Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz Celebrating J. Garnett and D. Marshall Seattle 2019 Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
Corona Problem Theorem (Carleson). Let f 1 ( z ) , ..., f n ( z ) be given functions in H ∞ ( D ) with || f k || ∞ ≤ 1 , k = 1 , 2 , ..., n , and verifying that for some δ > 0, | f 1 ( z ) | + | f 2 ( z ) | + ... | f n ( z ) | ≥ δ > 0 , Then, there exist { g k } n k = 0 ∈ H ∞ ( D ) , so that: n � f k g k = 1 k = 1 and || g k || ∞ ≤ C ( n , δ ) . The functions { f k } and { g k } are called corona data and corona solutions respectively. OPEN PROBLEM: Is Corona true for any domain in the plane? Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
Corona Problem Theorem (Carleson). Let f 1 ( z ) , ..., f n ( z ) be given functions in H ∞ ( D ) with || f k || ∞ ≤ 1 , k = 1 , 2 , ..., n , and verifying that for some δ > 0, | f 1 ( z ) | + | f 2 ( z ) | + ... | f n ( z ) | ≥ δ > 0 , Then, there exist { g k } n k = 0 ∈ H ∞ ( D ) , so that: n � f k g k = 1 k = 1 and || g k || ∞ ≤ C ( n , δ ) . The functions { f k } and { g k } are called corona data and corona solutions respectively. OPEN PROBLEM: Is Corona true for any domain in the plane? Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
Corona Problem Theorem (Carleson). Let f 1 ( z ) , ..., f n ( z ) be given functions in H ∞ ( D ) with || f k || ∞ ≤ 1 , k = 1 , 2 , ..., n , and verifying that for some δ > 0, | f 1 ( z ) | + | f 2 ( z ) | + ... | f n ( z ) | ≥ δ > 0 , Then, there exist { g k } n k = 0 ∈ H ∞ ( D ) , so that: n � f k g k = 1 k = 1 and || g k || ∞ ≤ C ( n , δ ) . The functions { f k } and { g k } are called corona data and corona solutions respectively. OPEN PROBLEM: Is Corona true for any domain in the plane? Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
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 ∈ H p , p ≥ 1 � | f | p d µ ≤ � f � p µ ( Q ) ≤ c l ( Q ) iff p D 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 | < ǫ k ; k < 1, and arc-lengh Γ is a Carleson measure. Corona decomposition Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
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 ∈ H p , p ≥ 1 � | f | p d µ ≤ � f � p µ ( Q ) ≤ c l ( Q ) iff p D 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 | < ǫ k ; k < 1, and arc-lengh Γ is a Carleson measure. Corona decomposition Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
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 ∈ H p , p ≥ 1 � | f | p d µ ≤ � f � p µ ( Q ) ≤ c l ( Q ) iff p D 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 | < ǫ k ; k < 1, and arc-lengh Γ is a Carleson measure. Corona decomposition Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
Interpolation Let B ( z ) and C ( z ) be Blaschke products with zeros ( b n ) and ( c n ) respectively, and such that | B ( z ) | + | C ( z ) | ≥ δ > 0 The following two problems are equivalent Solve Corona problem with corona data B ( z ) and C ( z ) . 1 Construct f ∈ H ∞ ( D ) , with � f � ≤ c ( δ ) such that f ( b n ) = 0 2 and f ( c n ) = 1 f = B h ; h ∈ H ∞ ( D ) 1 − f = C g g ∈ H ∞ ( D ) Then B h + C g = 1 Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
Interpolation Let B ( z ) and C ( z ) be Blaschke products with zeros ( b n ) and ( c n ) respectively, and such that | B ( z ) | + | C ( z ) | ≥ δ > 0 The following two problems are equivalent Solve Corona problem with corona data B ( z ) and C ( z ) . 1 Construct f ∈ H ∞ ( D ) , with � f � ≤ c ( δ ) such that f ( b n ) = 0 2 and f ( c n ) = 1 f = B h ; h ∈ H ∞ ( D ) 1 − f = C g g ∈ H ∞ ( D ) Then B h + C g = 1 Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
Interpolation Let B ( z ) and C ( z ) be Blaschke products with zeros ( b n ) and ( c n ) respectively, and such that | B ( z ) | + | C ( z ) | ≥ δ > 0 The following two problems are equivalent Solve Corona problem with corona data B ( z ) and C ( z ) . 1 Construct f ∈ H ∞ ( D ) , with � f � ≤ c ( δ ) such that f ( b n ) = 0 2 and f ( c n ) = 1 f = B h ; h ∈ H ∞ ( D ) 1 − f = C g g ∈ H ∞ ( D ) Then B h + C g = 1 Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
Solving interpolation By duality, if w n is the sequence of 0’s and 1’s � G ( w n ) w n � � G ∈ H 1 , � G � 1 ≤ 1 � � � f � ∞ = sup � ; � � ( B C ) ′ ( w n ) � Let Γ be the Carleson contour for C ( z ) with ǫ k < δ/ 2. Recall that | B ( z ) | + | C ( z ) | > δ . � � � � G ( c n ) 1 � G ( z ) � � � � � � f � ∞ = sup � = sup B ( z ) C ( z ) dz � � � � ( B C ) ′ ( c n ) 2 π i � � � Γ � < c ( δ ) | G ( z ) | ds < C ( δ ) � G � 1 ≤ c ( δ ) . Γ Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
Solving interpolation By duality, if w n is the sequence of 0’s and 1’s � G ( w n ) w n � � G ∈ H 1 , � G � 1 ≤ 1 � � � f � ∞ = sup � ; � � ( B C ) ′ ( w n ) � Let Γ be the Carleson contour for C ( z ) with ǫ k < δ/ 2. Recall that | B ( z ) | + | C ( z ) | > δ . � � � � G ( c n ) 1 � G ( z ) � � � � � � f � ∞ = sup � = sup B ( z ) C ( z ) dz � � � � ( B C ) ′ ( c n ) 2 π i � � � Γ � < c ( δ ) | G ( z ) | ds < C ( δ ) � G � 1 ≤ c ( δ ) . Γ Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
¯ ∂ - 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: f 1 and f 2 . First find solutions ϕ 1 and ϕ 2 NOT necessarily analytic. Set g 1 = ϕ 1 + b f 2 g 2 = ϕ 2 − b f 1 Want b such that ∂ϕ 1 + ¯ ¯ ∂ b f 2 = 0 ∂ϕ 2 − ¯ ¯ ∂ b f 1 = 0 Therefore, to solve corona it is enough to solve ∂ b = ϕ 2 ¯ ¯ ∂ϕ 1 − ϕ 1 ¯ ∂ϕ 2 f j / � | f j | 2 Wolff‘s choice: ϕ j = ¯ Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
¯ ∂ - 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: f 1 and f 2 . First find solutions ϕ 1 and ϕ 2 NOT necessarily analytic. Set g 1 = ϕ 1 + b f 2 g 2 = ϕ 2 − b f 1 Want b such that ∂ϕ 1 + ¯ ¯ ∂ b f 2 = 0 ∂ϕ 2 − ¯ ¯ ∂ b f 1 = 0 Therefore, to solve corona it is enough to solve ∂ b = ϕ 2 ¯ ¯ ∂ϕ 1 − ϕ 1 ¯ ∂ϕ 2 f j / � | f j | 2 Wolff‘s choice: ϕ j = ¯ Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
Solving ¯ ∂ -problem: particular case Let µ be a Carleson measure in D. Want to solve ¯ ∂ b = µ First try: b ( z ) = F ( z ) = 1 � d µ ( w ) π w − z C BUT F(z) might NOT be bounded. By duality � � 1 � � � � b � ∞ = sup F ( z ) G ( z ) dz � � 2 π i G ∈ H 1 , � G � 1 ≤ 1 � ∂ D � � � � � ¯ � � ≃ sup ∂ ( F G ) dx dy � ≤ sup | G | d µ ≤ c � G � 1 ≤ c � � � D D where the constant c depends on the Carleson constant of µ . Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
Solving ¯ ∂ -problem: particular case Let µ be a Carleson measure in D. Want to solve ¯ ∂ b = µ First try: b ( z ) = F ( z ) = 1 � d µ ( w ) π w − z C BUT F(z) might NOT be bounded. By duality � � 1 � � � � b � ∞ = sup F ( z ) G ( z ) dz � � 2 π i G ∈ H 1 , � G � 1 ≤ 1 � ∂ D � � � � � ¯ � � ≃ sup ∂ ( F G ) dx dy � ≤ sup | G | d µ ≤ c � G � 1 ≤ c � � � D D where the constant c depends on the Carleson constant of µ . Quasicircles, quasiconformal extensions and the Corona Theorem María José González Universidad de Cádiz
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