INTERPOLATION AND THE BELTRAMI EQUATION
Transcript of INTERPOLATION AND THE BELTRAMI EQUATION
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}
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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