4− n

i}̄ = ´(}) i}, |´(})| � n ‐G(}), 3 � n < 4

L!-isometry!("#̄) = "#, " ∈ $!,"(C)

Calderón-Zygmund: ! : "# → "#

Riesz-Thorin:

Beurling transform

i}̄ = (Lg− ´V)−4(´) = ´+ ´V´+ ´V(´V´) + · · ·

i ∈ Z4,sorf (C), 5 ≤ s < s3(N)

Vi(}) = −

4ヽ

�C

i(、)(、 − })5

gp(、)

vrph s3 > 5,

Bojarski (1957):

�V�s3 =4n

Critical Sobolev exponent

Sobolev embeddings3(N) =5N

N− 44N

Hölder exponent

} �→}|}|

|}|4N

(Mori)

dimK{} ∈ C : ü(}) = ü} ≤ 4+ ü −|4− ü|

n

1/K K1

Astala (1994): “this is the worst case scenario”

Conjecture (Iwaniec): �!�"#(C) = #− !, # � "

“holomorphic motions = quasiconformal maps”

Holomorphic motions

Mañé-Sad-Sullivan, Slodkowski’s !-lemma:

java animation by Aleksi Vähäkangas

Φ : D× H → C, H ⊂ C

} �→ Φ(゜, }) = Φ゜(}) lv lqmhfwlyh iru"doo ゜ ∈ D,

゜ �→ Φ(゜, }) lv krorprusklf iru"doo } ∈ H,

Φ(3, }) ≡ }.

{Φ゜(})} (extends to) an analytic family of qc maps

Complex interpolation

homeomorphisms

diffeomorphisms

quasiconformal maps Lp

L2

L!

complex interpolation

!-lemma:

quasiconformal maps = “complex interpolation class”

analytic dependence

L functionsBeltrami equa

holomorphic mi equation/rphic motions

interpolateBeurling

transformdimension/pressure

p-norm of gradient

end-point estimates

L!-isometry dim " 2Jacobian null-Lagrangian

non-vanishing

- injectivityJacobian positive

a.e.

applicationgain in regularity

Bojarski

sharp exponent, area/dimension

distortionAstala

sharp weight,quasiconvexityAstala-Iwaniec-Prause-Saksman

p

Interpolation theme

Sharp weighted integrability

Theorem:

i(}) = } +O(4/})i}̄ = ´(}) i}, |´(})| � n ‐び(}), 3 � n < 4,

5 ≤ s ≤ 4+ 4/n

4|び|

�

び

�

4− s|´(})|

4+ |´(})|

�

|Gi(})|s � 4

• sharp weight, sharp constants

• “localized integrability” at the borderline

• partial quasiconvexityrank-one convexity quasiconvexityvs

(JAMS, to appear)

Astala-Iwaniec-Prause-Saksman

Rank-one convexity vs quasiconvexity Morrey

! ∈ "+ #∞! (",R$)

�!

E(!") �

�!

E(#) = E(#)|!|!"#$ ! = %

E : R!×! → R

⇐

! � ! "verák

! = !

�

? Faraco-Székelyhidi: “localization”

local global

!" (#) =� "

!det # + ("−

"

!)�

�#�

�

!�

· |#|"−!

Burkholder: rank-one concave!"(#$) = !"($%, $%̄)

w �→ E(D + w[) frqyh{

(lower semicontinuity)(ellipticity of Euler-Lagrange)

BurkholderMartingale inequality

subordinated martingales

⇒ �!"�# � (#− !) �$"�#.

!"(#,$) =�

|#| − ("− !) |$|�

·�

|#| + |$|�"−!

E !"(#$, %$) � E !"(#$−!, %$−!) � . . . � "

!" ≺ #"

|}|s − (s − 4)s |z|s � fs Es(},z)

|[q − [q−4| ≤ |¥q − ¥q−4| a.s.

Quasiconvexity result

E !"(#$, %$) � !

Burkholder’s martingale inequality

Theorem:

! � !!"(#,$) =�

|#| − ("− !) |$|�

·�

|#| + |$|�"−!

!"(#$) = !"($%, $%̄)

!(") ∈ "+ #∞! ("), $%(&!) � !, " ∈ "

for

(equiv.) �び

Es (Gi) �

�び

Es (Lg) = |び|

full quasiconvexity �V�Os(C) = s− 4

!" ≺ #"

Interpolation lemma

U

10

!! !

!

践

non-vanishing 漸゜(}) �= 3

analytic family,

⇒

3 < s3, s4 � ∞, 践 ∈ (3, 4)

゜ ∈ X = {Uh ゜ > 3}漸゜(})

�漸゜�s3 � P3 hf Uh゜

�漸4�s4 � P4

�漸践�s践 � P4−践3 ·P践

4

s践=

− 践s3

+践s4

Hadamard Harnack

change p freeze p

subharmonic harmonic

duality log-convexity

!

!

!

!θ

cf. Riesz-Thorin

orj �漸践�s � D ·4s+ E

Proof of the lemma

Harnack

harmonicnon-vanishing

orj �漸゜�s � D ·4s+ E(゜)

orj �漸践�s践 = D ·4s践

+ E(践) = D ·4s3

+ E(践) + 践 · D�

4s4

−4s3

�

� 践�

D ·4s4

+ E(4)�

� 践 orj �漸4�s4 � 3

D ·4s3

+ E(践) � 践�

D ·4s3

+ E(4)�

Corollaries

Müller: LlogL integrability under J(z,f)#0

sharp integrability estimates

LlogL integrability: !(") ∈ "+ #∞! ("), $%&'%&%()$*+&,

!!"

!"

�

�

�

�

"="

Proof:

�

び

�

4 + orj |Gi(})|5�

M(}, i) �

�

び|Gi(})|5.

Exp integrability:

!!"

!"

�

�

�

�

"=∞

Proof:

|´(})| � ‐G(}), } ∈ C

4ヽ

�

G

�

4− |´|�

h|´|+Uh V´� 4.

H(D) =45

|D|5

det D+ log

�

det D�

− log |D|, det D > 3

quasiconvexity:

Mappings of integrable distortion

EXUNKROGHU LQWHJUDOV DQG TXDVLFRQIRUPDO PDSSLQJV 54

Frqyhujhqfh"wkhruhpv" lq" wkh"wkhru¦"ri" lqwhjudov" ohw"xv"sdvv" wr"wkh" olplw"zkhq

/ dv"iroorzv

GU

M } i 4 Gi } 5 g}

GU

M } i 4 4 Gi 5 g}GU

M } i 4 Gi 5 g}

GU

M } i 4 4 Gi 5 g}

GU

M } i 4 Gi 5 g}

Khuh"wkh 0whup"lv"mxvwlÛhg"e¦"Idwrx*v"wkhruhp"zkloh"wkh 0whup"e¦

wkh"Ohehvjxh"grplqdwhg"frqyhujhqfh/ zkhuh"zh"revhuyh" wkdw" wkh" lqwhjudqg" lv

grplqdwhg"srlqw0zlvh"e¦ M } i Gi 5 41 Wkh"olqhv"ri"frpsxwdwlrq"frqwlqxh

dv"iroorzv

GU

M } i 4 Gi 5 g}

GU

Gi 5 g}GU

Gi } 5 g}

Ilqdoo¦/ zh"revhuyh"wkdw

GU びM } i 4 Gi } 5 g}

GU びGi } 5 g}

zklfk"frpelqhg"zlwk"wkh"suhylrxv"hvwlpdwh"¦lhogv +4143,/ dv"ghvluhg1

Wkh"deryh"hqhuj¦"ixqfwlrqdo"fdq"eh"ixuwkhu"fxowlydwhg"e¦"dsso¦lqj"wkh"lqyhuvh

pds/ dv"ghvfulehg"dw"wkh"hqg"ri"Vhfwlrq"61 D vhdufk"ri"plqlpdo"uhjxodulw¦"lq"wkh

fruuhvsrqglqj"lqwhjudo"hvwlpdwhv"ohdgv"xv"wr"pdsslqjv"ri"lqwhjudeoh"glvwruwlrq1

Fruroodu¦ 7161 Ohw び ⊂ R5 eh"d"erxqghg"grpdlq/ dqg"vxssrvh k ∈ W

4,4orf (び) lv"d

krphrprusklvp k : び onto→ び vxfk"wkdw k(}) = } iru } ∈ ∂び1 Dvvxph k vdwlvÛhv"wkh

glvwruwlrq"lqhtxdolw¦

|Gk(})|5 ! N(})M(}, k), a.e in び,

zkhuh 4 ! N(}) < ∞ doprvw"hyhu¦zkhuh"lq び. Wkh"vpdoohvw"vxfk"ixqfwlrq/ ghqrwhg"e¦

N(}, k)/ lv"dvvxphg"wr"eh"lqwhjudeoh1 Wkhq

+718, 5∫び[log |Gk(})|− log M(}, k)] g} !

∫び[N(}, k)− M(}, k)] g}

Lq"sduwlfxodu/ log M(}, k) lv"lqwhjudeoh1 Djdlq"wkhuh"lv"d"zhdowk"ri"ixqfwlrqv/ wr"eh"ghvfulehg

lq"Vhfwlrq"8/ vdwlvi¦lqj +718, dv"dq"lghqwlw¦1

cf. Koskela-Onninen

Many extremals

!

expanding

linear on rank-one connections!"

Baernstein-Montgomery-Smith�

E(3,U)Es(Gj) = ヽ

� U

3

�

[ ヾ(w) ]s

ws−5

�

�

gw = ヽU5

ヾ(w)w

� ˙ヾ (w), ヾ(w) = r(w −

5s )

j(}) = ヾ(|}|)}|}|

Stretching vs Rotation

stretching rotation

quasiconformal bilipschitz

Grötzsch problem John’s problem

Hölder exponent rate of spiralling

log J(z,f) ! BMO arg f ! BMO

higher integrability exponential integrability

(linear) multifractifractal spectrum

harmonic dependece “conjugate harmonic”

z

Complex exponentsQ: What complex exponents $ can we take to make

i é} ∈ O4orf ?

0 2

“null-Lagrangians”

controls

rotation & stretching

joint multifractal spectrum

eccentricity = ellipticity coefficient = k|é|+ |é − 5| <

5n

foci =

•mappings of finite distortion (Eero)

• distortion of Hausdorff measures, removability (Ignacio)

• quasisymmetric maps, harmonic measure

• higher/even dimensions...

Outlook