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Quasicircles, quasiconformal extensions and the Corona Theorem

María José González Universidad de Cádiz Celebrating J. Garnett and D. Marshall Seattle 2019

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Corona Problem

Theorem (Carleson). Let f1(z), ..., fn(z) be given functions in H∞(D) with ||fk||∞ ≤ 1, k = 1, 2, ..., n, and verifying that for some δ > 0, |f1(z)| + |f2(z)| + ...|fn(z)| ≥ δ > 0, Then, there exist {gk}n

k=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 corona solutions respectively. OPEN PROBLEM: Is Corona true for any domain in the plane?

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Corona Problem

Theorem (Carleson). Let f1(z), ..., fn(z) be given functions in H∞(D) with ||fk||∞ ≤ 1, k = 1, 2, ..., n, and verifying that for some δ > 0, |f1(z)| + |f2(z)| + ...|fn(z)| ≥ δ > 0, Then, there exist {gk}n

k=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 corona solutions respectively. OPEN PROBLEM: Is Corona true for any domain in the plane?

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Corona Problem

Theorem (Carleson). Let f1(z), ..., fn(z) be given functions in H∞(D) with ||fk||∞ ≤ 1, k = 1, 2, ..., n, and verifying that for some δ > 0, |f1(z)| + |f2(z)| + ...|fn(z)| ≥ δ > 0, Then, there exist {gk}n

k=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 corona solutions respectively. OPEN PROBLEM: Is Corona true for any domain in the plane?

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Carleson’s Theorem

Carleson, L., Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547-559. Carleson measures: µ a positive measure in D , and f ∈ Hp, p ≥ 1

• D

|f|p dµ ≤ fp

p

iff µ(Q) ≤ c l(Q) for any Carleson cube Q ⊂ D. Carleson Contour: System of curves Γ where the analytic function is not too big, not too small, i.e. ǫ < |f| < ǫk; k < 1, and arc-lengh Γ is a Carleson measure. Corona decomposition

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Carleson’s Theorem

Carleson, L., Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547-559. Carleson measures: µ a positive measure in D , and f ∈ Hp, p ≥ 1

• D

|f|p dµ ≤ fp

p

iff µ(Q) ≤ c l(Q) for any Carleson cube Q ⊂ D. Carleson Contour: System of curves Γ where the analytic function is not too big, not too small, i.e. ǫ < |f| < ǫk; k < 1, and arc-lengh Γ is a Carleson measure. Corona decomposition

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Carleson’s Theorem

Carleson, L., Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547-559. Carleson measures: µ a positive measure in D , and f ∈ Hp, p ≥ 1

• D

|f|p dµ ≤ fp

p

iff µ(Q) ≤ c l(Q) for any Carleson cube Q ⊂ D. Carleson Contour: System of curves Γ where the analytic function is not too big, not too small, i.e. ǫ < |f| < ǫk; k < 1, and arc-lengh Γ is a Carleson measure. Corona decomposition

María José González Universidad de Cádiz Quasicircles, 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) = 0 and 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 Cádiz Quasicircles, 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) = 0 and 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 Cádiz Quasicircles, 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) = 0 and 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 Cádiz Quasicircles, 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, G1 ≤ 1 Let Γ be the Carleson contour for C(z) with ǫk < δ/2. Recall that |B(z)| + |C(z)| > δ. f∞ = sup

• G(cn)

(B C)′(cn)

• = sup
• 1

2πi

• Γ

G(z) B(z) C(z)dz

• < c(δ)
• Γ

|G(z)|ds < C(δ)G1 ≤ c(δ).

María José González Universidad de Cádiz Quasicircles, 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, G1 ≤ 1 Let Γ be the Carleson contour for C(z) with ǫk < δ/2. Recall that |B(z)| + |C(z)| > δ. f∞ = sup

• G(cn)

(B C)′(cn)

• = sup
• 1

2πi

• Γ

G(z) B(z) C(z)dz

• < c(δ)
• Γ

|G(z)|ds < C(δ)G1 ≤ c(δ).

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

¯ ∂- problem

Wolff,T. Published by Gamelin, T. W. (1980), Wolff’s proof of the corona theorem, Israel Journal of Mathematics, 37, 113-119 ¯ ∂- problem: Corona problem for two functions: f1 and f2. First find 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 Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

¯ ∂- problem

Wolff,T. Published by Gamelin, T. W. (1980), Wolff’s proof of the corona theorem, Israel Journal of Mathematics, 37, 113-119 ¯ ∂- problem: Corona problem for two functions: f1 and f2. First find 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 Cádiz Quasicircles, 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) = 1 π

• C

dµ(w) w − z BUT F(z) might NOT be bounded. By duality b∞ = sup

G∈H1,G1≤1

• 1

2πi

• ∂D

F(z)G(z) dz

• ≃ sup
• D

¯ ∂ (F G) dx dy

• ≤ sup
• D

|G| dµ ≤ c G1 ≤ c where the constant c depends on the Carleson constant of µ.

María José González Universidad de Cádiz Quasicircles, 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) = 1 π

• C

dµ(w) w − z BUT F(z) might NOT be bounded. By duality b∞ = sup

G∈H1,G1≤1

• 1

2πi

• ∂D

F(z)G(z) dz

• ≃ sup
• D

¯ ∂ (F G) dx dy

• ≤ sup
• D

|G| dµ ≤ c G1 ≤ c where the constant c depends on the Carleson constant of µ.

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 Antoni Zygmund, 1983, pp. 349-372. Denjoy domains with thick boundary: Domains having boundary E ⊂ R satisfying, for some ǫ > 0, |(x − r, x + r) ∩ E| ≥ ǫr for every x ⊂ E and r > 0. Jones, P . W. and Marshall, D. E., Critical points of Green’s function, 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 show that it is enough to solve the corona problem at the critical points 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 Antoni Zygmund, 1983, pp. 349-372. Denjoy domains with thick boundary: Domains having boundary E ⊂ R satisfying, for some ǫ > 0, |(x − r, x + r) ∩ E| ≥ ǫr for every x ⊂ E and r > 0. Jones, P . W. and Marshall, D. E., Critical points of Green’s function, 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 show that it is enough to solve the corona problem at the critical points 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 Antoni Zygmund, 1983, pp. 349-372. Denjoy domains with thick boundary: Domains having boundary E ⊂ R satisfying, for some ǫ > 0, |(x − r, x + r) ∩ E| ≥ ǫr for every x ⊂ E and r > 0. Jones, P . W. and Marshall, D. E., Critical points of Green’s function, 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 show that it is enough to solve the corona problem at the critical points 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 Denjoy

• domains. Acta Math. 155, 27-40 (1985).

NO assumption on thickness

1

Divide the domain Ω into two almost disjoint regions: Ω1 where ω(z, E) > ǫ and Ω2 = Ω \ Ω1(cǫ).

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 (diamond shape) D = {z; dist(z, Γ) < α y}. Then there is Ψ supported on D, which is 0 or 1 on each side of D, and such 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 whose boundary lies in a smooth curve. P . Am. Math. Soc. 100, 266-270 (1987). Handy, J.: The Corona Theorem on the Complement of Certain Square Cantor Sets. J. Anal. Math. 108, 1-18 (2009). Newdelman, B. M.: Homogeneous subsets of a lipschitz graph and the corona theorem. Publ. Mat. 55, 93-121 (2011). Open: Corona for the 1/3-Cantor set or Corona for domains with bounday contained on a chord- arc curve (no thickness).

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

More results

Moore, C.N.: The corona Theorem for domains whose boundary lies in a smooth curve. P . Am. Math. Soc. 100, 266-270 (1987). Handy, J.: The Corona Theorem on the Complement of Certain Square Cantor Sets. J. Anal. Math. 108, 1-18 (2009). Newdelman, B. M.: Homogeneous subsets of a lipschitz graph and the corona theorem. Publ. Mat. 55, 93-121 (2011). Open: Corona for the 1/3-Cantor set or Corona for domains with bounday contained on a chord- arc curve (no thickness).

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Different approach ( joint work with J.M. Salamanca)

Let ρ be a quasiconformal mapping on the plane with complex dilatation µ, µ∞ < 1.. ∂ρ − µ∂ρ = 0 A Jordan curve Γ passing through ∞ is a quasicircle if it is the image of the real line R under a quasiconformal mapping on the plane. Geometric properties of Γ ← → Conditions on µ.

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Cauchy Integral

Given a function f on a rectifiable curve Γ, define its Cauchy integral F(z) = CΓf(z) off Γ by F(z) = 1 2π

• Γ

f(w) w − z dw, 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 Plemelj formula states that f±(z) = ±1 2f(z) + 1 2π P .V.

• Γ

f(w) w − z dw, 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 Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Cauchy Integral

Given a function f on a rectifiable curve Γ, define its Cauchy integral F(z) = CΓf(z) off Γ by F(z) = 1 2π

• Γ

f(w) w − z dw, 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 Plemelj formula states that f±(z) = ±1 2f(z) + 1 2π P .V.

• Γ

f(w) w − z dw, 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 Cádiz Quasicircles, 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 in a rectifiable quasicircle Γ = ρ (∂R). Set Ω0 = ρ−1(Ω) and E0 = ρ−1(E). Note that Ω0 is a Denjoy domain. 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 Cádiz Quasicircles, 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 in a rectifiable quasicircle Γ = ρ (∂R). Set Ω0 = ρ−1(Ω) and E0 = ρ−1(E). Note that Ω0 is a Denjoy domain. 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 Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Dictionary

Idea: Translate Corona data in Ω to Corona data in the Denjoy domain 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 is analytic 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 this equation holds on all C. REMARK: H ∈ L∞(C) iff ˜ g ∈ H∞(Ω0).

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Dictionary

Idea: Translate Corona data in Ω to Corona data in the Denjoy domain 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 is analytic 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 this equation holds on all C. REMARK: H ∈ L∞(C) iff ˜ g ∈ H∞(Ω0).

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Dictionary

Idea: Translate Corona data in Ω to Corona data in the Denjoy domain 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 is analytic 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 this equation holds on all C. REMARK: H ∈ L∞(C) iff ˜ g ∈ H∞(Ω0).

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Solving ¯ ∂

Recall that f ∈ L∞(Γ) and g = f ◦ ρ. Want to solve ¯ ∂H = µ∂g H(z0) = 1 π

• C

¯ ∂H z − z0 dxdy = 1 π

• C

µ∂g z − 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 conformal mapping can be entended to a quasiconformal map with |µ|2/y1+ǫ Carleson measure. The converse is also true. Theorem: Under such conditions, H is bounded in C , and Corona holds. (We recover Moore‘s result). Idea: Show that the dictionary provides corona data on Ω0, use Garnett-Jones to find Corona solutions on Ω0, use the dictionary to get analytic functions in Ω. Modify them to get the corona solutions ( localization argument by T. Gamelin)

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Solving ¯ ∂

Recall that f ∈ L∞(Γ) and g = f ◦ ρ. Want to solve ¯ ∂H = µ∂g H(z0) = 1 π

• C

¯ ∂H z − z0 dxdy = 1 π

• C

µ∂g z − 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 conformal mapping can be entended to a quasiconformal map with |µ|2/y1+ǫ Carleson measure. The converse is also true. Theorem: Under such conditions, H is bounded in C , and Corona holds. (We recover Moore‘s result). Idea: Show that the dictionary provides corona data on Ω0, use Garnett-Jones to find Corona solutions on Ω0, use the dictionary to get analytic functions in Ω. Modify them to get the corona solutions ( localization argument by T. Gamelin)

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Solving ¯ ∂

Recall that f ∈ L∞(Γ) and g = f ◦ ρ. Want to solve ¯ ∂H = µ∂g H(z0) = 1 π

• C

¯ ∂H z − z0 dxdy = 1 π

• C

µ∂g z − 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 conformal mapping can be entended to a quasiconformal map with |µ|2/y1+ǫ Carleson measure. The converse is also true. Theorem: Under such conditions, H is bounded in C , and Corona holds. (We recover Moore‘s result). Idea: Show that the dictionary provides corona data on Ω0, use Garnett-Jones to find Corona solutions on Ω0, use the dictionary to get analytic functions in Ω. Modify them to get the corona solutions ( localization argument by T. Gamelin)

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

Solving ¯ ∂

Recall that f ∈ L∞(Γ) and g = f ◦ ρ. Want to solve ¯ ∂H = µ∂g H(z0) = 1 π

• C

¯ ∂H z − z0 dxdy = 1 π

• C

µ∂g z − 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 conformal mapping can be entended to a quasiconformal map with |µ|2/y1+ǫ Carleson measure. The converse is also true. Theorem: Under such conditions, H is bounded in C , and Corona holds. (We recover Moore‘s result). Idea: Show that the dictionary provides corona data on Ω0, use Garnett-Jones to find Corona solutions on Ω0, use the dictionary to get analytic functions in Ω. Modify them to get the corona solutions ( localization argument by T. Gamelin)

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

More conditions

Theorem: If µ satisfies any of the following two conditions, then Corona holds

1

• 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 Cádiz Quasicircles, 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 ball B(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 a quasiconformal extension satisfying condition (2), BUT it is NOT Dini smooth.

María José González Universidad de Cádiz Quasicircles, 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 ball B(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 a quasiconformal extension satisfying condition (2), BUT it is NOT Dini smooth.

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem

T HANK YOU DON AND J OHN for your Math and for your Kindness.

María José González Universidad de Cádiz Quasicircles, quasiconformal extensions and the Corona Theorem