Beurling and Grunsky.
Ilia Binder
2020 Virtual Bozeman Workshop
August 13, 2020
Ilia Binder Beurling and Grunsky. By Cauchy inequalities, Lp-convergence in Ap(Ω) implies locally uniform convergence, so Ap(Ω) is a closed subspace (a locally uniform limit of analytic functions is analytic). We will also need the conjugate space of anti-analytic functions:
p A (Ω) := Ap(Ω) = Lp(Ω) ∩ H(Ω)
We will mostly talk about p = 2.
Bergman Space
p Let Ω be a domain in C. The Bergman space A (Ω) is the subspace of analytic functions in Lp(Ω) Ap(Ω) := Lp(Ω) ∩ H(Ω) (Lp here is with respect to the area).
Ilia Binder Beurling and Grunsky. We will also need the conjugate space of anti-analytic functions:
p A (Ω) := Ap(Ω) = Lp(Ω) ∩ H(Ω)
We will mostly talk about p = 2.
Bergman Space
p Let Ω be a domain in C. The Bergman space A (Ω) is the subspace of analytic functions in Lp(Ω) Ap(Ω) := Lp(Ω) ∩ H(Ω) (Lp here is with respect to the area). By Cauchy inequalities, Lp-convergence in Ap(Ω) implies locally uniform convergence, so Ap(Ω) is a closed subspace (a locally uniform limit of analytic functions is analytic).
Ilia Binder Beurling and Grunsky. We will mostly talk about p = 2.
Bergman Space
p Let Ω be a domain in C. The Bergman space A (Ω) is the subspace of analytic functions in Lp(Ω) Ap(Ω) := Lp(Ω) ∩ H(Ω) (Lp here is with respect to the area). By Cauchy inequalities, Lp-convergence in Ap(Ω) implies locally uniform convergence, so Ap(Ω) is a closed subspace (a locally uniform limit of analytic functions is analytic). We will also need the conjugate space of anti-analytic functions:
p A (Ω) := Ap(Ω) = Lp(Ω) ∩ H(Ω)
Ilia Binder Beurling and Grunsky. Bergman Space
p Let Ω be a domain in C. The Bergman space A (Ω) is the subspace of analytic functions in Lp(Ω) Ap(Ω) := Lp(Ω) ∩ H(Ω) (Lp here is with respect to the area). By Cauchy inequalities, Lp-convergence in Ap(Ω) implies locally uniform convergence, so Ap(Ω) is a closed subspace (a locally uniform limit of analytic functions is analytic). We will also need the conjugate space of anti-analytic functions:
p A (Ω) := Ap(Ω) = Lp(Ω) ∩ H(Ω)
We will mostly talk about p = 2.
Ilia Binder Beurling and Grunsky. !∞ 1 n en := z pπ(n + 1) n=0 2 P n 2 is an orthonormal basis of A (D), and for f (z) = anz ∈ A (D): ∞ 2 2 1 X |an| kf k = . 2 π n + 1 n=0
2 2 If g(z) ∈ L (D), then its orthogonal projection to A (D) is given by ∞ X Pg(z) := < g, en > en(z) n=0 ∞ Z n Z 1 X n z 1 g(w) = g(w)w dA(w) = dA(w) π n + 1 π (1 − zw)2 n=0 D D The interchange of the integration and summation is justified because the sum P∞ w nzn n=0 n+1 converges uniformly on D for each fixed z.
Bergman Space in D
Let us now concentrate on Ω = D and p = 2.
Ilia Binder Beurling and Grunsky. 2 2 If g(z) ∈ L (D), then its orthogonal projection to A (D) is given by ∞ X Pg(z) := < g, en > en(z) n=0 ∞ Z n Z 1 X n z 1 g(w) = g(w)w dA(w) = dA(w) π n + 1 π (1 − zw)2 n=0 D D The interchange of the integration and summation is justified because the sum P∞ w nzn n=0 n+1 converges uniformly on D for each fixed z.
Bergman Space in D
Let us now concentrate on Ω = D and p = 2. !∞ 1 n en := z pπ(n + 1) n=0 2 P n 2 is an orthonormal basis of A (D), and for f (z) = anz ∈ A (D): ∞ 2 2 1 X |an| kf k = . 2 π n + 1 n=0
Ilia Binder Beurling and Grunsky. The interchange of the integration and summation is justified because the sum P∞ w nzn n=0 n+1 converges uniformly on D for each fixed z.
Bergman Space in D
Let us now concentrate on Ω = D and p = 2. !∞ 1 n en := z pπ(n + 1) n=0 2 P n 2 is an orthonormal basis of A (D), and for f (z) = anz ∈ A (D): ∞ 2 2 1 X |an| kf k = . 2 π n + 1 n=0
2 2 If g(z) ∈ L (D), then its orthogonal projection to A (D) is given by ∞ X Pg(z) := < g, en > en(z) n=0 ∞ Z n Z 1 X n z 1 g(w) = g(w)w dA(w) = dA(w) π n + 1 π (1 − zw)2 n=0 D D
Ilia Binder Beurling and Grunsky. Bergman Space in D
Let us now concentrate on Ω = D and p = 2. !∞ 1 n en := z pπ(n + 1) n=0 2 P n 2 is an orthonormal basis of A (D), and for f (z) = anz ∈ A (D): ∞ 2 2 1 X |an| kf k = . 2 π n + 1 n=0
2 2 If g(z) ∈ L (D), then its orthogonal projection to A (D) is given by ∞ X Pg(z) := < g, en > en(z) n=0 ∞ Z n Z 1 X n z 1 g(w) = g(w)w dA(w) = dA(w) π n + 1 π (1 − zw)2 n=0 D D The interchange of the integration and summation is justified because the sum P∞ w nzn n=0 n+1 converges uniformly on D for each fixed z.
Ilia Binder Beurling and Grunsky. We can assume ψ ∈ Σ0, so consider 1 φ(z) := ∈ S ψ(1/z) Then ∞ ∞ φ(z) − φ(w) X X n k φ(z) φ(w) log = − γ z w + log + log . z − w nk z w n=1 k=1
Now differentiate to get the Grunsky kernel:
∂2 φ(z)−φ(w) Φ(z, w) := ∂z∂w log z−w = 0 0 ∞ ∞ φ (z)φ (w) 1 X X n−1 k−1 − = − nkγ z w (φ(z) − φ(w))2 (z − w)2 nk n=1 k=1
Grunsky kernel
Let ψ ∈ Σ. Its Grunsky decomposition: ∞ ∞ ψ(z) − ψ(w) X X −n −k log = − γ z w . z − w nk n=1 k=1
Ilia Binder Beurling and Grunsky. Now differentiate to get the Grunsky kernel:
∂2 φ(z)−φ(w) Φ(z, w) := ∂z∂w log z−w = 0 0 ∞ ∞ φ (z)φ (w) 1 X X n−1 k−1 − = − nkγ z w (φ(z) − φ(w))2 (z − w)2 nk n=1 k=1
Grunsky kernel
Let ψ ∈ Σ. Its Grunsky decomposition: ∞ ∞ ψ(z) − ψ(w) X X −n −k log = − γ z w . z − w nk n=1 k=1
We can assume ψ ∈ Σ0, so consider 1 φ(z) := ∈ S ψ(1/z) Then ∞ ∞ φ(z) − φ(w) X X n k φ(z) φ(w) log = − γ z w + log + log . z − w nk z w n=1 k=1
Ilia Binder Beurling and Grunsky. Grunsky kernel
Let ψ ∈ Σ. Its Grunsky decomposition: ∞ ∞ ψ(z) − ψ(w) X X −n −k log = − γ z w . z − w nk n=1 k=1
We can assume ψ ∈ Σ0, so consider 1 φ(z) := ∈ S ψ(1/z) Then ∞ ∞ φ(z) − φ(w) X X n k φ(z) φ(w) log = − γ z w + log + log . z − w nk z w n=1 k=1
Now differentiate to get the Grunsky kernel:
∂2 φ(z)−φ(w) Φ(z, w) := ∂z∂w log z−w = 0 0 ∞ ∞ φ (z)φ (w) 1 X X n−1 k−1 − = − nkγ z w (φ(z) − φ(w))2 (z − w)2 nk n=1 k=1
Ilia Binder Beurling and Grunsky. Observe
∞ ∞ Z 1 X X n−1 k−1 m−1 Γ e = − √ nkγ z w w dA(w) = φ m−1 π πm nk n=1 k=1 D ∞ ∞ 1 X n−1 X √ − √ mπnγ z = − nmγ e π πm nm nm n−1 n=1 n=1
2 2 So Γφ maps A (D) to A (D), with the Grunsky matrix. Grunsky inequalities are equivalent to the fact that Γφ is a contraction for all φ ∈ S, i.e. kΓφf k2 ≤ kf k2. 2 2 Equivalently, a linear version Γφf := Γφf is a contraction from A (D) to A (D).
Grunsky operator
Grunsky operator is an anti-linear operator on L2(Ω) with kernel Φ(z, w):
1 Z φ0(z)φ0(w) 1 Γ f (z) := − f (w) dA(w). φ π (φ(z) − φ(w))2 (z − w)2 D
Ilia Binder Beurling and Grunsky. Grunsky inequalities are equivalent to the fact that Γφ is a contraction for all φ ∈ S, i.e. kΓφf k2 ≤ kf k2. 2 2 Equivalently, a linear version Γφf := Γφf is a contraction from A (D) to A (D).
Grunsky operator
Grunsky operator is an anti-linear operator on L2(Ω) with kernel Φ(z, w):
1 Z φ0(z)φ0(w) 1 Γ f (z) := − f (w) dA(w). φ π (φ(z) − φ(w))2 (z − w)2 D
Observe
∞ ∞ Z 1 X X n−1 k−1 m−1 Γ e = − √ nkγ z w w dA(w) = φ m−1 π πm nk n=1 k=1 D ∞ ∞ 1 X n−1 X √ − √ mπnγ z = − nmγ e π πm nm nm n−1 n=1 n=1
2 2 So Γφ maps A (D) to A (D), with the Grunsky matrix.
Ilia Binder Beurling and Grunsky. 2 2 Equivalently, a linear version Γφf := Γφf is a contraction from A (D) to A (D).
Grunsky operator
Grunsky operator is an anti-linear operator on L2(Ω) with kernel Φ(z, w):
1 Z φ0(z)φ0(w) 1 Γ f (z) := − f (w) dA(w). φ π (φ(z) − φ(w))2 (z − w)2 D
Observe
∞ ∞ Z 1 X X n−1 k−1 m−1 Γ e = − √ nkγ z w w dA(w) = φ m−1 π πm nk n=1 k=1 D ∞ ∞ 1 X n−1 X √ − √ mπnγ z = − nmγ e π πm nm nm n−1 n=1 n=1
2 2 So Γφ maps A (D) to A (D), with the Grunsky matrix. Grunsky inequalities are equivalent to the fact that Γφ is a contraction for all φ ∈ S, i.e. kΓφf k2 ≤ kf k2.
Ilia Binder Beurling and Grunsky. Grunsky operator
Grunsky operator is an anti-linear operator on L2(Ω) with kernel Φ(z, w):
1 Z φ0(z)φ0(w) 1 Γ f (z) := − f (w) dA(w). φ π (φ(z) − φ(w))2 (z − w)2 D
Observe
∞ ∞ Z 1 X X n−1 k−1 m−1 Γ e = − √ nkγ z w w dA(w) = φ m−1 π πm nk n=1 k=1 D ∞ ∞ 1 X n−1 X √ − √ mπnγ z = − nmγ e π πm nm nm n−1 n=1 n=1
2 2 So Γφ maps A (D) to A (D), with the Grunsky matrix. Grunsky inequalities are equivalent to the fact that Γφ is a contraction for all φ ∈ S, i.e. kΓφf k2 ≤ kf k2. 2 2 Equivalently, a linear version Γφf := Γφf is a contraction from A (D) to A (D).
Ilia Binder Beurling and Grunsky. Beurling transform of f : Z ∂ 1 fz (w) Green (Bf )(z) := (Cf )(z) = (Cfz )(z) = lim dA(w) = ∂z r→0 π |w−z|>r z − w −1 Z f (w) −1 Z f (w) lim dA(w) = p. v. dA(w) r→0 π (z − w)2 π (z − w)2 |w−z|>r C
The fundamental property: Bfz = fz , since ∂ ∂ ∂ ∂ B ◦ = ◦ C ◦ = . ∂z ∂z ∂z ∂z
Cauchy and Beurling transforms
∞ Let f ∈ C0 (C). Its Cauchy transform: 1 Z f (w) (Cf )(z) := dA(w). π z − w C
By Green, (Cfz ) = (Cf )z = f . Holds in weak sense for any distribution.
Ilia Binder Beurling and Grunsky. The fundamental property: Bfz = fz , since ∂ ∂ ∂ ∂ B ◦ = ◦ C ◦ = . ∂z ∂z ∂z ∂z
Cauchy and Beurling transforms
∞ Let f ∈ C0 (C). Its Cauchy transform: 1 Z f (w) (Cf )(z) := dA(w). π z − w C
By Green, (Cfz ) = (Cf )z = f . Holds in weak sense for any distribution. Beurling transform of f : Z ∂ 1 fz (w) Green (Bf )(z) := (Cf )(z) = (Cfz )(z) = lim dA(w) = ∂z r→0 π |w−z|>r z − w −1 Z f (w) −1 Z f (w) lim dA(w) = p. v. dA(w) r→0 π (z − w)2 π (z − w)2 |w−z|>r C
Ilia Binder Beurling and Grunsky. Cauchy and Beurling transforms
∞ Let f ∈ C0 (C). Its Cauchy transform: 1 Z f (w) (Cf )(z) := dA(w). π z − w C
By Green, (Cfz ) = (Cf )z = f . Holds in weak sense for any distribution. Beurling transform of f : Z ∂ 1 fz (w) Green (Bf )(z) := (Cf )(z) = (Cfz )(z) = lim dA(w) = ∂z r→0 π |w−z|>r z − w −1 Z f (w) −1 Z f (w) lim dA(w) = p. v. dA(w) r→0 π (z − w)2 π (z − w)2 |w−z|>r C
The fundamental property: Bfz = fz , since ∂ ∂ ∂ ∂ B ◦ = ◦ C ◦ = . ∂z ∂z ∂z ∂z
Ilia Binder Beurling and Grunsky. ∞ It is unitary for g of the form g = fz , f ∈ C0 : Z Z 2 |g(z)| dA(z) = fz (z)fz (z) dA(z) = C C Z Z Z Green 2 fz (z)f z (z) dA(z) = fz (z)f z (z) dA(z) = |(Bg)(z)| dA(z). C C C ∞ ∞ Since functions of the form fz , f ∈ C0 and of the form fz , f ∈ C0 are dense 2 in L (C) (the functions orthogonal to all of them must be anti-entire or entire 2 correspondingly), B extends to a unitary operator on L (C). In terms of Fourier transform: ([Bf )(ζ) = ζ fˆ(ζ). ζ By the standard Calderon-Zygmund theory, B : Lp 7→ Lp, 1 < p < ∞.
Properties of Beurling transform
B is symmetric: Z −1 Z Z f (z)g(w) f (z)(Bg)(z) dA(z) = lim dA(w) dA(z) = r→0 π (z − w)2 C C |w−z|>r −1 Z Z f (z)g(w) Z lim dA(z) dA(w) = (Bf )(w)g(w) dA(w). r→0 π (z − w)2 C |w−z|>r C
Ilia Binder Beurling and Grunsky. In terms of Fourier transform: ([Bf )(ζ) = ζ fˆ(ζ). ζ By the standard Calderon-Zygmund theory, B : Lp 7→ Lp, 1 < p < ∞.
Properties of Beurling transform
B is symmetric: Z −1 Z Z f (z)g(w) f (z)(Bg)(z) dA(z) = lim dA(w) dA(z) = r→0 π (z − w)2 C C |w−z|>r −1 Z Z f (z)g(w) Z lim dA(z) dA(w) = (Bf )(w)g(w) dA(w). r→0 π (z − w)2 C |w−z|>r C
∞ It is unitary for g of the form g = fz , f ∈ C0 : Z Z 2 |g(z)| dA(z) = fz (z)fz (z) dA(z) = C C Z Z Z Green 2 fz (z)f z (z) dA(z) = fz (z)f z (z) dA(z) = |(Bg)(z)| dA(z). C C C ∞ ∞ Since functions of the form fz , f ∈ C0 and of the form fz , f ∈ C0 are dense 2 in L (C) (the functions orthogonal to all of them must be anti-entire or entire 2 correspondingly), B extends to a unitary operator on L (C).
Ilia Binder Beurling and Grunsky. By the standard Calderon-Zygmund theory, B : Lp 7→ Lp, 1 < p < ∞.
Properties of Beurling transform
B is symmetric: Z −1 Z Z f (z)g(w) f (z)(Bg)(z) dA(z) = lim dA(w) dA(z) = r→0 π (z − w)2 C C |w−z|>r −1 Z Z f (z)g(w) Z lim dA(z) dA(w) = (Bf )(w)g(w) dA(w). r→0 π (z − w)2 C |w−z|>r C
∞ It is unitary for g of the form g = fz , f ∈ C0 : Z Z 2 |g(z)| dA(z) = fz (z)fz (z) dA(z) = C C Z Z Z Green 2 fz (z)f z (z) dA(z) = fz (z)f z (z) dA(z) = |(Bg)(z)| dA(z). C C C ∞ ∞ Since functions of the form fz , f ∈ C0 and of the form fz , f ∈ C0 are dense 2 in L (C) (the functions orthogonal to all of them must be anti-entire or entire 2 correspondingly), B extends to a unitary operator on L (C). In terms of Fourier transform: ([Bf )(ζ) = ζ fˆ(ζ). ζ
Ilia Binder Beurling and Grunsky. Properties of Beurling transform
B is symmetric: Z −1 Z Z f (z)g(w) f (z)(Bg)(z) dA(z) = lim dA(w) dA(z) = r→0 π (z − w)2 C C |w−z|>r −1 Z Z f (z)g(w) Z lim dA(z) dA(w) = (Bf )(w)g(w) dA(w). r→0 π (z − w)2 C |w−z|>r C
∞ It is unitary for g of the form g = fz , f ∈ C0 : Z Z 2 |g(z)| dA(z) = fz (z)fz (z) dA(z) = C C Z Z Z Green 2 fz (z)f z (z) dA(z) = fz (z)f z (z) dA(z) = |(Bg)(z)| dA(z). C C C ∞ ∞ Since functions of the form fz , f ∈ C0 and of the form fz , f ∈ C0 are dense 2 in L (C) (the functions orthogonal to all of them must be anti-entire or entire 2 correspondingly), B extends to a unitary operator on L (C). In terms of Fourier transform: ([Bf )(ζ) = ζ fˆ(ζ). ζ By the standard Calderon-Zygmund theory, B : Lp 7→ Lp, 1 < p < ∞.
Ilia Binder Beurling and Grunsky. 1 n ∂ Let f (z) := Dz . Then f = ∂z g(z), where 1 1 g(z) = z n+11 + z −n−11 . n + 1 D n + 1 C\D So ∂ Bf (z) = g(z) = −1 z −n−2. ∂z C\D 2 1 An important observation: if f ∈ A (D), then DBf ≡ 0 (since it is true for the basis functions z n). 1 By scaling and translation, if f (z) := D(a,r): r 2 Bf (z) = − 1 . (z − a)2 C\D(a,r)
A computation.
1,2 By approximation, Bfz = fz , holds for any f ∈ W .
Ilia Binder Beurling and Grunsky. 2 1 An important observation: if f ∈ A (D), then DBf ≡ 0 (since it is true for the basis functions z n). 1 By scaling and translation, if f (z) := D(a,r): r 2 Bf (z) = − 1 . (z − a)2 C\D(a,r)
A computation.
1,2 By approximation, Bfz = fz , holds for any f ∈ W . 1 n ∂ Let f (z) := Dz . Then f = ∂z g(z), where 1 1 g(z) = z n+11 + z −n−11 . n + 1 D n + 1 C\D So ∂ Bf (z) = g(z) = −1 z −n−2. ∂z C\D
Ilia Binder Beurling and Grunsky. 1 By scaling and translation, if f (z) := D(a,r): r 2 Bf (z) = − 1 . (z − a)2 C\D(a,r)
A computation.
1,2 By approximation, Bfz = fz , holds for any f ∈ W . 1 n ∂ Let f (z) := Dz . Then f = ∂z g(z), where 1 1 g(z) = z n+11 + z −n−11 . n + 1 D n + 1 C\D So ∂ Bf (z) = g(z) = −1 z −n−2. ∂z C\D 2 1 An important observation: if f ∈ A (D), then DBf ≡ 0 (since it is true for the basis functions z n).
Ilia Binder Beurling and Grunsky. A computation.
1,2 By approximation, Bfz = fz , holds for any f ∈ W . 1 n ∂ Let f (z) := Dz . Then f = ∂z g(z), where 1 1 g(z) = z n+11 + z −n−11 . n + 1 D n + 1 C\D So ∂ Bf (z) = g(z) = −1 z −n−2. ∂z C\D 2 1 An important observation: if f ∈ A (D), then DBf ≡ 0 (since it is true for the basis functions z n). 1 By scaling and translation, if f (z) := D(a,r): r 2 Bf (z) = − 1 . (z − a)2 C\D(a,r)
Ilia Binder Beurling and Grunsky. Yes! By the previous observation:
−1 Z f (w) 1 Z by symmetry dA(w) = f (w)(B1 )(w) dA(w) = π (z − w)2 πr 2 D(z,r) |w−z|>r C 1 Z 1 Z (Bf )(w)1 (w) dA(w) = Bf (w) dA(w) πr 2 D(z,r) Area( (z, r)) C D D(z,r) So −1 Z f (w) lim 2 dA(w) = Bf (z) r→0 π |w−z|>r (z − w) for all Lebesgue points of Bf .
Pointwise existence of Beurling transform: following Mateu and Verdera.
p p For f ∈ L (C) ∞ > p > 1, Bf ∈ L (C) is defined a.e. But is it equal to the principal value of the integral?
Ilia Binder Beurling and Grunsky. By the previous observation:
−1 Z f (w) 1 Z by symmetry dA(w) = f (w)(B1 )(w) dA(w) = π (z − w)2 πr 2 D(z,r) |w−z|>r C 1 Z 1 Z (Bf )(w)1 (w) dA(w) = Bf (w) dA(w) πr 2 D(z,r) Area( (z, r)) C D D(z,r) So −1 Z f (w) lim 2 dA(w) = Bf (z) r→0 π |w−z|>r (z − w) for all Lebesgue points of Bf .
Pointwise existence of Beurling transform: following Mateu and Verdera.
p p For f ∈ L (C) ∞ > p > 1, Bf ∈ L (C) is defined a.e. But is it equal to the principal value of the integral? Yes!
Ilia Binder Beurling and Grunsky. So −1 Z f (w) lim 2 dA(w) = Bf (z) r→0 π |w−z|>r (z − w) for all Lebesgue points of Bf .
Pointwise existence of Beurling transform: following Mateu and Verdera.
p p For f ∈ L (C) ∞ > p > 1, Bf ∈ L (C) is defined a.e. But is it equal to the principal value of the integral? Yes! By the previous observation:
−1 Z f (w) 1 Z by symmetry dA(w) = f (w)(B1 )(w) dA(w) = π (z − w)2 πr 2 D(z,r) |w−z|>r C 1 Z 1 Z (Bf )(w)1 (w) dA(w) = Bf (w) dA(w) πr 2 D(z,r) Area( (z, r)) C D D(z,r)
Ilia Binder Beurling and Grunsky. Pointwise existence of Beurling transform: following Mateu and Verdera.
p p For f ∈ L (C) ∞ > p > 1, Bf ∈ L (C) is defined a.e. But is it equal to the principal value of the integral? Yes! By the previous observation:
−1 Z f (w) 1 Z by symmetry dA(w) = f (w)(B1 )(w) dA(w) = π (z − w)2 πr 2 D(z,r) |w−z|>r C 1 Z 1 Z (Bf )(w)1 (w) dA(w) = Bf (w) dA(w) πr 2 D(z,r) Area( (z, r)) C D D(z,r) So −1 Z f (w) lim 2 dA(w) = Bf (z) r→0 π |w−z|>r (z − w) for all Lebesgue points of Bf .
Ilia Binder Beurling and Grunsky. p BΩ is always bounded on L (Ω) for 1 < p < ∞. If Area(C \ Ω) = 0, it is an isometry on L2(Ω), otherwise, it is a contraction. If Ω is a simply-connected domain, let φ : D 7→ Ω be its Riemann map. Let the p p p p p isometry T : L (Ω) 7→ L (D) (and A (Ω) 7→ A (D)) be defined by
p 0 2/p Tφ f (z) := φ (z) f (φ(z)).
p For g ∈ L (D), define the transferred Beurling transform: φ0(w) 2 Bp g := T p ◦ B ◦ T p−1 ◦ M g, h(w) := − φ φ Ω φ h |φ0(w)|
p Bφ is bounded for 1 < p < ∞, is a contraction for p = 2.
The transferred Beurling transform
p Let Ω ⊂ C be a domain. For f ∈ L (Ω), the Restricted Beurling transform of f is: Z −1 f (w) 1 BΩf (z) := p. v. 2 dA(w) = Ω(z)Bf (z). π Ω (z − w)
Ilia Binder Beurling and Grunsky. If Ω is a simply-connected domain, let φ : D 7→ Ω be its Riemann map. Let the p p p p p isometry T : L (Ω) 7→ L (D) (and A (Ω) 7→ A (D)) be defined by
p 0 2/p Tφ f (z) := φ (z) f (φ(z)).
p For g ∈ L (D), define the transferred Beurling transform: φ0(w) 2 Bp g := T p ◦ B ◦ T p−1 ◦ M g, h(w) := − φ φ Ω φ h |φ0(w)|
p Bφ is bounded for 1 < p < ∞, is a contraction for p = 2.
The transferred Beurling transform
p Let Ω ⊂ C be a domain. For f ∈ L (Ω), the Restricted Beurling transform of f is: Z −1 f (w) 1 BΩf (z) := p. v. 2 dA(w) = Ω(z)Bf (z). π Ω (z − w)
p BΩ is always bounded on L (Ω) for 1 < p < ∞. If Area(C \ Ω) = 0, it is an isometry on L2(Ω), otherwise, it is a contraction.
Ilia Binder Beurling and Grunsky. p For g ∈ L (D), define the transferred Beurling transform: φ0(w) 2 Bp g := T p ◦ B ◦ T p−1 ◦ M g, h(w) := − φ φ Ω φ h |φ0(w)|
p Bφ is bounded for 1 < p < ∞, is a contraction for p = 2.
The transferred Beurling transform
p Let Ω ⊂ C be a domain. For f ∈ L (Ω), the Restricted Beurling transform of f is: Z −1 f (w) 1 BΩf (z) := p. v. 2 dA(w) = Ω(z)Bf (z). π Ω (z − w)
p BΩ is always bounded on L (Ω) for 1 < p < ∞. If Area(C \ Ω) = 0, it is an isometry on L2(Ω), otherwise, it is a contraction. If Ω is a simply-connected domain, let φ : D 7→ Ω be its Riemann map. Let the p p p p p isometry T : L (Ω) 7→ L (D) (and A (Ω) 7→ A (D)) be defined by
p 0 2/p Tφ f (z) := φ (z) f (φ(z)).
Ilia Binder Beurling and Grunsky. φ0(w) 2 h(w) := − |φ0(w)|
p Bφ is bounded for 1 < p < ∞, is a contraction for p = 2.
The transferred Beurling transform
p Let Ω ⊂ C be a domain. For f ∈ L (Ω), the Restricted Beurling transform of f is: Z −1 f (w) 1 BΩf (z) := p. v. 2 dA(w) = Ω(z)Bf (z). π Ω (z − w)
p BΩ is always bounded on L (Ω) for 1 < p < ∞. If Area(C \ Ω) = 0, it is an isometry on L2(Ω), otherwise, it is a contraction. If Ω is a simply-connected domain, let φ : D 7→ Ω be its Riemann map. Let the p p p p p isometry T : L (Ω) 7→ L (D) (and A (Ω) 7→ A (D)) be defined by
p 0 2/p Tφ f (z) := φ (z) f (φ(z)).
p For g ∈ L (D), define the transferred Beurling transform:
p p p−1 Bφg := Tφ ◦ BΩ ◦ Tφ ◦ Mhg,
Ilia Binder Beurling and Grunsky. p Bφ is bounded for 1 < p < ∞, is a contraction for p = 2.
The transferred Beurling transform
p Let Ω ⊂ C be a domain. For f ∈ L (Ω), the Restricted Beurling transform of f is: Z −1 f (w) 1 BΩf (z) := p. v. 2 dA(w) = Ω(z)Bf (z). π Ω (z − w)
p BΩ is always bounded on L (Ω) for 1 < p < ∞. If Area(C \ Ω) = 0, it is an isometry on L2(Ω), otherwise, it is a contraction. If Ω is a simply-connected domain, let φ : D 7→ Ω be its Riemann map. Let the p p p p p isometry T : L (Ω) 7→ L (D) (and A (Ω) 7→ A (D)) be defined by
p 0 2/p Tφ f (z) := φ (z) f (φ(z)).
p For g ∈ L (D), define the transferred Beurling transform: φ0(w) 2 Bp g := T p ◦ B ◦ T p−1 ◦ M g, h(w) := − φ φ Ω φ h |φ0(w)|
Ilia Binder Beurling and Grunsky. The transferred Beurling transform
p Let Ω ⊂ C be a domain. For f ∈ L (Ω), the Restricted Beurling transform of f is: Z −1 f (w) 1 BΩf (z) := p. v. 2 dA(w) = Ω(z)Bf (z). π Ω (z − w)
p BΩ is always bounded on L (Ω) for 1 < p < ∞. If Area(C \ Ω) = 0, it is an isometry on L2(Ω), otherwise, it is a contraction. If Ω is a simply-connected domain, let φ : D 7→ Ω be its Riemann map. Let the p p p p p isometry T : L (Ω) 7→ L (D) (and A (Ω) 7→ A (D)) be defined by
p 0 2/p Tφ f (z) := φ (z) f (φ(z)).
p For g ∈ L (D), define the transferred Beurling transform: φ0(w) 2 Bp g := T p ◦ B ◦ T p−1 ◦ M g, h(w) := − φ φ Ω φ h |φ0(w)|
p Bφ is bounded for 1 < p < ∞, is a contraction for p = 2.
Ilia Binder Beurling and Grunsky. Collect what we know about p = 2: 1 Z φ0(z)φ0(w) 1 (Γ g)(z) = − g(w) dA(w) φ π (φ(z) − φ(w))2 (z − w)2 D So 2 2 Γφg = Bφg − BDg 2 2 But on A ( ), B2 ≡ 0, so for g ∈ A ( ): D D D 2 Γφg = Bφg.
2 Since Bφ is a contraction, it proves the Grunsky inequality.
The integral form of transferred Beurling transform and Grunsky inequality.
In the integral form
1 Z φ0(z)2/pφ0(w)2−2/p Bp g (z) = p. v. g(w) dA(w). φ π (φ(z) − φ(w))2 D
Ilia Binder Beurling and Grunsky. So 2 2 Γφg = Bφg − BDg 2 2 But on A ( ), B2 ≡ 0, so for g ∈ A ( ): D D D 2 Γφg = Bφg.
2 Since Bφ is a contraction, it proves the Grunsky inequality.
The integral form of transferred Beurling transform and Grunsky inequality.
In the integral form
1 Z φ0(z)2/pφ0(w)2−2/p Bp g (z) = p. v. g(w) dA(w). φ π (φ(z) − φ(w))2 D
Collect what we know about p = 2: 1 Z φ0(z)φ0(w) 1 (Γ g)(z) = − g(w) dA(w) φ π (φ(z) − φ(w))2 (z − w)2 D
Ilia Binder Beurling and Grunsky. 2 2 But on A ( ), B2 ≡ 0, so for g ∈ A ( ): D D D 2 Γφg = Bφg.
2 Since Bφ is a contraction, it proves the Grunsky inequality.
The integral form of transferred Beurling transform and Grunsky inequality.
In the integral form
1 Z φ0(z)2/pφ0(w)2−2/p Bp g (z) = p. v. g(w) dA(w). φ π (φ(z) − φ(w))2 D
Collect what we know about p = 2: 1 Z φ0(z)φ0(w) 1 (Γ g)(z) = − g(w) dA(w) φ π (φ(z) − φ(w))2 (z − w)2 D So 2 2 Γφg = Bφg − BDg
Ilia Binder Beurling and Grunsky. 2 Since Bφ is a contraction, it proves the Grunsky inequality.
The integral form of transferred Beurling transform and Grunsky inequality.
In the integral form
1 Z φ0(z)2/pφ0(w)2−2/p Bp g (z) = p. v. g(w) dA(w). φ π (φ(z) − φ(w))2 D
Collect what we know about p = 2: 1 Z φ0(z)φ0(w) 1 (Γ g)(z) = − g(w) dA(w) φ π (φ(z) − φ(w))2 (z − w)2 D So 2 2 Γφg = Bφg − BDg 2 2 But on A ( ), B2 ≡ 0, so for g ∈ A ( ): D D D 2 Γφg = Bφg.
Ilia Binder Beurling and Grunsky. The integral form of transferred Beurling transform and Grunsky inequality.
In the integral form
1 Z φ0(z)2/pφ0(w)2−2/p Bp g (z) = p. v. g(w) dA(w). φ π (φ(z) − φ(w))2 D
Collect what we know about p = 2: 1 Z φ0(z)φ0(w) 1 (Γ g)(z) = − g(w) dA(w) φ π (φ(z) − φ(w))2 (z − w)2 D So 2 2 Γφg = Bφg − BDg 2 2 But on A ( ), B2 ≡ 0, so for g ∈ A ( ): D D D 2 Γφg = Bφg.
2 Since Bφ is a contraction, it proves the Grunsky inequality.
Ilia Binder Beurling and Grunsky. 2 φ0(w) − |φ0(w)| |w| < 1 h(w) := 2 ψ0(w) − |ψ0(w)| |w| > 1 And, in integral form
1 Z φ0(z)2/pφ0(w)2−2/p Bp g (z) = p. v. g(w) dA(w)+ φ,ψ π (φ(z) − φ(w))2 D 1 Z φ0(z)2/pψ0(w)2−2/p g(w) dA(w); |z| < 1. π (φ(z) − ψ(w))2 C\D and similar expression for |z| > 1. A compression for p = 2.
Generalized Grunsky inequality: transferred Beurling.
Essentially the same, but we need two univalent maps ψ : C \ D 7→ Ω−, φ : D 7→ Ω+, with Ω− ∩ Ω+ = ∅. Ω := Ω− ∪ Ω+. Then define ( φ0(z)2/pf (φ(z)) |z| < 1 T p f (z) := . φ,ψ ψ0(z)2/pf (ψ(z)) |z| > 1
p p p −1 Bφ,ψg(z) := Tφ,ψ ◦ BΩ ◦ Tφ,ψ ◦ Mhg,
Ilia Binder Beurling and Grunsky. And, in integral form
1 Z φ0(z)2/pφ0(w)2−2/p Bp g (z) = p. v. g(w) dA(w)+ φ,ψ π (φ(z) − φ(w))2 D 1 Z φ0(z)2/pψ0(w)2−2/p g(w) dA(w); |z| < 1. π (φ(z) − ψ(w))2 C\D and similar expression for |z| > 1. A compression for p = 2.
Generalized Grunsky inequality: transferred Beurling.
Essentially the same, but we need two univalent maps ψ : C \ D 7→ Ω−, φ : D 7→ Ω+, with Ω− ∩ Ω+ = ∅. Ω := Ω− ∪ Ω+. Then define ( φ0(z)2/pf (φ(z)) |z| < 1 T p f (z) := . φ,ψ ψ0(z)2/pf (ψ(z)) |z| > 1
0 2 − φ (w) |w| < 1 p p p −1 |φ0(w)| B g(z) := T ◦ BΩ ◦ T ◦ Mhg, h(w) := 2 φ,ψ φ,ψ φ,ψ ψ0(w) − |ψ0(w)| |w| > 1
Ilia Binder Beurling and Grunsky. A compression for p = 2.
Generalized Grunsky inequality: transferred Beurling.
Essentially the same, but we need two univalent maps ψ : C \ D 7→ Ω−, φ : D 7→ Ω+, with Ω− ∩ Ω+ = ∅. Ω := Ω− ∪ Ω+. Then define ( φ0(z)2/pf (φ(z)) |z| < 1 T p f (z) := . φ,ψ ψ0(z)2/pf (ψ(z)) |z| > 1
0 2 − φ (w) |w| < 1 p p p −1 |φ0(w)| B g(z) := T ◦ BΩ ◦ T ◦ Mhg, h(w) := 2 φ,ψ φ,ψ φ,ψ ψ0(w) − |ψ0(w)| |w| > 1 And, in integral form
1 Z φ0(z)2/pφ0(w)2−2/p Bp g (z) = p. v. g(w) dA(w)+ φ,ψ π (φ(z) − φ(w))2 D 1 Z φ0(z)2/pψ0(w)2−2/p g(w) dA(w); |z| < 1. π (φ(z) − ψ(w))2 C\D and similar expression for |z| > 1.
Ilia Binder Beurling and Grunsky. Generalized Grunsky inequality: transferred Beurling.
Essentially the same, but we need two univalent maps ψ : C \ D 7→ Ω−, φ : D 7→ Ω+, with Ω− ∩ Ω+ = ∅. Ω := Ω− ∪ Ω+. Then define ( φ0(z)2/pf (φ(z)) |z| < 1 T p f (z) := . φ,ψ ψ0(z)2/pf (ψ(z)) |z| > 1
0 2 − φ (w) |w| < 1 p p p −1 |φ0(w)| B g(z) := T ◦ BΩ ◦ T ◦ Mhg, h(w) := 2 φ,ψ φ,ψ φ,ψ ψ0(w) − |ψ0(w)| |w| > 1 And, in integral form
1 Z φ0(z)2/pφ0(w)2−2/p Bp g (z) = p. v. g(w) dA(w)+ φ,ψ π (φ(z) − φ(w))2 D 1 Z φ0(z)2/pψ0(w)2−2/p g(w) dA(w); |z| < 1. π (φ(z) − ψ(w))2 C\D and similar expression for |z| > 1. A compression for p = 2.
Ilia Binder Beurling and Grunsky. By differentiating ∂z ∂w and comparing we see that the operator
B2 B2 B2 Γφ,ψ := φ,ψ − D − C\D
has the generalized Grunsky matrix on A(D) ⊕ A(C \ D).It is a contraction, since both B2 and B2 vanish on A( ) ⊕ A( \ ). And this is exactly C\D D D C D generalized Grunsky inequality!
Generalized Grunsky inequality.
Generalized Grunsky: ∞ ∞ ψ(z) − ψ(w) X X −n −k log = − γ z w . z − w −n,−k n=1 k=1 ∞ ∞ φ(z) − φ(w) X X n k φ(z) φ(w) log = − γ z w + log + log . z − w n,k z w n=1 k=1 ∞ ∞ ψ(z) − φ(w) X X −n k ψ(z) log = − γ z w + log . z −n,k z n=1 k=1
Ilia Binder Beurling and Grunsky. It is a contraction, since both B2 and B2 vanish on A( ) ⊕ A( \ ). And this is exactly C\D D D C D generalized Grunsky inequality!
Generalized Grunsky inequality.
Generalized Grunsky: ∞ ∞ ψ(z) − ψ(w) X X −n −k log = − γ z w . z − w −n,−k n=1 k=1 ∞ ∞ φ(z) − φ(w) X X n k φ(z) φ(w) log = − γ z w + log + log . z − w n,k z w n=1 k=1 ∞ ∞ ψ(z) − φ(w) X X −n k ψ(z) log = − γ z w + log . z −n,k z n=1 k=1
By differentiating ∂z ∂w and comparing we see that the operator
B2 B2 B2 Γφ,ψ := φ,ψ − D − C\D
has the generalized Grunsky matrix on A(D) ⊕ A(C \ D).
Ilia Binder Beurling and Grunsky. And this is exactly generalized Grunsky inequality!
Generalized Grunsky inequality.
Generalized Grunsky: ∞ ∞ ψ(z) − ψ(w) X X −n −k log = − γ z w . z − w −n,−k n=1 k=1 ∞ ∞ φ(z) − φ(w) X X n k φ(z) φ(w) log = − γ z w + log + log . z − w n,k z w n=1 k=1 ∞ ∞ ψ(z) − φ(w) X X −n k ψ(z) log = − γ z w + log . z −n,k z n=1 k=1
By differentiating ∂z ∂w and comparing we see that the operator
B2 B2 B2 Γφ,ψ := φ,ψ − D − C\D
has the generalized Grunsky matrix on A(D) ⊕ A(C \ D).It is a contraction, since both B2 and B2 vanish on A( ) ⊕ A( \ ). C\D D D C D
Ilia Binder Beurling and Grunsky. Generalized Grunsky inequality.
Generalized Grunsky: ∞ ∞ ψ(z) − ψ(w) X X −n −k log = − γ z w . z − w −n,−k n=1 k=1 ∞ ∞ φ(z) − φ(w) X X n k φ(z) φ(w) log = − γ z w + log + log . z − w n,k z w n=1 k=1 ∞ ∞ ψ(z) − φ(w) X X −n k ψ(z) log = − γ z w + log . z −n,k z n=1 k=1
By differentiating ∂z ∂w and comparing we see that the operator
B2 B2 B2 Γφ,ψ := φ,ψ − D − C\D
has the generalized Grunsky matrix on A(D) ⊕ A(C \ D).It is a contraction, since both B2 and B2 vanish on A( ) ⊕ A( \ ). And this is exactly C\D D D C D generalized Grunsky inequality!
Ilia Binder Beurling and Grunsky. Indeed, 1 Z 1 (If )0(w) = f (ζ)dA(ζ) = (Pf )(w) = f (w) π 2 D ((1 − ζw))
φ0(z)φ0(w) 1 For a conformal φ, apply it to f (w) := φ(z)−φ(w) − z−w :
z(φ(z) − φ(w)) log = (If )(w) = (z − w)φ(z) 1 Z φ0(z)φ0(ζ) 1 w − dA(ζ) π φ(z) − φ(ζ) z − ζ D 1 − ζw
Grunsky identity
1 Let f ∈ A (D). Then 1 Z w Z (If )(w) := f (ζ) dA(ζ) = f (ζ) dζ π D 1 − ζw [0,w]
Ilia Binder Beurling and Grunsky. φ0(z)φ0(w) 1 For a conformal φ, apply it to f (w) := φ(z)−φ(w) − z−w :
z(φ(z) − φ(w)) log = (If )(w) = (z − w)φ(z) 1 Z φ0(z)φ0(ζ) 1 w − dA(ζ) π φ(z) − φ(ζ) z − ζ D 1 − ζw
Grunsky identity
1 Let f ∈ A (D). Then 1 Z w Z (If )(w) := f (ζ) dA(ζ) = f (ζ) dζ π D 1 − ζw [0,w] Indeed, 1 Z 1 (If )0(w) = f (ζ)dA(ζ) = (Pf )(w) = f (w) π 2 D ((1 − ζw))
Ilia Binder Beurling and Grunsky. Grunsky identity
1 Let f ∈ A (D). Then 1 Z w Z (If )(w) := f (ζ) dA(ζ) = f (ζ) dζ π D 1 − ζw [0,w] Indeed, 1 Z 1 (If )0(w) = f (ζ)dA(ζ) = (Pf )(w) = f (w) π 2 D ((1 − ζw))
φ0(z)φ0(w) 1 For a conformal φ, apply it to f (w) := φ(z)−φ(w) − z−w :
z(φ(z) − φ(w)) log = (If )(w) = (z − w)φ(z) 1 Z φ0(z)φ0(ζ) 1 w − dA(ζ) π φ(z) − φ(ζ) z − ζ D 1 − ζw
Ilia Binder Beurling and Grunsky. Adding the last two identities: z(φ(z) − φ(w)) 1 Z φ0(z)φ0(ζ) w log + log(1 − zw) = dA(ζ). (z − w)φ(z) π φ(z) − φ(ζ) D 1 − ζw
Apply ∂w ∂z and apply to a test function η to get
Z φ0(z)φ0(w) 1 p. v. − η(w) dA(w) = (φ(z) − φ(w))2 (z − w)2 D 1 Z Z φ0(ζ)φ0(z) 1 p. v. η(w) dA(ζ) dA(w) π (φ(ζ) − φ(z))2 D D 1 − ζw or Γ = B2 − B2 = B2 P – a contraction on L2( ), maps it to A2( )! φ φ D φ D D
Grunsky identity
For f (ζ) := log(1 − ζw), f (0) = 0, f = 0, f = − w , so, by generalized ζ ζ 1−ζw Cauchy: 1 Z w log(1 − zw) = dA(ζ) π D (z − ζ)(1 − ζw)
Ilia Binder Beurling and Grunsky. Apply ∂w ∂z and apply to a test function η to get
Z φ0(z)φ0(w) 1 p. v. − η(w) dA(w) = (φ(z) − φ(w))2 (z − w)2 D 1 Z Z φ0(ζ)φ0(z) 1 p. v. η(w) dA(ζ) dA(w) π (φ(ζ) − φ(z))2 D D 1 − ζw or Γ = B2 − B2 = B2 P – a contraction on L2( ), maps it to A2( )! φ φ D φ D D
Grunsky identity
For f (ζ) := log(1 − ζw), f (0) = 0, f = 0, f = − w , so, by generalized ζ ζ 1−ζw Cauchy: 1 Z w log(1 − zw) = dA(ζ) π D (z − ζ)(1 − ζw) Adding the last two identities: z(φ(z) − φ(w)) 1 Z φ0(z)φ0(ζ) w log + log(1 − zw) = dA(ζ). (z − w)φ(z) π φ(z) − φ(ζ) D 1 − ζw
Ilia Binder Beurling and Grunsky. or Γ = B2 − B2 = B2 P – a contraction on L2( ), maps it to A2( )! φ φ D φ D D
Grunsky identity
For f (ζ) := log(1 − ζw), f (0) = 0, f = 0, f = − w , so, by generalized ζ ζ 1−ζw Cauchy: 1 Z w log(1 − zw) = dA(ζ) π D (z − ζ)(1 − ζw) Adding the last two identities: z(φ(z) − φ(w)) 1 Z φ0(z)φ0(ζ) w log + log(1 − zw) = dA(ζ). (z − w)φ(z) π φ(z) − φ(ζ) D 1 − ζw
Apply ∂w ∂z and apply to a test function η to get
Z φ0(z)φ0(w) 1 p. v. − η(w) dA(w) = (φ(z) − φ(w))2 (z − w)2 D 1 Z Z φ0(ζ)φ0(z) 1 p. v. η(w) dA(ζ) dA(w) π (φ(ζ) − φ(z))2 D D 1 − ζw
Ilia Binder Beurling and Grunsky. Grunsky identity
For f (ζ) := log(1 − ζw), f (0) = 0, f = 0, f = − w , so, by generalized ζ ζ 1−ζw Cauchy: 1 Z w log(1 − zw) = dA(ζ) π D (z − ζ)(1 − ζw) Adding the last two identities: z(φ(z) − φ(w)) 1 Z φ0(z)φ0(ζ) w log + log(1 − zw) = dA(ζ). (z − w)φ(z) π φ(z) − φ(ζ) D 1 − ζw
Apply ∂w ∂z and apply to a test function η to get
Z φ0(z)φ0(w) 1 p. v. − η(w) dA(w) = (φ(z) − φ(w))2 (z − w)2 D 1 Z Z φ0(ζ)φ0(z) 1 p. v. η(w) dA(ζ) dA(w) π (φ(ζ) − φ(z))2 D D 1 − ζw or Γ = B2 − B2 = B2 P – a contraction on L2( ), maps it to A2( )! φ φ D φ D D
Ilia Binder Beurling and Grunsky. 2 So it is an analytic kernel, and for any f ∈ L (D), p 2 B − B + − 1 CM 00 0 f is an analytic function φ D p φ /φ So it does not change under projection to analytic functions: p 2 p 2 B − B + − 1 CM 00 0 = PB + P − 1 CM 00 0 φ D p φ /φ φ p φ /φ In the case p = 2, it gives a version of Grusky identity:
2 2 Bφ − BD = PBφ
Skewed Grunsky identities
Expand p-Beurling kernel near diagonal:
φ0(z)2/pφ0(w)2−2/p 1 2 φ00(w) − + − 1 = O(1) (φ(w) − φ(z))2 (z − w)2 p (w − z)φ0(w)
Ilia Binder Beurling and Grunsky. So it does not change under projection to analytic functions: p 2 p 2 B − B + − 1 CM 00 0 = PB + P − 1 CM 00 0 φ D p φ /φ φ p φ /φ In the case p = 2, it gives a version of Grusky identity:
2 2 Bφ − BD = PBφ
Skewed Grunsky identities
Expand p-Beurling kernel near diagonal:
φ0(z)2/pφ0(w)2−2/p 1 2 φ00(w) − + − 1 = O(1) (φ(w) − φ(z))2 (z − w)2 p (w − z)φ0(w)
2 So it is an analytic kernel, and for any f ∈ L (D), p 2 B − B + − 1 CM 00 0 f is an analytic function φ D p φ /φ
Ilia Binder Beurling and Grunsky. In the case p = 2, it gives a version of Grusky identity:
2 2 Bφ − BD = PBφ
Skewed Grunsky identities
Expand p-Beurling kernel near diagonal:
φ0(z)2/pφ0(w)2−2/p 1 2 φ00(w) − + − 1 = O(1) (φ(w) − φ(z))2 (z − w)2 p (w − z)φ0(w)
2 So it is an analytic kernel, and for any f ∈ L (D), p 2 B − B + − 1 CM 00 0 f is an analytic function φ D p φ /φ So it does not change under projection to analytic functions: p 2 p 2 B − B + − 1 CM 00 0 = PB + P − 1 CM 00 0 φ D p φ /φ φ p φ /φ
Ilia Binder Beurling and Grunsky. Skewed Grunsky identities
Expand p-Beurling kernel near diagonal:
φ0(z)2/pφ0(w)2−2/p 1 2 φ00(w) − + − 1 = O(1) (φ(w) − φ(z))2 (z − w)2 p (w − z)φ0(w)
2 So it is an analytic kernel, and for any f ∈ L (D), p 2 B − B + − 1 CM 00 0 f is an analytic function φ D p φ /φ So it does not change under projection to analytic functions: p 2 p 2 B − B + − 1 CM 00 0 = PB + P − 1 CM 00 0 φ D p φ /φ φ p φ /φ In the case p = 2, it gives a version of Grusky identity:
2 2 Bφ − BD = PBφ
Ilia Binder Beurling and Grunsky. Let −2 ∂2G (z, w) 1 L (z, w) := Ω = − l (z, w) Ω π ∂z∂w π(z − w)2 Ω
2 2 Since BDTφf = 0 for f ∈ A (Ω), we have Z LΩ(z, w)f (w) dA(w) ≡ 0 Ω which means that on A2(Ω), Z ΓΩf = lΩ(z, w)f (w) dA(w) Ω
Grunsky operator in the domain Ω.
−1 Z f (w) ΓΩf := BΩf = p. v. 2 dA(w) π Ω (z − w) is an anti-linear contraction of A2(Ω).
Ilia Binder Beurling and Grunsky. 2 2 Since BDTφf = 0 for f ∈ A (Ω), we have Z LΩ(z, w)f (w) dA(w) ≡ 0 Ω which means that on A2(Ω), Z ΓΩf = lΩ(z, w)f (w) dA(w) Ω
Grunsky operator in the domain Ω.
−1 Z f (w) ΓΩf := BΩf = p. v. 2 dA(w) π Ω (z − w) is an anti-linear contraction of A2(Ω). Let −2 ∂2G (z, w) 1 L (z, w) := Ω = − l (z, w) Ω π ∂z∂w π(z − w)2 Ω
Ilia Binder Beurling and Grunsky. which means that on A2(Ω), Z ΓΩf = lΩ(z, w)f (w) dA(w) Ω
Grunsky operator in the domain Ω.
−1 Z f (w) ΓΩf := BΩf = p. v. 2 dA(w) π Ω (z − w) is an anti-linear contraction of A2(Ω). Let −2 ∂2G (z, w) 1 L (z, w) := Ω = − l (z, w) Ω π ∂z∂w π(z − w)2 Ω
2 2 Since BDTφf = 0 for f ∈ A (Ω), we have Z LΩ(z, w)f (w) dA(w) ≡ 0 Ω
Ilia Binder Beurling and Grunsky. Grunsky operator in the domain Ω.
−1 Z f (w) ΓΩf := BΩf = p. v. 2 dA(w) π Ω (z − w) is an anti-linear contraction of A2(Ω). Let −2 ∂2G (z, w) 1 L (z, w) := Ω = − l (z, w) Ω π ∂z∂w π(z − w)2 Ω
2 2 Since BDTφf = 0 for f ∈ A (Ω), we have Z LΩ(z, w)f (w) dA(w) ≡ 0 Ω which means that on A2(Ω), Z ΓΩf = lΩ(z, w)f (w) dA(w) Ω
Ilia Binder Beurling and Grunsky.