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Quasicircles, quasiconformal extensions andthe Corona Theorem
María José GonzálezUniversidad de Cádiz
Celebrating J. Garnett and D. MarshallSeattle 2019
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Corona Problem
Theorem (Carleson). Let f1(z), ..., fn(z) be given functions inH∞(D) with ||fk ||∞ ≤ 1, k = 1,2, ...,n, and verifying that forsome δ > 0,
|f1(z)|+ |f2(z)|+ ...|fn(z)| ≥ δ > 0,
Then, there exist gknk=0 ∈ H∞(D), so that:
n∑k=1
fkgk = 1
and ||gk ||∞ ≤ C(n, δ).
The functions fk and gk are called corona data and coronasolutions respectively.
OPEN PROBLEM: Is Corona true for any domain in the plane?
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Corona Problem
Theorem (Carleson). Let f1(z), ..., fn(z) be given functions inH∞(D) with ||fk ||∞ ≤ 1, k = 1,2, ...,n, and verifying that forsome δ > 0,
|f1(z)|+ |f2(z)|+ ...|fn(z)| ≥ δ > 0,
Then, there exist gknk=0 ∈ H∞(D), so that:
n∑k=1
fkgk = 1
and ||gk ||∞ ≤ C(n, δ).
The functions fk and gk are called corona data and coronasolutions respectively.
OPEN PROBLEM: Is Corona true for any domain in the plane?
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Corona Problem
Theorem (Carleson). Let f1(z), ..., fn(z) be given functions inH∞(D) with ||fk ||∞ ≤ 1, k = 1,2, ...,n, and verifying that forsome δ > 0,
|f1(z)|+ |f2(z)|+ ...|fn(z)| ≥ δ > 0,
Then, there exist gknk=0 ∈ H∞(D), so that:
n∑k=1
fkgk = 1
and ||gk ||∞ ≤ C(n, δ).
The functions fk and gk are called corona data and coronasolutions respectively.
OPEN PROBLEM: Is Corona true for any domain in the plane?
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Carleson’s Theorem
Carleson, L., Interpolation by bounded analytic functions andthe corona problem, Ann. of Math. 76 (1962), 547-559.
Carleson measures: µ a positive measure in D , andf ∈ Hp, p ≥ 1∫
D|f |p dµ ≤ ‖f‖pp iff µ(Q) ≤ c l(Q)
for any Carleson cube Q ⊂ D.
Carleson Contour: System of curves Γ where the analyticfunction is not too big, not too small, i.e. ε < |f | < εk ; k < 1,and arc-lengh Γ is a Carleson measure.
Corona decomposition
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Carleson’s Theorem
Carleson, L., Interpolation by bounded analytic functions andthe corona problem, Ann. of Math. 76 (1962), 547-559.
Carleson measures: µ a positive measure in D , andf ∈ Hp, p ≥ 1∫
D|f |p dµ ≤ ‖f‖pp iff µ(Q) ≤ c l(Q)
for any Carleson cube Q ⊂ D.
Carleson Contour: System of curves Γ where the analyticfunction is not too big, not too small, i.e. ε < |f | < εk ; k < 1,and arc-lengh Γ is a Carleson measure.
Corona decomposition
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Carleson’s Theorem
Carleson, L., Interpolation by bounded analytic functions andthe corona problem, Ann. of Math. 76 (1962), 547-559.
Carleson measures: µ a positive measure in D , andf ∈ Hp, p ≥ 1∫
D|f |p dµ ≤ ‖f‖pp iff µ(Q) ≤ c l(Q)
for any Carleson cube Q ⊂ D.
Carleson Contour: System of curves Γ where the analyticfunction is not too big, not too small, i.e. ε < |f | < εk ; k < 1,and arc-lengh Γ is a Carleson measure.
Corona decomposition
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Interpolation
Let B(z) and C(z) be Blaschke products with zeros (bn) and(cn) respectively, and such that |B(z)|+ |C(z)| ≥ δ > 0
The following two problems are equivalent
1 Solve Corona problem with corona data B(z) and C(z).
2 Construct f ∈ H∞(D), with ‖f‖ ≤ c(δ) such that f (bn) = 0and f (cn) = 1
f = B h; h ∈ H∞(D)
1− f = C g g ∈ H∞(D)
Then B h + C g = 1
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Interpolation
Let B(z) and C(z) be Blaschke products with zeros (bn) and(cn) respectively, and such that |B(z)|+ |C(z)| ≥ δ > 0
The following two problems are equivalent
1 Solve Corona problem with corona data B(z) and C(z).
2 Construct f ∈ H∞(D), with ‖f‖ ≤ c(δ) such that f (bn) = 0and f (cn) = 1
f = B h; h ∈ H∞(D)
1− f = C g g ∈ H∞(D)
Then B h + C g = 1
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Interpolation
Let B(z) and C(z) be Blaschke products with zeros (bn) and(cn) respectively, and such that |B(z)|+ |C(z)| ≥ δ > 0
The following two problems are equivalent
1 Solve Corona problem with corona data B(z) and C(z).
2 Construct f ∈ H∞(D), with ‖f‖ ≤ c(δ) such that f (bn) = 0and f (cn) = 1
f = B h; h ∈ H∞(D)
1− f = C g g ∈ H∞(D)
Then B h + C g = 1
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Solving interpolation
By duality, if wn is the sequence of 0’s and 1’s
‖f‖∞ = sup
∣∣∣∣∑ G(wn) wn
(B C)′(wn)
∣∣∣∣ ; G ∈ H1, ‖G‖1 ≤ 1
Let Γ be the Carleson contour for C(z) with εk < δ/2.Recall that |B(z)|+ |C(z)| > δ.
‖f‖∞ = sup
∣∣∣∣∑ G(cn)
(B C)′(cn)
∣∣∣∣ = sup
∣∣∣∣ 12πi
∫Γ
G(z)
B(z) C(z)dz∣∣∣∣
< c(δ)
∫Γ|G(z)|ds < C(δ)‖G‖1 ≤ c(δ).
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Solving interpolation
By duality, if wn is the sequence of 0’s and 1’s
‖f‖∞ = sup
∣∣∣∣∑ G(wn) wn
(B C)′(wn)
∣∣∣∣ ; G ∈ H1, ‖G‖1 ≤ 1
Let Γ be the Carleson contour for C(z) with εk < δ/2.Recall that |B(z)|+ |C(z)| > δ.
‖f‖∞ = sup
∣∣∣∣∑ G(cn)
(B C)′(cn)
∣∣∣∣ = sup
∣∣∣∣ 12πi
∫Γ
G(z)
B(z) C(z)dz∣∣∣∣
< c(δ)
∫Γ|G(z)|ds < C(δ)‖G‖1 ≤ c(δ).
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
∂- problem
Wolff,T. Published by Gamelin, T. W. (1980), Wolff’s proof of thecorona theorem, Israel Journal of Mathematics, 37, 113-119
∂- problem: Corona problem for two functions: f1 and f2. Firstfind solutions ϕ1 and ϕ2 NOT necessarily analytic. Set
g1 = ϕ1 + b f2
g2 = ϕ2 − b f1
Want b such that∂ϕ1 + ∂b f2 = 0
∂ϕ2 − ∂b f1 = 0
Therefore, to solve corona it is enough to solve
∂b = ϕ2∂ϕ1 − ϕ1∂ϕ2
Wolff‘s choice: ϕj = fj/∑|fj |2
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
∂- problem
Wolff,T. Published by Gamelin, T. W. (1980), Wolff’s proof of thecorona theorem, Israel Journal of Mathematics, 37, 113-119
∂- problem: Corona problem for two functions: f1 and f2. Firstfind solutions ϕ1 and ϕ2 NOT necessarily analytic. Set
g1 = ϕ1 + b f2
g2 = ϕ2 − b f1
Want b such that∂ϕ1 + ∂b f2 = 0
∂ϕ2 − ∂b f1 = 0
Therefore, to solve corona it is enough to solve
∂b = ϕ2∂ϕ1 − ϕ1∂ϕ2
Wolff‘s choice: ϕj = fj/∑|fj |2
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Solving ∂-problem: particular case
Let µ be a Carleson measure in D.
Want to solve∂b = µ
First try:
b(z) = F (z) =1π
∫C
dµ(w)
w − z
BUT F(z) might NOT be bounded. By duality
‖b‖∞ = supG∈H1,‖G‖1≤1
∣∣∣∣ 12πi
∫∂D
F (z)G(z) dz∣∣∣∣
' sup
∣∣∣∣∫D∂ (F G) dx dy
∣∣∣∣ ≤ sup
∫D|G| dµ ≤ c ‖G‖1 ≤ c
where the constant c depends on the Carleson constant of µ.
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Solving ∂-problem: particular case
Let µ be a Carleson measure in D.
Want to solve∂b = µ
First try:
b(z) = F (z) =1π
∫C
dµ(w)
w − z
BUT F(z) might NOT be bounded. By duality
‖b‖∞ = supG∈H1,‖G‖1≤1
∣∣∣∣ 12πi
∫∂D
F (z)G(z) dz∣∣∣∣
' sup
∣∣∣∣∫D∂ (F G) dx dy
∣∣∣∣ ≤ sup
∫D|G| dµ ≤ c ‖G‖1 ≤ c
where the constant c depends on the Carleson constant of µ.
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Multiply connected domains
Carleson, L., On H∞ in multiply connected domains.Conference on Harmonic Analysis in Honor of AntoniZygmund, 1983, pp. 349-372.
Denjoy domains with thick boundary: Domains havingboundary E ⊂ R satisfying, for some ε > 0,
|(x − r , x + r) ∩ E | ≥ εrfor every x ⊂ E and r > 0.
Jones, P. W. and Marshall, D. E., Critical points of Green’sfunction, harmonic measure, and the corona problem, Ark.för Mat. 23 (1985), no. 2, 281-314.
They extend the result to thick Cantor sets and they showthat it is enough to solve the corona problem at the criticalpoints of the Green function.
Idea: Construct an explicit projection from H∞(D) onto H∞Γ (Ω).María José González Universidad de Cádiz
Quasicircles, quasiconformal extensions and the Corona Theorem
Multiply connected domains
Carleson, L., On H∞ in multiply connected domains.Conference on Harmonic Analysis in Honor of AntoniZygmund, 1983, pp. 349-372.
Denjoy domains with thick boundary: Domains havingboundary E ⊂ R satisfying, for some ε > 0,
|(x − r , x + r) ∩ E | ≥ εrfor every x ⊂ E and r > 0.
Jones, P. W. and Marshall, D. E., Critical points of Green’sfunction, harmonic measure, and the corona problem, Ark.för Mat. 23 (1985), no. 2, 281-314.
They extend the result to thick Cantor sets and they showthat it is enough to solve the corona problem at the criticalpoints of the Green function.
Idea: Construct an explicit projection from H∞(D) onto H∞Γ (Ω).María José González Universidad de Cádiz
Quasicircles, quasiconformal extensions and the Corona Theorem
Multiply connected domains
Carleson, L., On H∞ in multiply connected domains.Conference on Harmonic Analysis in Honor of AntoniZygmund, 1983, pp. 349-372.
Denjoy domains with thick boundary: Domains havingboundary E ⊂ R satisfying, for some ε > 0,
|(x − r , x + r) ∩ E | ≥ εrfor every x ⊂ E and r > 0.
Jones, P. W. and Marshall, D. E., Critical points of Green’sfunction, harmonic measure, and the corona problem, Ark.för Mat. 23 (1985), no. 2, 281-314.
They extend the result to thick Cantor sets and they showthat it is enough to solve the corona problem at the criticalpoints of the Green function.
Idea: Construct an explicit projection from H∞(D) onto H∞Γ (Ω).María José González Universidad de Cádiz
Quasicircles, quasiconformal extensions and the Corona Theorem
No thickness
Garnett, J.B., Jones, P.W.: The corona theorem for Denjoydomains. Acta Math. 155, 27-40 (1985).
NO assumption on thickness1 Divide the domain Ω into two almost disjoint regions: Ω1
where ω(z,E) > ε and Ω2 = Ω \ Ω1(cε).2 Solve Corona in each of the components of Ω1, because
they are simply connected (Max. principle).3 Find explicit solutions in Ω2 using that f (z) is analytic.4 Glue them together by solving a ∂-problem: First find a
Carleson contour Γ in Ω1 ∩ Ω2, define the region (diamondshape) D = z; dist(z, Γ) < α y. Then there is Ψsupported on D, which is 0 or 1 on each side of D, andsuch that |y∇ψ| is a Carleson mesaure. Set
ϕj = Gj,1(1−Ψ) + Gj,2Ψ
Solve the corresponding ∂-problem ( No duality, by hand!!!)María José González Universidad de Cádiz
Quasicircles, quasiconformal extensions and the Corona Theorem
More results
Moore, C.N.: The corona Theorem for domains whoseboundary lies in a smooth curve. P. Am. Math. Soc. 100,266-270 (1987).Handy, J.: The Corona Theorem on the Complement ofCertain Square Cantor Sets. J. Anal. Math. 108, 1-18(2009).Newdelman, B. M.: Homogeneous subsets of a lipschitzgraph and the corona theorem. Publ. Mat. 55, 93-121(2011).
Open: Corona for the 1/3-Cantor set or Corona fordomains with bounday contained on a chord- arc curve (nothickness).
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
More results
Moore, C.N.: The corona Theorem for domains whoseboundary lies in a smooth curve. P. Am. Math. Soc. 100,266-270 (1987).Handy, J.: The Corona Theorem on the Complement ofCertain Square Cantor Sets. J. Anal. Math. 108, 1-18(2009).Newdelman, B. M.: Homogeneous subsets of a lipschitzgraph and the corona theorem. Publ. Mat. 55, 93-121(2011).
Open: Corona for the 1/3-Cantor set or Corona fordomains with bounday contained on a chord- arc curve (nothickness).
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Different approach ( joint work with J.M. Salamanca)
Let ρ be a quasiconformal mapping on the plane withcomplex dilatation µ, ‖µ‖∞ < 1..
∂ρ− µ∂ρ = 0
A Jordan curve Γ passing through∞ is a quasicircle if it isthe image of the real line R under a quasiconformalmapping on the plane.
Geometric properties of Γ←→ Conditions on µ.
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Cauchy Integral
Given a function f on a rectifiable curve Γ, define its Cauchyintegral F (z) = CΓf (z) off Γ by
F (z) =1
2π
∫Γ
f (w)
w − zdw , z /∈ Γ.
If F+ and F− are the restrictions of F to Ω+ and Ω−, and if f+and f− denote their boundary values, then the classical Plemeljformula states that
f±(z) = ±12
f (z) +1
2πP.V.
∫Γ
f (w)
w − zdw , z ∈ Γ.
The singular integral is also called the Cauchy integral.
In particular the jump of F is f . Write j(F ) = f .
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Cauchy Integral
Given a function f on a rectifiable curve Γ, define its Cauchyintegral F (z) = CΓf (z) off Γ by
F (z) =1
2π
∫Γ
f (w)
w − zdw , z /∈ Γ.
If F+ and F− are the restrictions of F to Ω+ and Ω−, and if f+and f− denote their boundary values, then the classical Plemeljformula states that
f±(z) = ±12
f (z) +1
2πP.V.
∫Γ
f (w)
w − zdw , z ∈ Γ.
The singular integral is also called the Cauchy integral.
In particular the jump of F is f . Write j(F ) = f .
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Passing to a Denjoy domain
Consider a domain of the form
Ω = C \ E
where E ⊂ Γ is a compact set with positive length contained ina rectifiable quasicircle Γ = ρ (∂R).
Set Ω0 = ρ−1(Ω) and E0 = ρ−1(E). Note that Ω0 is a Denjoydomain.
Define the space:
H∞(Ω0, µ) = g = f ρ : f ∈ H∞(Ω)
Then,∂f = 0 on Ω⇔ ∂g − µ∂g = 0 on Ω0
The jump of g across E0 is given by j(g) = j(f ) ρ.
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Passing to a Denjoy domain
Consider a domain of the form
Ω = C \ E
where E ⊂ Γ is a compact set with positive length contained ina rectifiable quasicircle Γ = ρ (∂R).
Set Ω0 = ρ−1(Ω) and E0 = ρ−1(E). Note that Ω0 is a Denjoydomain.
Define the space:
H∞(Ω0, µ) = g = f ρ : f ∈ H∞(Ω)
Then,∂f = 0 on Ω⇔ ∂g − µ∂g = 0 on Ω0
The jump of g across E0 is given by j(g) = j(f ) ρ.
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Dictionary
Idea: Translate Corona data in Ω to Corona data in the Denjoydomain Ω0.
Let f ∈ H∞(Ω) and g = f ρ ∈ H∞(Ω0, µ).
Set g = CR(j(g)), where j(g) is the jump of g. Then g isanalytic in Ω0, but NOT necessarily bounded.Define
H = g − g
Then∂H = µ ∂g
Since H has no jump across E0, we can consider that thisequation holds on all C.
REMARK: H ∈ L∞(C) iff g ∈ H∞(Ω0).
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Dictionary
Idea: Translate Corona data in Ω to Corona data in the Denjoydomain Ω0.
Let f ∈ H∞(Ω) and g = f ρ ∈ H∞(Ω0, µ).
Set g = CR(j(g)), where j(g) is the jump of g. Then g isanalytic in Ω0, but NOT necessarily bounded.Define
H = g − g
Then∂H = µ ∂g
Since H has no jump across E0, we can consider that thisequation holds on all C.
REMARK: H ∈ L∞(C) iff g ∈ H∞(Ω0).
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Dictionary
Idea: Translate Corona data in Ω to Corona data in the Denjoydomain Ω0.
Let f ∈ H∞(Ω) and g = f ρ ∈ H∞(Ω0, µ).
Set g = CR(j(g)), where j(g) is the jump of g. Then g isanalytic in Ω0, but NOT necessarily bounded.Define
H = g − g
Then∂H = µ ∂g
Since H has no jump across E0, we can consider that thisequation holds on all C.
REMARK: H ∈ L∞(C) iff g ∈ H∞(Ω0).
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Solving ∂
Recall that f ∈ L∞(Γ) and g = f ρ. Want to solve ∂H = µ∂g
H(z0) =1π
∫C
∂Hz − z0
dxdy =1π
∫C
µ∂gz − z0
dxdy
for all z0 ∈ C.
PROBLEM: Find conditions on µ so that H ∈ L∞(C).
Theorem: If the quasicircle is C1+α, then the conformalmapping can be entended to a quasiconformal map with|µ|2/y1+ε Carleson measure. The converse is also true.
Theorem: Under such conditions, H is bounded in C , andCorona holds. (We recover Moore‘s result).
Idea: Show that the dictionary provides corona data on Ω0, useGarnett-Jones to find Corona solutions on Ω0, use thedictionary to get analytic functions in Ω. Modify them to get thecorona solutions ( localization argument by T. Gamelin)
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Solving ∂
Recall that f ∈ L∞(Γ) and g = f ρ. Want to solve ∂H = µ∂g
H(z0) =1π
∫C
∂Hz − z0
dxdy =1π
∫C
µ∂gz − z0
dxdy
for all z0 ∈ C.
PROBLEM: Find conditions on µ so that H ∈ L∞(C).
Theorem: If the quasicircle is C1+α, then the conformalmapping can be entended to a quasiconformal map with|µ|2/y1+ε Carleson measure. The converse is also true.
Theorem: Under such conditions, H is bounded in C , andCorona holds. (We recover Moore‘s result).
Idea: Show that the dictionary provides corona data on Ω0, useGarnett-Jones to find Corona solutions on Ω0, use thedictionary to get analytic functions in Ω. Modify them to get thecorona solutions ( localization argument by T. Gamelin)
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Solving ∂
Recall that f ∈ L∞(Γ) and g = f ρ. Want to solve ∂H = µ∂g
H(z0) =1π
∫C
∂Hz − z0
dxdy =1π
∫C
µ∂gz − z0
dxdy
for all z0 ∈ C.
PROBLEM: Find conditions on µ so that H ∈ L∞(C).
Theorem: If the quasicircle is C1+α, then the conformalmapping can be entended to a quasiconformal map with|µ|2/y1+ε Carleson measure. The converse is also true.
Theorem: Under such conditions, H is bounded in C , andCorona holds. (We recover Moore‘s result).
Idea: Show that the dictionary provides corona data on Ω0, useGarnett-Jones to find Corona solutions on Ω0, use thedictionary to get analytic functions in Ω. Modify them to get thecorona solutions ( localization argument by T. Gamelin)
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Solving ∂
Recall that f ∈ L∞(Γ) and g = f ρ. Want to solve ∂H = µ∂g
H(z0) =1π
∫C
∂Hz − z0
dxdy =1π
∫C
µ∂gz − z0
dxdy
for all z0 ∈ C.
PROBLEM: Find conditions on µ so that H ∈ L∞(C).
Theorem: If the quasicircle is C1+α, then the conformalmapping can be entended to a quasiconformal map with|µ|2/y1+ε Carleson measure. The converse is also true.
Theorem: Under such conditions, H is bounded in C , andCorona holds. (We recover Moore‘s result).
Idea: Show that the dictionary provides corona data on Ω0, useGarnett-Jones to find Corona solutions on Ω0, use thedictionary to get analytic functions in Ω. Modify them to get thecorona solutions ( localization argument by T. Gamelin)
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
More conditions
Theorem: If µ satisfies any of the following two conditions, thenCorona holds
1 ∫0
µ∗(t)|t |
log(1/|t |)dt <∞
where µ∗(t) = sup|µ(z)| : 0 < |Im(z)| < |t |
2 ∫R
σ(y)
|y |3/2 dy <∞
where σ(y) = (∫R |µ(x + iy)|2 dx)1/2, a.a. y ∈ R, and
|µ(z0)| .∫|z−z0|<C|y0|
|µ(z)|dxdy , z0 ∈ C \ R
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Smooth curves
Remark : In both cases, for all a ∈ R,∫C
|µ(z)||z − a|
dxdy|y |
< M
and therefore the quasicircle is smooth (Gutlyanskii and Martio)
Example: Let h be the conformal map taking D onto the ballB(9/10,1/10),h(z) = (9 + z)/10. Consider:
g(z) = 2z +1− z
log(1− z)
and set f = g h.
Then f is conformal in D and, Γ = f (∂D) is smooth. It has aquasiconformal extension satisfying condition (2), BUT it isNOT Dini smooth.
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
Smooth curves
Remark : In both cases, for all a ∈ R,∫C
|µ(z)||z − a|
dxdy|y |
< M
and therefore the quasicircle is smooth (Gutlyanskii and Martio)
Example: Let h be the conformal map taking D onto the ballB(9/10,1/10),h(z) = (9 + z)/10. Consider:
g(z) = 2z +1− z
log(1− z)
and set f = g h.
Then f is conformal in D and, Γ = f (∂D) is smooth. It has aquasiconformal extension satisfying condition (2), BUT it isNOT Dini smooth.
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem
T HANK YOU DON AND JOHN
for your Math and for your Kindness.
María José González Universidad de CádizQuasicircles, quasiconformal extensions and the Corona Theorem