The Corona Problem in the Dirichlet Space
Brett D. Wick
Georgia Institute of Technology School of Mathematics
MSRI Summer Graduate Program: The Dirichlet Space: Connections between Operator Theory, Function Theory, and Complex Analysis June 27, 2011 to July 1, 2011
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 1 / 131 Lecture Outlines & Topics Covered
Motivations for the Problem ∞ The Corona Problem for H (D) Carleson measures and ∂-problems; Wolff’s proof of the Corona Problem; Jones’ constructive solution to ∂b = µ;
The Corona Problem for MD Xiao’s Theorem on the Dirichlet space The Corona Problem for Multiplier Algebras with the Complete Nevanlinna-Pick Property Reproducing kernel Hilbert function spaces with Complete Nevanlinna-Pick kernel; The Baby Corona Problem & The Corona Problem; Toeplitz Corona Theorem; The Corona Problem in Several Variables
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 2 / 131 Motivations for the Problem
Motivations for the Problem
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 3 / 131 Motivations for the Problem Where Did the Name Come From?
The Beer Problem?
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 4 / 131 Motivations for the Problem Commutative Banach Algebras
A (commutative) Banach algebra A is a complex (commutative) algebra A that is also a Banach space under a norm that satisfies
kfgkA ≤ kf kA kgkA f , g ∈ A.
We will also assume that there is an identity element 1 ∈ A and that our algebra is commutative. An element f ∈ A is invertible if there exists an element g ∈ A such that fg = 1 and write f −1 for g. We let
A−1 = {f ∈ A : f −1 exists}.
Finally, we need to consider the non-zero multiplicative linear functionals on the algebra A. These are simply the complex homomorphisms m : A → C. Note that we trivially have that m(1) = 1.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 5 / 131 Motivations for the Problem Commutative Banach Algebras
Lemma 1
Every complex homomorphism from A to C is a continuous linear functional with norm at most 1. Namely,
kmk = sup |m(f )| ≤ 1. f ∈A,kf kA≤1
Proof: If m is unbounded or if kmk > 1 then we can find an element
f ∈ A with kf kA < 1 but m(f ) = 1. Since kf kA < 1 we have
∞ ∞ ∞ 1 X f n ≤ X kf nk ≤ X kf kn = . A A 1 − kf k n=0 A n=0 n=0 A P∞ n −1 Note that we have (1 − f ) n=0 f = 1, so 1 − f ∈ A . However, 1 = m(1) = m((1 − f )(1 − f )−1) = m((1 − f )−1)(m(1) − m(f )) = 0.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 6 / 131 Motivations for the Problem Commutative Banach Algebras
Lemma 2 Suppose that M is a maximal ideal of A. Then M is the kernel of a multiplicative linear functional m : A → C. Conversely, suppose that m : A → C is a multiplicative linear functional. Then ker m is a maximal ideal. Proof: For the converse, it is immediate that the ker m is an ideal. The maximality follows since the codimension of any linear functional is 1. Namely, dim(A\ ker m) = 1. In the other direction, one first shows that the maximal ideal M is closed. Then one shows that the quotient algebra B = A/M satisfies
B = C1 where 1 = 1 + M denotes the unit in the quotient algebra. The quotient mapping will then define the multiplicative linear functional, and the kernel of this mapping with then be M. B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 7 / 131 Motivations for the Problem Commutative Banach Algebras
Let MA denote the set of complex homomorphisms of A (called the maximal ideal space of the Banach algebra). ∗ By Lemma1, we have that MA ⊂ Ball1 (A ). ∗ Endow MA with the weak-∗ topology of A (called the Gelfand topology). Namely, the basic neighborhood of a m0 ∈ MA is determined by > 0 and by elements f1,..., fn ∈ A such that
∗ V = {m ∈ A : kmk ≤ 1, |m(fj ) − m0(fj )| < , 1 ≤ j ≤ n} .
Because
∗ MA = {m ∈ A : kmk ≤ 1, m(fg) = m(f )m(g), f , g ∈ A}
∗ we have that MA is a weak-∗ closed subset of the unit ball of A . The Banach-Alaoglu Theorem gives that the ball of A∗ is weak-∗ compact, which implies that MA is a compact Hausdorff space.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 8 / 131 Motivations for the Problem Where Did the Name Come From? ∞ The Corona Problem for H (D) ∞ The Banach algebra H (D) is the collection of all analytic functions on the disc such that
kf k ∞ := sup |f (z)| < ∞. H (D) z∈D ∞ Let ϕ : H (D) → C be a non-zero multiplicative linear functional. Namely, ϕ(fg) = ϕ(f )ϕ(g) and ϕ(f + g) = ϕ(f ) + ϕ(g). As we have seen for any multiplicative linear functional
sup |ϕ(f )| ≤ kf kH∞( ) . ∞ D f ∈H (D) To each z ∈ D we can associate a multiplicative linear functional on ∞ H (D): ϕz (f ) := f (z) (point evaluation at z).
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 9 / 131 Motivations for the Problem Where Did the Name Come From? ∞ The Corona Problem for H (D)
Every multiplicative linear functional ϕ determines a maximal (proper) ∞ ∞ ideal of H (D): ker ϕ = {f ∈ H (D): ϕ(f ) = 0}. ∞ Conversely, if M is a maximal (proper) ideal of H (D) then M = ker ϕ for some multiplicative linear functional.
∞ The maximal ideal space of H ( ), M ∞ , is the collection of all D H (D) multiplicative linear functionals ϕ. We then have that the maximal ideal space is contained in the unit ball of ∞ the dual space H (D). If we put the weak-∗ topology on this space then M ∞ is a compact Hausdorff space. H (D)
The proceeding discussion then shows that ⊂ M ∞ . D H (D)
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 10 / 131 Motivations for the Problem Where Did the Name Come From? ∞ The Corona Problem for H (D) ∞ One then defines the Corona of H ( ) to be M ∞ \ . D H (D) D In 1941, Kakutani asked if there was a Corona in the maximal ideal space ∞ M ∞ of H ( ), i.e. whether or not the disc was dense in M ∞ ? H (D) D D H (D)
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 11 / 131 Motivations for the Problem Where Did the Name Come From? ∞ The Corona Problem for H (D)
Using basic functional analysis, Kakutani’s question can be phrased as the following question about analytic functions on the unit disc: Theorem 1
The open disc D is dense in MH∞ if and only if the following condition ∞ holds: If f1,..., fn ∈ H (D) and if
max |fj (z)| ≥ δ > 0 1≤j≤n
∞ then there exists g1, ··· , gn ∈ H (D) such that
f1g1 + ··· + fngn = 1.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 12 / 131 Motivations for the Problem Proof of Theorem1
Proof: Suppose that D is dense in MH∞ . Then, by continuity we have that max |m(fj )| ≥ δ 1≤j≤n
for all m ∈ MH∞ . This implies that {f1,..., fn} is in no proper ideal of ∞ H (D). Hence the ideal generated by {f1,..., fn} must contain the ∞ constant function 1 and so there exists g1,..., gn ∈ H (D) such that
1 = f1g1 + ··· + fngn.
Conversely, suppose that D is not dense in MH∞ . Then for some m0 ∈ MH∞ has a neighborhood disjoint from D and this neighborhood has the form n V = ∩j=1{m : |m(fj )| < δ} ∞ where δ > 0 and f1,..., fn ∈ H (D) with m0(fj ) = 0.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 13 / 131 Motivations for the Problem Proof of Theorem1, Continued
Since D ∩ V = ∅ we have that
max |fj (z)| ≥ δ > 0. 1≤j≤n
But, it is not possible that we have
1 = f1g1 + ··· + fngn
since we have that m0(fj ) = 0 for all 1 ≤ j ≤ n. Exercise 2 Let A( ) = {f ∈ Hol( ) ∩ C( ): sup |f (z)| < ∞} denote the disc D D D z∈D algebra. Show that M = . A(D) D
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 14 / 131 Motivations for the Problem Where Did the Name Come From? ∞ The Corona Problem for H (D)
Kakutani’s question was settled in 1962 by Carleson: = M ∞ . D H (D) Theorem 3 (Carleson’s Corona Theorem) N ∞ Let {fj }j=1 ∈ H (D) satisfy
N X 2 0 < δ ≤ |fj (z)| ≤ 1, ∀z ∈ D. j=1
Then there are functions {g }N in H∞ ( ) with Lennart Carleson j j=1 D
N X fj (z) gj (z) = 1 ∀z ∈ D and kgj k∞ ≤ Cδ,N . j=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 15 / 131 Motivations for the Problem Reasons to Care about the Corona Problem
2 2 1 Angles between Invariant Subspaces: Kθ := H (D) θH (D)
The angle between Kθ1 and Kθ2 is positive if and only if there exists ψ1 and ψ2 such that ψ1θ1 + ψ2θ2 = 1.
∞ 2 Interpolation in H (D): Let B(z) be a Blaschke product with zeros ∞ {zk } = Z, and let f ∈ H (D) be such that |f (z)| + |B(z)| ≥ δ > 0 ∀z ∈ D. ∞ Then there exists p, q ∈ H (D) such that pf + qB = 1 if and only if there is a solution to the interpolation problem 1 p(zk ) = ∀zk ∈ Z. f (zk )
3 Control Theory
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 16 / 131 The Operator Corona Problem Reformulation of the Problem
n Let Ω be a domain in C . Let E and E∗ be separable complex Hilbert spaces. H∞ (Ω) is the collection of all bounded operator-valued functions. E∗→E
F (z): E → E and kF k ∞ := sup kF (z)k ∗ H (Ω) E∗→E E∗→E z∈Ω
∞ Question 4 (HE∗→E (Ω)-Operator Corona Problem) Let F ∈ H∞ (Ω). Can we find, preferably local, necessary and sufficient E∗→E conditions on F so that it has an analytic left inverse? Namely, what conditions imply the existence of a function G ∈ H∞ (Ω) such that E→E∗ G(z)F (z) ≡ I ∀z ∈ Ω.
A simple necessary condition is: I ≥ F ∗(z)F (z) ≥ δ2I ∀z ∈ Ω.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 17 / 131 The Operator Corona Problem Connection to the Usual Corona Problem
Let Ω = D, the unit disc in the complex plane. Take T F (z) = (f1(z),..., fN (z)) in the Operator Corona Problem to recover: Question 5 (Corona Problem) ∞ Suppose that f1,..., fN ∈ H (D) with
N X 2 1 ≥ |fj (z)| ≥ δ > 0 ∀z ∈ D. j=1
∞ Do there exist gj ∈ H (D) such that
N X fj (z)gj (z) ≡ 1 ∀z ∈ D? j=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 18 / 131 The Operator Corona Problem Known Results for the disc D
When E∗ = C and dim E < ∞: In 1962 Carleson demonstrated that the simple necessary condition is sufficient; In 1967 H¨ormander gave another proof of the result using ∂-equations; In 1979 Wolff gave a simpler proof of Carleson’s result. When E∗ = C, dim E = ∞: Rosenblum, Tolokonnikov, and Uchiyama independently gave proofs.
When dim E∗ < ∞ and dim E = ∞: (Matrix Corona Problem) Fuhrmann and Vasyunin independently demonstrated this.
When dim E = dim E∗ = ∞: (Operator Corona Problem) In 1988 Treil constructed a counter example which indicates that the necessary condition is no longer sufficient. In 2004 he gave another construction which demonstrated the same phenomenon.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 19 / 131 The Operator Corona Problem Known Results for General Domains and Several Variables
In 1970 Gamelin showed that the H∞(Ω)-Corona problem is true when the domain Ω ⊂ C has “holes.” More generally, if Ω ⊂ C is a Denjoy domain, then the Corona Theorem is true by a result of Garnett and Jones from 1985.
The Story is Much Different in Several Complex Variables Simple cases where the maximal ideal space of the algebra can be n identified with a compact subset of C . For example, the Ball Algebra n A(Bn) or Polydisc Algebra A(D ). There are counterexamples to Corona Theorems in several complex variables due to Cole, Sibony and Fornaess, but for domains that are very complicated geometrically. n n ∞ When n ≥ 2 and Ω ⊂ C (e.g. Bn or D ) the H (Ω)-Corona Problem is open.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 20 / 131 The Operator Corona Problem Take Away Point
There are NO known domains in Ω ⊂ C for which the H∞(Ω)-Corona problem fails. There are no KNOWN domains in Ω ⊂ Cn for which the H∞(Ω)-Corona problem holds.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 21 / 131 ∞ The Corona Problem for H (D)
The Corona Problem for H∞(D)
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 22 / 131 ∞ The Corona Problem for H (D) Carleson Corona Problem
Our goal is now to prove the following important theorem of Carleson Theorem 6 (Carleson) ∞ Suppose that f1,..., fn ∈ H (D) and there exists a δ > 0 such that
1 ≥ max {|fj (z)|} ≥ δ > 0. 1≤j≤n
∞ Then there exists g1,..., gn ∈ H (D) such that
1 = f1(z)g1(z) + ··· + fn(z)gn(z) ∀z ∈ D
and
kgj k ∞ ≤ Cδ,n ∀j = 1,..., n. H (D) An obvious remark is that the condition on the functions f is clearly necessary. We can’t have all the functions simultaneously vanish if they can generate the function 1. B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 23 / 131 ∞ The Corona Problem for H (D) Motivating the ∂-problem
First consider the case of two functions so that we can see the connections between this problem and the ∂-problem we will study. Suppose we have ∞ two functions f1, f2 ∈ H (D) such that
max(|f1(z)| , |f2(z)|) ≥ δ.
Define the following functions
f1(z) f2(z) ϕ1(z) = 2 2 ϕ2(z) = 2 2 . |f1(z)| + |f2(z)| |f1(z)| + |f2(z)|
The hypotheses on f1 and f2 imply that the functions ϕ1 and ϕ2 are in fact bounded and smooth on D. Note that
1 = f1(z)ϕ1(z) + f2(z)ϕ2(z) ∀z ∈ D
but the functions ϕ1 and ϕ2 are in general not analytic. B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 24 / 131 ∞ The Corona Problem for H (D) Motivating the ∂-problem
Now, observe for any function r we have that the functions
g1 = ϕ1 + rf2 g2 = ϕ2 − rf1 also solve the problem f1g1 + f2g2 = 1. Our goal is to select a good choice of function r so that the resulting choice will make g1 and g2 be analytic and bounded. Now, we have that g1 is analytic if and only if
0 = ∂g1 = ∂ϕ1 + f2∂r.
Similarly, g2 is analytic if and only if
0 = ∂g2 = ∂ϕ2 − f1∂r.
Using these two equations and the condition that f1ϕ1 + f2ϕ2 = 1 gives that the function r must satisfy the equation
∂r = ϕ1∂ϕ2 − ϕ2∂ϕ1.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 25 / 131 ∞ The Corona Problem for H (D) Carleson measures and ∂-problems Solving ∂-Equations
1 Set ∂ = 2 (∂x + i∂y ). Recall that a function h is analytic if ∂h = 0.
We need to solve slightly more general differential equations: ∂F = G. Theorem 7
Suppose that G is a smooth compactly supported function in D. Then 1 Z 1 1 Z G(ξ) F (z) = G(ξ) dξ ∧ dξ = dA(ξ) 2πi D z − ξ π D z − ξ solves ∂F = G. Natural question to ask: Can we solve ∂F = G but with some estimates?
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 26 / 131 ∞ The Corona Problem for H (D) Carleson measures and ∂-problems Differential Forms and Stokes’ Theorem
Recall that Stokes Theorem can (roughly) be stated as Z Z ω = dω. ∂Ω Ω Here Ω is a nice domain, ∂Ω is the boundary of Ω, ω is a differential form, and d is the exterior differential. In two variables df = ∂x dx + ∂y dy Recall also that dx ∧ dy = −dy ∧ dx and that dx ∧ dx = dy ∧ dy = 0. Writing this in the variables z and z we have df = ∂fdz + ∂fdz
1 1 where ∂ = 2 (∂x + i∂y ) ∂ = 2 (∂x − i∂y ) and dz = dx + idy dz = dx − idy dz ∧ dz = −2idx ∧ dy.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 27 / 131 ∞ The Corona Problem for H (D) Carleson measures and ∂-problems Proof of Theorem7
Fix > 0 and z ∈ D let D(z) = {ξ ∈ D : |z − ξ| ≥ }. Note that ∂D = T ∪ {ξ : |ξ − z| = }. Suppose that ϕ is a smooth compactly supported function in D. Then, we have 1 Z 1 1 Z ϕ(ξ) ∂ϕ(ξ) dξ ∧ dξ = − ∂ dξ ∧ dξ 2πi D ξ − z 2πi D ξ − z 1 Z ϕ(ξ) Z ϕ(ξ) = dξ + dξ 2πi |ξ−z|= ξ − z T ξ − z 1 Z ϕ(ξ) = dξ. 2πi |ξ−z|= ξ − z
Here we have used the fact that the support of ϕ ⊂ D to conclude that the last integral is 0. We have also used the following computation ϕ(ξ) ϕ(ξ) ϕ(ξ) d dξ = ∂ dξ ∧ dξ + ∂ dξ ∧ dξ. ξ − z ξ − z ξ − z
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 28 / 131 ∞ The Corona Problem for H (D) Carleson measures and ∂-problems Proof of Theorem7, Continued
Now note that as → 0 we have that 1 Z ϕ(ξ) dξ → ϕ(z). 2πi |ξ−z|= ξ − z
This says that 1 Z 1 ∂ϕ(ξ) dξ ∧ dξ = ϕ(z). 2πi D ξ − z So, if we have a solution to the problem ∂F = G then one solution should be given by 1 Z 1 F (z) = G(ξ) dξ ∧ dξ. 2πi D ξ − z Note that this solution is continuous in the complex plane and smooth in the disc since it is the convolution of a continuous function and a bounded function. We now show that we do indeed have ∂F = G.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 29 / 131 ∞ The Corona Problem for H (D) Carleson measures and ∂-problems Proof of Theorem7, Continued
First, note that Z Z Z F ∂ϕdz ∧ dz + ∂F ϕdz ∧ dz = ∂(F ϕ)dz ∧ dz D D ZD = F ϕdz = 0. T Here again we have used the support of ϕ and the analogous computations from above. This then implies Z Z F ∂ϕdz ∧ dz = − ∂F ϕdz ∧ dz. D D
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 30 / 131 ∞ The Corona Problem for H (D) Carleson measures and ∂-problems Proof of Theorem7, Continued
Using these computations we see that Z Z ∂F ϕdz ∧ dz = − F ∂ϕdz ∧ dz D D Z 1 Z 1 = − G(ξ) dξ ∧ dξ ∂ϕdz ∧ dz D 2πi D ξ − z Z 1 Z 1 = − G ∂ϕ dz ∧ dz dξ ∧ dξ D 2πi D ξ − z Z = Gϕdξ ∧ dξ. D Since this is true for all smooth compactly supported ϕ in D we have that
∂F = G
as claimed.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 31 / 131 ∞ The Corona Problem for H (D) Carleson measures and ∂-problems
The above Theorem demonstrates that it is possible to solve equations for the form ∂F = G. However, we will want to solve the equation with some norm control, in particular we want to solve the equation and obtain estimates on kF k∞ in terms of information from G. To accomplish this, we will assume the the function G “generates” 2 Carleson measures for H (D). We now prove a result of Wolff that gives the desired estimates.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 32 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem ∂-problems and Carleson measures
Theorem 8 (Wolff)
Suppose that G(z) is bounded and smooth on the disc D. Further, assume that the measures 1 1 |G(z)|2 log dA(z) and |∂G(z)| log dA(z) ∈ CM(H2( )) |z| |z| D
Then there exists a continuous function b(z) on D, smooth on D such that
∂b = G
and there exists a constant such that 2 2 1 1 kbk ∞ |G(z)| log dA(z) + |∂G(z)| log dA(z) . L (T) . |z| CM(H2) |z| CM(H2)
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 33 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Proof of Theorem8
By Theorem7 above, we clearly have one solution to the problem ∂b = G. We can obtain lots of solutions by adding functions that are in the kernel of the operator ∂ and any function h in the disc algebra A(D) allows us to have that b + h also satisfies that ∂(b + h) = G. The goal is to select a good choice of the function h that allows us to obtain the estimates we
seek. A duality argument show that if we take kkj kH2 ≤ 1 then Z n o 2 2 inf kbk∞ : ∂b = G = sup F k1k2dm : k1 ∈ H (D), k2 ∈ H0 (D) . T Here we have that the function F is defined as in the Theorem7. Since we are supposing that G is bounded and smooth, we have that F is smooth on the D and continuous on D. A density argument lets us further assume that the functions k1 and k2 are smooth across the boundary of D (just consider dilates of the functions fr (z) = f (rz) and apply a normal family argument). B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 34 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Proof of Theorem8
Now, we apply Green’s Theorem to the function F k1k2. First, we compute the Laplacian of the function F k1k2 since it will appear in Green’s Theorem. ∆(F k1k2) = 4∂ ∂F k1k2 + F ∂(k1k2)
0 0 = 4 ∂Gk1k2 + G k1k2 + k1k2 .
Here we have used that ∂F = G and that k1k2 is anti-holomorphic. Substituting in this information we find: Z Z 1 F k1k2dm = F (0)k1(0)k2(0) + ∆(F k1k2) log dA T D |z| Z 1 = 2 ∂Gk1k2 log dA(z) D |z| Z 0 0 1 + 2 G k1k2 + k1k2 log dA(z) D |z| = I + II. B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 35 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Proof of Theorem8
We estimate each of these integrals separately. First, consider the integral corresponding to I. Making obvious estimates, we have Z Z 1 1 ∂Gk1k2 log dA(z) ≤ |k1| |k2| |∂G| log dA(z) D |z| D |z| Z 1 2 1 2 ≤ |k1| |∂G| log dA(z) D |z| Z 1 2 1 2 × |k2| |∂G| log dA(z) D |z| 2 1 ≤ |∂G(z)| log dA(z) kk1kH2 kk2kH2 . |z| CM(H2)
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 36 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Proof of Theorem8
0 Turning to term II, it suffices to handle the term k1k2 since the other will follow by symmetry. Z 1 Z 1 0 0 Gk1k2 log dA(z) ≤ Gk1k2 log dA(z) D |z| D |z| Z 1 0 2 1 2 ≤ k1 log dA(z) D |z| Z 1 2 2 1 2 × |k2| |G| log dA(z) D |z|
2 1 ≤ |G(z)| log kk1kH2 kk2kH2 |z| CM(H2) Thus, we see that we have the following estimate for term II Z 0 0 1 2 1 G k1k2 + k1k2 log dA(z) . |G(z)| log kk1kH2 kk2kH2 . D |z| |z| CM(H2)
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 37 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Proof of Theorem8
Putting the estimates for terms I and II together gives us that Z 2 1 F k1k2dm . |G(z)| log kk1kH2( ) kk2kH2( ) |z| 2 D D T CM(H (D)) 2 1 + |∂G(z)| log dA(z) kk1kH2( ) kk2kH2( ) . |z| 2 D D CM(H (D)) This last estimate then proves that
n R 2 2 o sup F k1k2dm : k1 ∈ H ( ), k2 ∈ H ( ) T D 0 D . 2 |G(z)|2 log 1 + |∂G(z)| log 1 dA(z) . |z| 2 |z| 2 CM(H (D)) CM(H (D))
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 38 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Wolff’s Proof of the Corona Problem
Theorem 9 (Carleson)
∞ Suppose that f1,..., fn ∈ H (D) and there exists a δ > 0 such that
1 ≥ max {|fj (z)|} ≥ δ > 0. 1≤j≤n
∞ Then there exists g1,..., gn ∈ H (D) such that
1 = f1(z)g1(z) + ··· + fn(z)gn(z) ∀z ∈ D
and
kgj k ∞ ≤ Cδ,n ∀j = 1,..., n. H (D) We will now use Wolff’s ∂-problem with estimates to prove this Theorem.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 39 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Proof of Theorem9
Without loss of generality, we may assume that the functions are analytic in a neighborhood of the closed disc D (Reduce to a normal families argument). Define the functions f (z) ϕ (z) = j ∀z ∈ . j Pn 2 D j=1 |fj (z)|
The hypotheses on the functions fj clearly give that |ϕj (z)| ≤ Cn,δ. These functions are in general not analytic, and so we must correct them. We set n X gj (z) = ϕj (z) + aj,k (z)fk (z) k=1
where the functions aj,k (z) are to be determined, but we will require that aj,k (z) = −ak,j (z). Note this alternating condition implies that n n n n X X X X fj (z)gj (z) = fj (z)ϕj (z) + aj,k (z)fj (z)fk (z) = 1. j=1 j=1 j=1 k=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 40 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Proof of Theorem9, Continued
To have the alternating characteristic of aj,k we set
aj,k (z) = bj,k (z) − bk,j (z)
for some yet to be determined functions. We will chose the functions bj,k to be solutions to the follow ∂ problem:
∂bj,k = ϕj ∂ϕk := Gj,k . Using this, we see that n X ∂gj = ∂ϕj + fk ∂aj,k k=1 n X = ∂ϕj + fk ∂bj,k − ∂bk,j k=1 n X = ∂ϕj + fk ϕj ∂ϕk − ϕk ∂ϕj k=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 41 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Proof of Theorem9, Continued
n ! n X X = ∂ϕj + ϕj ∂ fk ϕk − ∂ϕj fk ϕk k=1 k=1
= ∂ϕj + ϕj ∂1 − ∂ϕj 1 = 0.
So the functions gj are analytic. To prove that |gj (z)| ≤ Cn,δ are bounded, it suffices to prove the functions bj,k are bounded by Cn,δ. With this in mind, and having the result of Wolff at our disposal, we must show that the measures 1 1 |G (z)|2 log dA(z) and |∂G (z)| log dA(z) j,k |z| j,k |z|
2 are H (D)-Carleson measures.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 42 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Proof of Theorem9, Continued
Claim 10 ∞ Let fj ∈ H (D) and Gj,k be defined as above. Then the measures 1 1 |G (z)|2 log dA(z) and |∂G (z)| log dA(z) j,k |z| j,k |z|
are dominated by (up to a constant Cn,δ) by the following measure
n 2 1 X f 0(z) log dA(z). j |z| j=1
Exercise 11 ∞ 0 2 1 Show that for f ∈ H (D) that we have |f (z)| log |z| dA(z) is a 2 2 H (D)-Carleson measure. Hint: Use the alternate norm for H (D) and think about the product rule for derivatives.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 43 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Proof of Theorem9, Continued
2 Consider the expression |Gj,k | . Note that by the hypotheses on fj we have that |ϕj | ≤ Cn,δ and so, 2 2 |Gj,k | ≤ Cn,δ ∂ϕk . Now, if we compute we see that 0 Pn 0 f fk fj f ∂ϕ = k − j=1 j k Pn 2 2 |fj | Pn 2 j=1 j=1 |fj | Pn 0 0 fj (fj f − fk f ) = j=1 k j . Pn 22 j=1 |fj | Using this, we see that 2 Pn 2 Pn 0 n 2 j=1 |fj | j=1 fj 2 2 X 0 |Gj,k | ≤ Cn,δ ∂ϕk ≤ Cn,δ 2 ≤ Cn,δ fj . Pn 2 j=1 |fj | j=1 B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 44 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Proof of Theorem9, Continued
Next consider |∂Gj,k |. First, observe that
∂Gj,k = ∂ϕj ∂ϕk + ϕj ∂∂ϕk
By the computations above, we have that
Pn 0 0 j=1 fj (fj fk − fk fj ) ∂ϕk = . Pn 22 j=1 |fj |
Direct computation also gives,
Pn 0 fj l=1 fl fl ∂ϕj = − . Pn 22 j=1 |fj |
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 45 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Proof of Theorem9, Continued
Finally, we have that Pn 0 0 0 Pn f 0f Pn f f 0 − f f 0 j=1 fj (fj fk − fk fj ) l=1 l l l=1 l k k l ∂∂ϕk = − 2 . Pn 22 Pn 23 j=1 |fj | j=1 |fj |
Now consider the term ∂ϕj ∂ϕk . It is obvious that we can dominate this expression by n X 0 0 X 0 2 Cn,δ fj fk ≤ Cn,δ fk . j,k k=1
Similarly, we have that ϕj ∂∂ϕk can be dominated by an identical expression. This then proves the claim. An application of Wolff’s Theorem, Theorem8, then gives the bounded solution we seek.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 46 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem
Here is an exercise to see if you really understand the proof of the Corona Theorem just given Exercise 12 ∞ Suppose that f1,..., fn, g ∈ H (D) such that
n X |g(z)| ≤ |fj (z)| . j=1
∞ Show that there exists g1,..., gn ∈ H (D) such that
n 3 X g = fj gj . j=1
Hint: Mimic Wolff’s proof of the Corona Theorem but start with ψj = gϕj , with ϕj as defined above.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 47 / 131 ∞ The Corona Problem for H (D) Wolff’s Theorem Suggested Method for Proving Corona Problems
Step 1 (Find “trivial” Solutions): Use the condition PN 2 0 < δ ≤ j=1 |fj (z)| ≤ 1 to construct smooth and bounded solutions to the problem.
Step 2 (Use Algebra): Write any solution to the problem as
N X gj = ϕj + ak,j fk k=1
where the ϕj are the smooth solutions from Step 1 and the ak,j are some yet to be determined functions.
Step 3 (Find Bounds on the Solutions): ∂-equation, Nehari’s Theorem
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 48 / 131 Jones’ Constructive Proof of ∂b = µ for Carleson Measures Constructive Proofs of ∂-problems
We now give a more constructive method to solve the ∂ problem. We have seen from above, that to solve the equation ∂F = G we can set 1 Z 1 F (z) = G(ξ) dξ ∧ dξ. 2πi D z − ξ However, we can also use another kernel to accomplish this. Choose a function K(z, ξ) that is analytic in z, K(z, z) = 1 and that is smooth, then: 1 Z 1 F (z) = K(z, ξ)G(ξ) dξ ∧ dξ. 2πi D z − ξ solves ∂F = G. Exercise 13 Show that this is true. Hint: K(z, ξ) = 1 + (z − ξ)K˜ (z, ξ).
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 49 / 131 Jones’ Constructive Proof of ∂b = µ for Carleson Measures Constructive Proofs of ∂-problems
Theorem 14 (Jones)
Let µ ∈ CM(H2) and σ = |µ| . Set S (µ)(z) = R K (σ, z, ζ) dµ (ζ) kµkCM(H2) D with
2i 1 − |ζ|2 K (σ, z, ζ) ≡ π (z − ζ) 1 − ζz ( ) Z 1 + ωz 1 + ωζ × exp − + dσ (ω) , |ω|≥|ζ| 1 − ωz 1 − ωζ
we have that:
1 1 S (µ) ∈ Lloc (D), ∂S (µ) = µ in the sense of distributions; R 2 |K (σ, x, ζ)| d |µ| (ζ) kµk 2 for all x ∈ = ∂ , D . CM(H ) T D so kS (µ)k ∞ kµk 2 . L (T) . CM(H )
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 50 / 131 Jones’ Constructive Proof of ∂b = µ for Carleson Measures Constructive Proofs of ∂-problems
Note that the kernel K(σ, z, ξ) is analytic in z,
2i 1 K(σ, z, ξ) = K˜ (σ, z, ξ). π z − ξ
The kernel K˜ (σ, z, ξ) is smooth with
2 (Z ) ˜ 1 − |z| 1 + ωz 1 + ωz K(σ, z, z) = 2 exp − + dσ(ω) = 1. 1 − |z| |ω|≥|z| 1 − ωz 1 − ωz
So ∂S(µ) = µ follows from the exercise above. Remains to show that we have the desired estimates.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 51 / 131 Jones’ Constructive Proof of ∂b = µ for Carleson Measures Proof of Theorem 14
Proof: Observe that if we prove Z |K (σ, x, ζ)| d |µ| (ζ) . kµkCM(H2) ∀x ∈ T = ∂D D then Z |S(µ)(z)| ≤ |K (σ, z, ζ)| d |µ| (ζ) . kµkCM(H2) . D Note that for the measure σ we have ! Z 1 + wζ Z 1 + wζ Re dσ(w) = Re dσ(w) |w|≥|ζ| 1 − wζ |w|≥|ζ| 1 − wζ Z 1 − |ζ|2 ≤ 2 2 dσ(w) D |1 − wζ| 2 ≤ 2 k˜ = 2, ζ 2 H (D)
2 1 where k˜ (z) = (1−|ζ| ) 2 is the normalized reproducing kernel for H2( ). ζ 1−ζz D B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 52 / 131 Jones’ Constructive Proof of ∂b = µ for Carleson Measures Proof of Theorem 14, Continued
It will suffice to control the boundary values of the function S(µ), and so we can take z ∈ T. Using the estimate from above and that z ∈ T we have ( ) 2 1 − |ξ|2 Z 1 + ωz 1 + ωξ |K(σ, z, ξ)| = exp − + dσ(ω) 2 π |ω|≥|ξ| 1 − ωz 1 − ωξ 1 − ξz ( ) 2 1 − |ξ|2 Z 1 + ωz = exp Re − dσ(ω) 2 π |ω|≥|ξ| 1 − ωz 1 − ξz ( ) Z 1 + ωz × exp Re dσ(ω) |ω|≥|ξ| 1 − ωz ( ) 2 1 − |ξ|2 Z 1 − |ω|2 ≤ e2 exp − dσ(ω) . 2 2 π |ω|≥|ξ| |1 − ωz| 1 − ξz
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 53 / 131 Jones’ Constructive Proof of ∂b = µ for Carleson Measures Proof of Theorem 14, Continued
We can then use this estimate on the kernel K(σ, z, ζ) to give Z
|S (µ)(z)| = K (σ, z, ζ) dµ (ζ) D Z ≤ kµkCM(H2) |K (σ, z, ζ)| dσ(ζ) D Z 2 2 2 1 − |ζ| ≤ e kµk 2 CM(H ) 2 π D 1 − ζz ( ) Z 1 − |ω|2 × exp − 2 dσ(ω) dσ(ζ). |ω|≥|ζ| |1 − ωz| It remains to show that ( ) Z 1 − |ζ|2 Z 1 − |ω|2 exp − dσ(ω) dσ(ζ) ≤ 1. 2 2 D |ω|≥|ζ| |1 − ωz| 1 − ζz
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 54 / 131 Jones’ Constructive Proof of ∂b = µ for Carleson Measures Proof of Theorem 14, Continued
PN First, suppose that we have dσ = j=1 aj δζj with |ζj | ≤ |ζj+1|. Further, 2 1−|ζj | set βj = aj 2 and |1−ζj z| N X tj = βk , and so βj = tj − tj−1. k=j If we evaluate the above integral for this measure and use the resulting notation, we see that the integral becomes N Z ∞ X −tj −t (tj − tj−1)e ≤ e dt = 1. j=1 0 A standard measure theory argument then finishes the proof that ( ) Z 1 − |ζ|2 Z 1 − |ω|2 exp − dσ(ω) dσ(ζ) ≤ 1. 2 2 D |ω|≥|ζ| |1 − ωz| 1 − ζz
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 55 / 131 The Corona Problem for MD
Corona Problem for MD
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 56 / 131 The Corona Problem for MD Refreshers about the Dirichlet Space
The Dirichlet space D is the Hilbert space of functions f ∈ Hol (D) for which Z 2 2 1 0 2 kf kD = |f (0)| + f (z) dxdy < ∞. π D The multiplier algebra of a space D is the collection of functions such that
MD := {ϕ ∈ Hol(D): kϕf kD ≤ C kf kD ∀f ∈ D}
Which is normed by kϕk := inf{C : kϕf k ≤ C kf k ∀f ∈ D}. MD D D Proposition 15
∞ MD H ( ) with kϕk ∞ ≤ kϕk . ( D H (D) MD
∞ While the observation that MD ( H (D) is useful, it would be better to have a full characterization of this algebra in terms of objects that we already have seen.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 57 / 131 The Corona Problem for MD Refreshers about the Dirichlet Space
Define the space X to be the collection of functions such that |b0|2 dA(z) is a Carleson measure for the Dirichlet space D. Namely, a function b ∈ X if and only if Z 2 0 2 2 |f (z)| b (z) dA(z) ≤ C kf kD ∀f ∈ D. D
0 2 We can norm this space X by setting dµb := |b (z)| dA(z) and Z 2 2 2 2 kbkX = |b(0)| + inf C : |f (z)| dµb(z) ≤ C kf kD . D
Proposition 16
∞ MD = H (D) ∩ X Moreover, kbk ≈ kbk + kbk MD ∞ X
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 58 / 131 The Corona Problem for MD
∂-problem on MD Since we need a boundary estimate, we will work with another more convenient norm for the Dirichlet space and its multiplier algebra. Z Z 2 2 2 f (z) − f (w) kf k 1 = kf kL2( ) + dm(z)dm(w) W 2 ( ) T T T T z − w
∞ We then have that ϕ ∈ MD if and only if f ∈ H (D) and Z Z 2 f (z) − f (w) dm(z)dm(w) ≤ C Cap(E) E ⊂ T. E T z − w
Theorem 17 (Xiao)
If |g(z)|2 dA(z) ∈ CM(D) then there is a function f such that ∂f = g and
kf k |g(z)|2 dA(z) . M 1 . CM(D) W 2 (T)
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 59 / 131 The Corona Problem for MD Sketch of Proof of Theorem 17
2 First note that |g(z)| dA(z) ∈ CM(H (D)) (straightforward computation); Apply Jones’ Theorem, Theorem 14 to see that ∂f = g with
kgk ∞ 1; L (T) . It suffices to show that for any collection of intervals {Ik } ⊂ T that we have Z 2 |∇f (z)| dA(z) ≤ C Cap (∪k Ik ) . ∪k T (Ik ) Simple computations then give that
Z |g(w)| |∇f (z)| . 2 dA(w). D |1 − wz|
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 60 / 131 The Corona Problem for MD Sketch of Proof of Theorem 17, Continued
Lemma 18 (Rochberg & Wu)
1 Let α ∈ (−∞, 2 ] and β > max{−1, −1 − 2α} and β+3 β+3 b > max{ 2 , 2 − α}. Also, let Z f (w) 2 b−2 Tf (z) = b (1 − |w| ) dA(w). D |1 − wz|
2 2 β If |f (z)| (1 − |z| ) dA(z) is an Dα-Carleson measure, then 2 2 β |Tf (z)| (1 − |z| ) dA(z) is also an Dα-Carleson measure.
This lemma shows that it is possible to produce new Carleson measures from old Carleson measures via “spreading them out”. This finishes the proof of Theorem 17. There is an alternate way to conclude the estimates at this point using singular integral theory (Beurling operator).
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 61 / 131 The Corona Problem for MD
The Corona Theorem in MD
Theorem 19 (Tolokonnikov, Xiao, Trent)
Suppose that f1,..., fn ∈ MD are such that
N X 2 0 < δ ≤ |fj (z)| ≤ 1 ∀z ∈ D. j=1
Then there exists g1,..., gN ∈ MD such that PN j=1 fj (z)gj (z) = 1 for all z ∈ D; PN kgj k ≤ C . j=1 MD δ,N
There are three different proofs of this theorem now. The order above is chronological. The condition that fj do not simultaneously vanish is clearly necessary.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 62 / 131 The Corona Problem for MD Proof of Theorem 19
Proof: For j = 1,..., N, define
f (z) ϕ (z) = j . j PN 2 j=1 |fj (z)| Then we have that N X 1 = fj (z)ϕj (z). j=1
Suppose that we can find functions bjk such that (i) ∂bjk = ϕj ∂ϕk ; (ii) kbjk k 1. MD . Then we define N X gj (z) = ϕj (z) + (bjk (z) − bkj (z)) fk (z). k=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 63 / 131 The Corona Problem for MD Proof of Theorem 19, Continued
These functions are analytic since, N X ∂gj = ∂ϕj + ∂bjk − ∂bkj fk k=1 N X = ∂ϕj + ϕj ∂ϕk − ϕk ∂ϕj fk k=1 N ! N ! X X = ∂ϕj − ϕk fk ∂ϕj + ϕj ∂ fk ϕk k=1 k=1
= ∂ϕj − ∂ϕj + ϕj ∂1 = 0.
Moreover, because of the estimate on bjk it is clear that gj ∈ MD. Finally, with these functions we have N X fj (z)gj (z) = 1 j=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 64 / 131 The Corona Problem for MD Proof of Theorem 19, Continued
The only issue that remains is to demonstrate that it is in fact possible to find such solutions bjk , and to do so, it suffices to demonstrate that the functions ϕj ∂ϕk are in fact Carleson measures for the Dirichlet space D. Computations from a previous lecture give
P 0 0 f l fl fl fk − fk fl ϕ ∂ϕ = j . j k P 2 2 |fj (z)| P 2 j j |fj (z)|
This then implies that
N N 2 1 X 0 2 X 0 2 ϕj ∂ϕk . |fl | . |fl | . P 23 j |fj (z)| l=1 l=1
The last estimate follows since the functions fl ∈ MD. An application of Theorem 17 provides the solutions and estimates we seek.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 65 / 131 Corona Problem for Multiplier Algebras with CNP Kernels
Corona Problem for Multiplier Algebras with Complete Nevanlinna-Pick Kernels
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 66 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Abstract Setup
n Ω ⊂ C open set H an irreducible complete Nevanlinna-Pick reproducing kernel Hilbert function space on Ω:
For each λ ∈ Ω there exists a function kλ ∈ H such that
f (λ) = hf , kλiH ∀f ∈ H; The Nevanlinna-Pick problem is solvable for scalars and matrices; The kernel kλ(·) is irreducible if it doesn’t vanish, i.e., kλ(µ) 6= 0;
MH will denote the multiplier algebra of H, i.e. ϕ ∈ MH if ϕf ∈ H for all f ∈ H and kϕk = kM k MH ϕ H→H
We now formulate two different Corona questions for the space H and MH that turn out to guide the study of questions in this area.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 67 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Baby Corona Problem for H
Question 20 (Baby Corona Problem for H)
Given f1,..., fN ∈ MH satisfying
N X 2 0 < δ ≤ |fj (z)| ≤ 1 ∀z ∈ Ω j=1
and h ∈ H. Does there exist a constant Cn,N,δ and functions k1,..., kN ∈ H satisfying
N X 2 2 kkj kH ≤ Cn,N,δ khkH , j=1 N X kj (z) fj (z) = h (z) ∀z ∈ Ω? j=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 68 / 131 Corona Problem for Multiplier Algebras with CNP Kernels
Corona Problem for MH
Question 21 (Corona Problem for Multiplier Algebras MH)
Given f1,..., fN ∈ MH satisfying
N X 2 0 < δ ≤ |fj (z)| ≤ 1 ∀z ∈ Ω. j=1
Are there functions g1,..., gN ∈ MH and a constant Cn,N,δ such that:
N X kgj k ≤ C MH n,N,δ j=1 N X gj (z) fj (z) = 1 ∀z ∈ Ω? j=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 69 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Density in Maximal Ideal Space
Recall that we showed the equivalence between the Corona problem for ∞ H (D) and the density of D in the maximal ideal space. Repeating the argument given from a previous lecture we have the following theorem for the multiplier algebra MH of a reproducing kernel Hilbert space H. Theorem 22 (Density in Maximal Ideal Space)
The set Ω is dense in MH if and only if the following condition holds: If f1,..., fn ∈ MH and if max |fj (z)| ≥ δ > 0 1≤j≤n
then there exists g1, ··· , gn ∈ MH such that
f1g1 + ··· + fngn = 1.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 70 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Corona implies Baby Corona
Indeed, if the Corona Problem is true and we take h ∈ H, we can see that the Baby Corona Problem follows. Suppose f1,..., fN ∈ MH, and there exists g1,..., gN ∈ MH such that
N N X X kgj k ≤ C gj (z) fj (z) = 1 ∀z ∈ Ω. MH n,N,δ j=1 j=1
Multiplying the second equation by h, we find
N N X X h(z) = gj (z) fj (z) h(z) = kj (z) fj (z) ∀z ∈ Ω. j=1 j=1
Since g1,..., gN ∈ MH we then have that kj := gj h ∈ H with kkj k ≤ kgj k khk , so the claimed estimates follow as well. H MH H
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 71 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Baby Corona implies Corona?
It turns out that in certain situations the Baby Corona Problem is equivalent to the Corona Problem. Recall that the Baby Corona Problem is the following question: Given f1,..., fN ∈ MH satisfying
2 2 |f1 (z)| + ··· + |fN (z)| ≥ δ > 0, z ∈ Ω, (1)
and h ∈ H. Are there functions k1,..., kN ∈ H such that 1 kk k2 + ··· + kk k2 ≤ khk2 , (2) 1 H N H δ H k1 (z) f1 (z) + ··· + kN (z) fN (z) = h (z) , ∀z ∈ Ω?
We now will rephrase the Baby Corona Problem in operator theory language.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 72 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Baby Corona implies Corona?
More succinctly, (2) is equivalent to the operator lower bound
∗ Mf Mf − δIH ≥ 0, (3)
N PN where f ≡ (f1,..., fN ), Mf : ⊕ H → H by Mf g = α=1 gj fj , and N M∗h = M∗ h . f fj j=1 N N N For f = (fj )j=1 ∈ ⊕ H and h ∈ H, define Mf h = (fj h)j=1 and
kf kMult(H,⊕N H) = kMf kH→⊕N H = sup kMf hk⊕N H . khkH≤1
r 2 Note that max M ≤ kf k ≤ PN M . 1≤j≤N fj Mult H,⊕N H j=1 fj MH ( ) MH
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 73 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Equivalence of (2) and (3)
To see this note that (3) is equivalent to
∗ ∗ ∗ δ hh, hiH ≤ hh, Mf Mf hiH = hMf h, Mf hi⊕N H . (4)
From functional analysis, we obtain that the bounded map N ⊥ Mf : ⊕ H → H is onto. If N = ker Mf , then Mdf : N → H is ∗ ⊥ invertible. Now (4) implies that Mdf : H → N is invertible and that −1 −1 ∗ 1 1 Md ≤ √ . By duality we then have Md ≤ √ . Thus f δ f δ ⊥ given h ∈ H, there is k ∈ N satisfying Mf k = h and
2 2 −1 1 2 kkk⊕N H = Mdf h ≤ khkH , ⊕N H δ which is (2).
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 74 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Equivalence of (2) and (3), Continued
Conversely, using (2) we compute that
∗ ∗ kMf hk⊕N H = sup hp, Mf hi⊕N H = sup |hMf p, hiH| kpk⊕N H≤1 kpk⊕N H≤1 * + k khk2 √ H ≥ Mf , h = ≥ δ khkH , kkk N kkk N ⊕ H H ⊕ H which is (4), and hence (3). Next we note that (1) is necessary for (3) as
can be seen by testing (4) on reproducing kernels kz :
∗ ∗ 2 δ hkz , kz iH ≤ hMf kz , Mf kz i⊕N H = |f (z)| hkz , kz iH
N ∗ since M kz = fα (z)kz . f α=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 75 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Baby Corona implies Corona? Toeplitz Corona Theorem
Theorem 23 (Toeplitz Corona Theorem, (Agler and McCarthy))
n Let H be a Hilbert function space in an open set Ω in C with an irreducible complete Nevanlinna-Pick kernel. Let > 0 and let f1,..., fN ∈ MH. Then the following are equivalent: PN (i) There exists g1,..., gN ∈ MH such that j=1 fj gj = 1 and 1 kgkMult(H,⊕N H) ≤ ; PN (ii) For any h ∈ H, there exists k1,..., kN ∈ H such that h = j=1 kj fj PN 2 1 2 and j=1 kkj kH ≤ 2 khkH. Moral: If the Hilbert space has a reproducing kernel with enough structure, then the Corona Problem and the Baby Corona Problem are the same question.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 76 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Suggested Method for Proving Corona Problems
Step 1 (Baby Corona Problem): Show that for functions PN 2 f1,..., fN ∈ MH such that 0 < δ ≤ j=1 |fj (z)| ≤ 1 for all z ∈ Ω and h ∈ H that it is possible to find k1,..., kN ∈ H such that PN (i) j=1 fj (z)kj (z) = h(z) for all z ∈ Ω; PN 2 2 (ii) j=1 kkj kH ≤ Cδ,N,n khkH.
Step 2 (Toeplitz Corona Problem): Suppose that f1,..., fN ∈ MH are 2 ∗ such that I ≤ Mf Mf ≤ I. Then find g1,..., gN ∈ MH such that PN (i) j=1 fj (z)gj (z) = 1 for all z ∈ Ω; (ii) kgk ≤ 1 . Mult(H,⊕N H) 2
Step 1 is generally “easy”. Step 2 is generally hard and requires some magic.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 77 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23 (i) implies (ii)
Multiply Mf g = 1 by h to get Mf gh = h. Set k = gh = (g1h,..., gN h) and then
2 2 kkk⊕N H = kghk⊕N H 2 2 = kg1hkH + ··· + kgN hkH
= kMg hk⊕N H ≤ kgk2 khk2 Mult(H,⊕N H) H 1 ≤ khk2 . 2 H
The other direction is a little more difficult.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 78 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23 (ii) implies (i): Preliminaries
An important point to notice is that we can transform operator bounds ∗ into bounds on the kernel functions. For example, Mf Mf − IH ≥ 0 for H in (3), can be recast in terms of kernel functions, namely
{hf (ζ) , f (λ)i N − } k (ζ, λ) ≥ 0. (5) C
PJ Indeed, if we let h = i=1 ξi kxi in (4) we obtain J J X X D E ξi ξj k (xj , xi ) = ξi ξj kx , kx i j H i,j=1 i,j=1 N * J J + X X X ≤ ξi fα (xi )kxi , ξj fα (xj )kxj α=1 i=1 j=1 H J ( N ) X X = ξi ξj fα (xi )fα (xj ) k (xj , xi ) . i,j=1 α=1 B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 79 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23 (ii) implies (i): Preliminaries
A similar calculation shows that the operator upper bound ∗ IH − Mf Mf ≥ 0
which is equivalent to kMf k⊕N H→H ≤ 1, can be recast in terms of kernel functions as
{1 − hf (ζ) , f (λ)i N } k (ζ, λ) ≥ 0. (6) C
We need to understand kf kMulti(H,⊕N H) = kMf kH→⊕N H in terms of kernel functions and must consider N × N matrix kernel functions. Recall N N N that Mf : H → ⊕ H by Mf h = (fαh)α=1. Then for g ∈ ⊕ MH, N * N + X X ∗ hMf h, gi⊕N H = hfαh, gαiH = h, Mfα gα , α=1 α=1 H and so ∗g = PN M∗ g . Mf α=1 fα α B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 80 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23 (ii) implies (i): Preliminaries
2 We can then see that kMf kH→⊕N H ≤ 1, which is the same as ∗ 2 kMf k⊕N H→H ≤ 1, is equivalent to
N N 2 X 2 X ∗ ∗ ∗ 0 ≤ kg k − M g = hg, gi N − h g, gi (7) α H fα α ⊕ H Mf Mf H α=1 α=1 H ∗ = I⊕N H − Mf Mf g, g ⊕N H . Which is the operator bound
∗ I⊕N H − Mf Mf ≥ 0. (8)
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 81 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23 (ii) implies (i): Preliminaries
To obtain an equivalent kernel estimate, let N N PJ α g = (gα) = ξ kx α so that α=1 i=1 i i α=1
N N J ∗ X ∗ X X α α g = M g = ξ f (x )k α . Mf fα α i α i xi α=1 α=1 i=1 If we substitute this in (7) we obtain
N J X X α β α β β α ∗ ∗ ξi ξj fα (xi )fβ xj k xj , xi = hMf g, Mf giH α,β=1 i,j=1
≤ hg, gi⊕N H N J X α X α β β α = δβ ξi ξj k xj , xi . α,β=1 i,j=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 82 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23 (ii) implies (i): Preliminaries
This can be rearranged to give
N J X X α β hn α α βo β αi ξi ξj δβ − fα (xi )fβ xj k xj , xi ≥ 0. α,β=1 i,j=1
N If we view f (ζ) ∈ B C, C we can rewrite this last expression as
∗ {I N − f (ζ) f (λ) } k (ζ, λ) ≥ 0. (9) C This is the required matrix-valued kernel equivalence of the multiplier bound appearing in the Toeplitz Corona Theorem.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 83 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23 (ii) implies (i)
Since k is an irreducible complete Nevanlinna-Pick kernel, we can find a Hilbert space K and a map b :Ω → K with b (λ0) = 0 and such that 1 k (ζ, λ) = . (10) 1 − hb (ζ) , b (λ)iK
From (3) we now obtain (5):
K (ζ, λ) ≡ {hf (ζ) , f (λ)i N − } k (ζ, λ) ≥ 0. C
By a kernel-valued version of the Lax-Milgram Theorem, we can factor the
left hand side K (ζ, λ) as hG (ζ) , G (λ)iE where G :Ω → E for some auxiliary space E. Indeed, define F :Ω → E by F (ζ) = Kζ so that
K (ζ, λ) = hKλ, Kζ iE = hF (λ) , F (ζ)iE .
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 84 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23, Continued (ii) implies (i)
Now fix an orthonormal basis {eα}α for K and define a conjugate linear operator Γ by ! X X Γ cαeα = cαeα. α α Then G = Γ ◦ F satisfies
K (ζ, λ) = hF (λ) , F (ζ)iE = hΓ ◦ F (ζ) , Γ ◦ F (λ)iE = hG (ζ) , G (λ)iE .
Hence
hf (ζ) , f (λ)i N − = [1 − hb (ζ) , b (λ)i ] hG (ζ) , G (λ)i , C K E or equivalently,
hf (ζ) , f (λ)i N + hb (ζ) , b (λ)i hG (ζ) , G (λ)i = + hG (ζ) , G (λ)i . C K E E (11)
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 85 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23, Continued (ii) implies (i)
Define the space N1 and N2 by: ( ! ) f (λ) N = Span u : u ∈ , λ ∈ Ω ⊂ N ⊕ (K ⊗ E) , 1 b (λ) ⊗ G (λ) C C ( √ ! ) N = Span u : u ∈ , λ ∈ Ω ⊂ ⊕ E. 2 G (λ) C C
We rewrite (11) in terms of inner products of direct sums of Hilbert spaces,
hf (ζ) , f (λ)i N + hb (ζ) ⊗ G (ζ) , b (λ) ⊗ G (λ)i √C √ K⊗E = , + hG (ζ) , G (λ)i , C E
Then we can interpret this as saying that the map from N1 to N2 is an isometry!
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 86 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23, Continued (ii) implies (i)
Using (ii) we have obtained (11) that defines a linear isometry V 0 from the linear span N1 of the elements f (λ) u ⊕ (b (λ) ⊗ G (λ)) u onto the subspace N2. 0 N Extend this isometry V to an isometry V from all of C ⊕ (K ⊗ E) onto C ⊕ E, where we add an infinite-dimensional summand to E if necessary. Decompose the extended isometry V as a block matrix " # " # " # AB N V = : C → C . (12) CD K ⊗ E E Since V is an onto isometry we obtain the formulas, " # " #" # A∗A + C ∗CA∗B + C ∗D A∗ C ∗ AB = (13) B∗A + D∗CB∗B + D∗D B∗ D∗ CD " # I N 0 ∗ C = V V = I N ⊕(K⊗E) = . C 0 IK⊗E
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 87 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23, Continued (ii) implies (i)
Then (12) on the subspace N1 becomes √ Af (λ) + B [b (λ) ⊗ G (λ)] = , (14) Cf (λ) + D [b (λ) ⊗ G (λ)] = G (λ) .
N Now define g :Ω → B C, C by
∗ −1 g (λ) = A + B b (λ) ⊗ I − DEb(λ) C , (15)
where Eb is the map Eb : E → K ⊗ E given by
Ebv = b ⊗ v, v ∈ E. (16)
∗ Observe that Eb (c ⊗ w) = hc, biK w, so that ∗ Eb Ec = hc, biK IE . (17)
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 88 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23, Continued (ii) implies (i)
From this we conclude that I − DEb(λ) is invertible. Indeed, (10) shows that hb, biK < 1 and (17) then implies that Eb(λ) is a strict contraction. ∗ ∗ From the equation B B + D D = IK⊗E in (13) we see that D is a
contraction, which altogether implies DEb(λ) < 1. ∗ Thus g (λ) satisfies ∗ −1 g (λ) f (λ) = Af (λ) + B b (λ) ⊗ I − DEb(λ) Cf (λ) (18) = Af (λ) + B [b (λ) ⊗ G (λ)] √ = , Which is a restatement of (part) of (i). To see the estimate in (i) holds, we must show that g (λ) is a contractive multiplier, i.e. that (9) holds: ∗ {I N − g (ζ) g (λ) } k (ζ, λ) ≥ 0. (19) C
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 89 / 131 Corona Problem for Multiplier Algebras with CNP Kernels Proof of Theorem 23, Continued (ii) implies (i)
We then compute
∗ −1 g (λ) = A + BEb(λ) I − DEb(λ) C, −1 ∗ ∗ ∗ ∗ ∗ ∗ g (ζ) = A + C I − Eb(ζ)D Eb(ζ)B .
and then using (13) we obtain
∗ I N − g (ζ) g (λ) = C −1 −1 ∗ ∗ ∗ 1 − hb (ζ) , b (λ)iK C I − Eb(ζ)D I − DEb(λ) C.
∗ This is then enough to conclude that I N − g (ζ) g (λ) ≥ 0. C
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 90 / 131 Corona Problem for Multiplier Algebras with CNP Kernels ∞ ∞ Baby Corona for H (Bn) versus Corona for H (Bn) We know that the Corona Problem always implies the Baby Corona Problem. By the Toeplitz Corona Theorem, we know that, under certain conditions on the reproducing kernel, these problems are in fact equivalent. But, what happens if we don’t have these conditions? Theorem 3 (Equivalence between Corona and Baby Corona, (Amar 2003)) N ∞ N ∞ Let {gj }j=1 ⊆ H (Bn). Then there exists {fj }j=1 ⊆ H (Bn) with
N N 1 X f (z)g (z) = 1 ∀z ∈ and sup X |g (z)|2 ≤ j j Bn j 2 j=1 z∈Bn j=1
if and only if µ µ ∗ 2 Mg (Mg ) ≥ Iµ
for all probability measures µ on ∂Bn.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 91 / 131 Corona Problem for Multiplier Algebras with CNP Kernels ∞ ∞ Baby Corona for H (Bn) versus Corona for H (Bn) This is a great theorem since it suggests how to attack the Corona ∞ Problem for H (Bn). But, the difficulty is that one must solve the Baby Corona Problem for every probability measure on ∂Bn. Instead, it is possible to reduce this to a class of probability measures for which the methods of harmonic analysis and operator theory are more amenable. Theorem 4 (Trent, BDW (2008)) H H∗ 2 Assume that Mg Mg ≥ Iw for all w ∈ W. Then there exists a ∞ f1,..., fN ∈ H (Bn), so that
N N 1 X f (z)g (z) = 1 ∀z ∈ and sup X |f (z)|2 ≤ . j j Bn j 2 j=1 z∈Bn j=1
∞ This reduces the H (Bn) Corona Problem to a certain “weighted” Baby Corona Problem. B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 92 / 131 The Corona Problem in Several Variables
The Corona Problem in Several Variables
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 93 / 131 The Corona Problem in Several Variables Besov-Sobolev Spaces on the Unit Ball
σ The space B2 (Bn) is the collection of holomorphic functions f on the unit ball Bn such that
1 (m−1 ) 2 2 Z m+σ 2 X (k) 2 (m) f (0) + 1 − |z| f (z) dλn (z) < ∞,
k=0 Bn
2−n−1 where dλn (z) = 1 − |z| dV (z) is the invariant measure on n Bn and m + σ > 2 . Various choices of σ give important examples of classical function spaces: σ = 0: Corresponds to the Dirichlet Space; 1 σ = 2 : Drury-Arveson Hardy Space; n σ = 2 : Classical Hardy Space; n σ > 2 : Bergman Spaces.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 94 / 131 The Corona Problem in Several Variables Besov-Sobolev Spaces
σ The spaces B2 (Bn) are examples of reproducing kernel Hilbert spaces. σ Namely, for each point λ ∈ Bn there exists a function kλ ∈ B2 (Bn) such that f (λ) = hf , k i σ λ B2
It isn’t too difficult to compute (or show) that the kernel function kλ is given by 1 k (z) = λ 2σ 1 − λz
σ = 1 : Drury-Arveson Hardy Space; k (z) = 1 2 λ 1−λz σ = n : Classical Hardy Space; k (z) = 1 2 λ (1−λz)n σ = n+1 : Bergman Space; k (z) = 1 2 λ (1−λz)n+1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 95 / 131 The Corona Problem in Several Variables
Multiplier Algebras of Besov-Sobolev Spaces M σ ( ) B2 Bn
σ We are interested in the multiplier algebras, M σ ( ), for B ( ).A B2 Bn 2 Bn function ϕ belongs to M σ ( ) if B2 Bn σ kϕf k σ ≤ C kf k σ ∀f ∈ B2 ( n) B2 (Bn) B2 (Bn) B kϕk = inf{C : above inequality holds}. MBσ ( n) 2 B ∞ σ It is easy to see that M σ ( ) = H ( ) ∩ X ( ). B2 Bn Bn 2 Bn σ Where X2 (Bn) is the collection of functions ϕ such that for all σ f ∈ B2 (Bn): Z m+σ 2 2 2 (m) 2 |f (z)| 1 − |z| ϕ (z) dλn (z) ≤ C kf k σ , (‡) B2 (Bn) Bn
with kϕk σ = inf{C :(‡) holds}. X2 (Bn) Thus, we have
kϕk ≈ kϕk ∞ + kϕk σ . MBσ ( n) H ( n) X ( n) 2 B B 2 B
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 96 / 131 The Corona Problem in Several Variables
The Corona Problem for M σ ( ) B2 Bn
We wish to study a generalization of Carleson’s Corona Theorem to higher dimensions and additional function spaces. Question 24 (Corona Problem) N 2 Given f ,..., f ∈ M σ ( ) satisfying 0 < δ ≤ P |f (z)| ≤ 1 for all 1 N B2 Bn j=1 j z ∈ Bn. Does there exist a constant Cn,σ,N,δ and functions g ,..., g ∈ M σ ( ) satisfying 1 N B2 Bn
N X kgj k ≤ Cn,σ,N,δ MBσ (Bn) j=1 2 N X gj (z) fj (z) = 1, z ∈ Bn? j=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 97 / 131 The Corona Problem in Several Variables The Baby Corona Problem
It is easy to see that the Corona Problem for M σ ( ) implies a “simpler” B2 Bn question that one can consider. Question 25 (Baby Corona Problem) N 2 Given f ,..., f ∈ M σ ( ) satisfying 0 < δ ≤ P |f (z)| ≤ 1 for all 1 N B2 Bn j=1 j σ z ∈ Bn and h ∈ B2 (Bn). Does there exist a constant Cn,σ,N,δ and σ functions k1,..., kN ∈ B2 (Bn) satisfying
N X 2 2 kkj k σ ≤ Cn,σ,N,δ khkBσ( ) , B2 (Bn) 2 Bn j=1 N X kj (z) fj (z) = h (z) , z ∈ Bn? j=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 98 / 131 The Corona Problem in Several Variables σ Baby Corona Theorem for Bp (Bn)
Theorem 26 (S¸. Costea, E. Sawyer, BDW)
σ Let 0 ≤ σ and 1 < p < ∞. Given f1,..., fN ∈ MBp (Bn) satisfying
N X 2 0 < δ ≤ |fj (z)| ≤ 1, z ∈ Bn, j=1
σ there is a constant Cn,σ,N,p,δ such that for each h ∈ Bp (Bn) there are σ k1,..., kN ∈ Bp (Bn) satisfying
N X p p kkj k σ ≤ Cn,σ,N,p,δ khk σ , Bp (Bn) Bp (Bn) j=1 N X gj (z) kj (z) = h (z) , z ∈ Bn. j=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 99 / 131 The Corona Problem in Several Variables
The Corona Theorem for M σ ( ) B2 Bn
Corollary 27 (S¸. Costea, E. Sawyer, BDW) 1 Let 0 ≤ σ ≤ . Given g ,..., g ∈ M σ ( ) satisfying 2 1 N B2 Bn
N X 2 0 < δ ≤ |gj (z)| ≤ 1, z ∈ Bn, j=1
there is a constant C and there are functions f ,..., f ∈ M σ ( ) n,σ,N,δ 1 N B2 Bn satisfying
N X kfj k ≤ Cn,σ,N,δ MBσ (Bn) j=1 2 N X gj (z) fj (z) = 1, z ∈ Bn. j=1
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 100 / 131 The Corona Problem in Several Variables
The Corona Theorem for M σ ( ) B2 Bn
The proof of this Corollary follows from the main Theorem very easily. 1 σ When 0 ≤ σ ≤ 2 the spaces B2 (Bn) are reproducing kernel Hilbert spaces with a complete Nevanlinna-Pick kernel. By the Toeplitz Corona Theorem, we then have that the Baby Corona Problem is equivalent to the full Corona Problem. The result then follows. An additional corollary of the above result is the following: Corollary 28 1 For 0 ≤ σ ≤ 2 , the unit ball Bn is dense in the maximal ideal space of M σ ( ). B2 Bn
This is because the density of the the unit ball Bn in the maximal ideal space of M σ ( ) is equivalent to the Corona Theorem above. B2 Bn
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 101 / 131 Sketch of Proofs Sketch of Proof of the Baby Corona Theorem
σ Given g1,..., gN ∈ MBp (Bn) satisfying N X 2 0 < δ ≤ |gj (z)| ≤ 1, z ∈ Bn, j=1
Set ϕ (z) = gj (z) h(z). We have that PN g (z)ϕ (z) = h(z). j P |g (z)|2 j=1 j j j j This solution is smooth and satisfies the correct estimates, but is far from analytic. In order to have an analytic solution, we will need to solve a sequence of ∂-equations: For η a ∂-closed (0, q) form, we want to solve the equation ∂ψ = η for ψ a (0, q − 1) form. To accomplish this, we will use the Koszul complex. This gives an algorithmic way of solving the ∂-equations for each (0, q) with 1 ≤ q ≤ n after starting with a (0, n) form.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 102 / 131 Sketch of Proofs Sketch of Proof of the Baby Corona Theorem
This produces a correction to the initial guess of ϕj , call it ξj , and set fj = ϕj − ξj . By the Koszul complex we will have that each fj is in fact analytic. P Algebraic properties of the Koszul complex give that j fj gj = h. However, now the estimates that we seek are in doubt. To guarantee the estimates, we have to look closer at the solution operator to the ∂-equation on ∂-closed (0, q) forms. Following the work of Øvrelid and Charpentier, one can compute that the solution operator is an integral operator that that takes (0, q) forms to (0, q − 1) forms with integral kernel:
q−1 (1 − wz)n−q 1 − |w|2 (w − z ) ∀ 1 ≤ q ≤ n. 4 (w, z)n j j Here 4 (w, z) = |1 − wz|2 − 1 − |w|2 1 − |z|2 .
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 103 / 131 Sketch of Proofs Sketch of Proof of the Baby Corona Theorem
One then needs to show that these solution operators map the p Besov-Sobolev spaces Bσ(Bn) to themselves. This is accomplished by a couple of key facts: The Besov-Sobolev spaces are very “flexible” in terms of the norm that one can use. One need only take the parameter m sufficiently high. We show that these operators are very well behaved on “real variable” p versions of the space Bσ(Bn). These, of course, contain the space that we are interested in. p To show that the solution operators are bounded on L (Bn; dV ) the original proof uses the Schur Test. To handle the boundedness on p Bσ(Bn), we can also use the Schur test but this requires a little more work to handle the derivative. Finally, key to this approach, properties of the kernel and the unit ball are exploited to achieve the desired estimates. We now explain some (but not all!) of the ingredients behind this sketch.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 104 / 131 Sketch of Proofs The Koszul Complex
Define ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ = − i , = + i ∂zk 2 ∂xk ∂yk ∂zk 2 ∂xk ∂yk
where zk = xk + iyk and dzk = dxk + idyk , dzk = dxk − idyk . Let
n ∂f ∂f = X dz . ∂z k k=1 k
Given a (0, 1)-form η (z) = η1 (z) dz1 + ··· + ηn (z) dzn in the ball Bn, the ∂-equation for η is
∂f = η in the ball Bn. (20)
P I J More generally, we can let η = |I|=p,|J|=q+1 ηI,J (z) dz ∧ dz be a (p, q + 1)-form in the ball and ask for a (p, q)-form f to satisfy (20).
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 105 / 131 Sketch of Proofs The Koszul Complex
N 2 PN 2 If f = (fj )j=1 satisfies |f | = j=1 |fj | ≥ 1, let
!N N 1 f fj 1 Ω0 = = = Ω0 (j) , 2 2 j=1 |f | |f | j=1
N which we view as a 1-tensor (in C ) of (0, 0)-forms with components Ω1 (j) = fj . 0 |f |2 1 Then g = Ω0h satisfies f · g = h, but in general fails to be analytic. The Koszul complex provides a scheme when f , h are holomorphic for solving a sequence of ∂ equations that result in a correction term 2 Λf Γ0 that when subtracted from f above yields an analytic solution to f · g = h.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 106 / 131 Sketch of Proofs Lifting of Forms
N N The 1-tensor of (0, 1)-forms ∂Ω = ∂ fj = ∂Ω1 (j) is 0 |f |2 0 j=1 j=1 given by N 1 fj 1 X ∂Ω0 (j) = ∂ 2 = 4 fk {fk ∂fj − ∂fk fj }. |f | |f | k=1 A key fact is that this 1-tensor of (0, 1)-forms can be written as
" N #N 1 2 X 2 ∂Ω0 = Λf Ω1 ≡ Ω1 (j, k) fk , k=1 j=1 2 where the 2-tensor Ω1 of (0, 1)-forms is given by " #N N 2 h 2 i {fk ∂fj − ∂fk fj } Ω1 = Ω1 (j, k) = . j,k=1 4 |f | j,k=1 1 2 2 The form ∂Ω0 has been factored as Λf Ω1 where Ω1 is antisymmetric. B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 107 / 131 Sketch of Proofs Solving the complex ...
We can repeat this process and by induction we have
q+1 q+2 ∂Ωq = Λf Ωq+1, 0 ≤ q ≤ n,
q+1 where Ωq is an alternating (q + 1)-tensor of (0, q)-forms. n+1 Recall that h is holomorphic. When q = n we have that Ωn h is ∂-closed since every (0, n)-form is ∂-closed. This allows us to begin solving a chain of ∂ equations
q q q+1 ∂Γq−2 = Ωq−1h − Λf Γq−1,
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 108 / 131 Sketch of Proofs ...using that the forms are closed
n+1 Since Ωn h is ∂-closed and alternating, there is an alternating n+1 (n + 1)-tensor Γn−1 of (0, n − 1)-forms satisfying
n+1 n+1 ∂Γn−1 = Ωn h.
n n+1 Now note that the n-tensor Ωn−1h − Λf Γn−1 of (0, n − 1)-forms is ∂-closed:
n n+1 n n+1 n+1 n+1 ∂ Ωn−1h − Λf Γn−1 = ∂Ωn−1h−∂Λf Γn−1 = Λf Ωn h−Λf Ωn h = 0.
n Thus there is an alternating n-tensor Γn−2 of (0, n − 2)-forms satisfying n n n+1 ∂Γn−2 = Ωn−1h − Λf Γn−1.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 109 / 131 Sketch of Proofs The Bezout equation
n+2 With the convention that Γn ≡ 0, induction shows that there are q+2 alternating (q + 2)-tensors Γq of (0, q)-forms for 0 ≤ q ≤ n satisfying
q+1 q+2 ∂ Ωq h − Λf Γq = 0, 0 ≤ q ≤ n, (21) q+1 q+1 q+2 ∂Γq−1 = Ωq h − Λf Γq , 1 ≤ q ≤ n. Now 1 2 g ≡ Ω0h − Λf Γ0 2 is holomorphic by the first line in (21) with q = 0, and since Γ0 is 2 2 antisymmetric, we compute that Λf Γ0 · f = Γ0 (f , f ) = 0 and 1 2 f · g = Ω0h · g − Λf Γ0 · f = h − 0 = h.
Thus g = (g1, g2,..., gN ) is an N-vector of holomorphic functions satisfying f · g = h.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 110 / 131 Sketch of Proofs Charpentier’s solution kernels
n n We begin with some notation. Denote by 4 : C × C → [0, ∞) the map: 4 (w, z) = |1 − wz|2 − 1 − |w|2 1 − |z|2 (22) = 1 − |z|2 |w − z|2 + |z(w − z)|2 = 1 − |w|2 |w − z|2 + |w(w − z)|2 2 2 = |1 − wz| |ϕw (z)| 2 2 = |1 − wz| |ϕz (w)|
Here (1 − |z|2)(1 − |w|2) |ϕz (w)| = 1 − . |1 − wz|2
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 111 / 131 Sketch of Proofs The Cauchy-Leray Form
The Cauchy-Leray form
1 n µ(ξ, w, z) ≡ X(−1)i−1ξ [∧ dξ ] ∧n d(w − z ), (ξ(w − z))n i j6=i j i=1 i i i=1
n n n is a closed form on C × C × C . One then lifts the form µ via a section n n n n n s : C × C → C to give a closed form on C × C : 1 n s∗µ (w, z) ≡ X(−1)i−1s (w, z)[∧ ds ]∧n d (w − z ) . (s (w, z)(w − z))n i j6=i j i=1 i i i=1 Fix s to be the following section used by Charpentier:
s(w, z) ≡ w(1 − wz) − z(1 − |w|2).
We compute that s(w, z)(w − z) = 4 (w, z) by (22).
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 112 / 131 Sketch of Proofs Charpentier’s Forms
∗ Define the Cauchy Kernel on Bn × Bn by Cn (w, z) ≡ s µ(w, z) where s is Charpentier’s section.
Definition 29 p,q For 0 ≤ p ≤ n and 0 ≤ q ≤ n − 1 we let Cn be the component of Cn (w, z) that has bidegree (p, q) in z and bidegree (n − p, n − q − 1) in w.
p,q Thus if η is a (p, q + 1)-form in w, then Cn ∧ η is a (p, q)-form in z and Vn a multiple of the volume form in w. Let ωn (z) = j=1 dzj . For n a q positive integer and 0 ≤ q ≤ n − 1 let Pn denote the collection of all permutations ν on {1,..., n} that map to {iν, Jν, Lν} where Jν is an increasing multi-index with card(Jν) = n − q − 1 and card(Lν) = q. Let ν ≡ sgn (ν) ∈ {−1, 1} denote the signature of the permutation ν.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 113 / 131 Sketch of Proofs Explicit Formulas for Charpentier Kernels
Theorem 30 (Charpentier)
Let n be a positive integer and suppose that 0 ≤ q ≤ n − 1. Then
0,q X q q Cn (w, z) = (−1) Φn (w, z) sgn (ν)(wiν − ziν ) (23) q ν∈Pn ^ ^ ^ × dwj dzl ωn (w) . j∈Jν l∈Lν
n−1−q 2 q q (1−wz) (1−|w| ) where Φn (w, z) ≡ 4(w,z)n for 0 ≤ q ≤ n − 1.
0,q We can rewrite the formula for Cn (w, z) in (23) as
0,q q X X µ(k,J) J (J∪{k})c Cn (w, z) = Φn (w, z) (−1) (zk − wk ) dz ∧dw ∧ωn (w) , |J|=q k∈/J
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 114 / 131 Sketch of Proofs Ameliorated Kernels
We now wish to define right inverses with improved behaviour at the boundary. We consider the case when the right side f of the ∂ equation is a (p, q + 1)-form in Bn. As usual for a positive integer s > n we will ”project” the formula p,q p,q ∂Cs f = f in Bs for a ∂-closed form f in Bs to a formula ∂Cn,s f = f in Bn for a ∂-closed form f in Bn. p,q To accomplish this we define ameliorated operators Cn,s by p,q p,q Cn,s = RnCs Es ,
where for n < s,Es (Rn) is the extension (restriction) operator that takes P I J forms Ω = ηI,J dw ∧ dw in Bn (Bs ) and extends (restricts) them to Bs (Bn) by X I J X I J Es ηI,J dw ∧ dw ≡ (ηI,J ◦ R) dw ∧ dw , X I J X I J Rn ηI,J dw ∧ dw ≡ (ηI,J ◦ E) dw ∧ dw . I,J⊂{1,2,...,n}
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 115 / 131 Sketch of Proofs Ameliorated Kernels, Continued
s n Here R is the natural orthogonal projection from C to C and E is the n s natural embedding of C into C . In other words, we extend a form by taking the coefficients to be constant in the extra variables, and we restrict a form by discarding all wedge products of differentials involving the extra variables and restricting the coefficients accordingly. p,q For s > n we observe that the operator Cn,s has integral kernel Z p,q p,q 0 0 C (w, z) ≡ √ C w, w , (z, 0) dV w , z, w ∈ n, n,s 2 s B 1−|w| Bs−n
s−n where Bs−n denotes the unit ball in C with respect to the orthogonal s n s−n decomposition C = C ⊕ C , and dV denotes Lebesgue measure.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 116 / 131 Sketch of Proofs Ameliorated Kernels, Continued
0 If f (w) is a ∂-closed form on Bn then f (w, w ) = f (w) is a ∂-closed form on Bs and we have for z ∈ Bn, Z p,q 0 0 f (z) = f (z, 0) = ∂ Cs w, w , (z, 0) f (w) dV (w) dV w Bs Z (Z ) = ∂ √ Cp,q w, w 0 , (z, 0) dV w 0 f (w) dV (w) 2 s Bn 1−|w| Bs−n Z p,q = ∂ Cn,s (w, z) f (w) dV (w) . Bn We have proved the following: Theorem 31
For all s > n and ∂-closed forms f in Bn, we have
p,q ∂Cn,s f = f in Bn.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 117 / 131 Sketch of Proofs Ameliorated Kernels, Continued
Theorem 32 0,q Suppose that s > n and 0 ≤ q ≤ n − 1. Then we have Cn,s (w, z) is given by
j 2 !s−n n−q−1 1 − |w|2 1 − |z|2 0,q 1 − |w| X C (w, z) cj,n,s n 1 − wz 2 j=0 |1 − wz|
Note that the numerator and denominator are balanced in the sense that the sum of the exponents in the denominator minus the corresponding sum in the numerator (counting 4 (w, z) double) is s + n + j − (s + j − 1) = n + 1, the exponent of the invariant measure of the ball Bn.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 118 / 131 Sketch of Proofs Invariant Derivatives
Define ∂ ∂ ∂ ∂ ∇z = , ..., and ∇z = , ..., ∂z1 ∂zn ∂z1 ∂zn Recall that the gradient with invariant length given by
0 0 0 ∇e f (a) = (f ◦ ϕa) (0) = f (a) ϕa (0) 1 0 2 2 2 = −f (a) 1 − |a| Pa + 1 − |a| Qa .
We want an analogue of this operator on the tree Tn. We define for z ∈ Bn, 0 0 Daf (z) = f (z) ϕa (0) 1 0 2 2 2 = −f (z) 1 − |a| Pa + 1 − |a| Qa .
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 119 / 131 Sketch of Proofs A Tree Seminorm
Lemma 33
Let a, b ∈ Bn satisfy β (a, b) ≤ C. There is a positive constant Cm depending only on C and m such that
−1 m m m Cm |Db f (z)| ≤ |Da f (z)| ≤ Cm |Db f (z)| ,
for all f ∈ Hol (Bn).
Definition 34 Suppose σ ≥ 0, 1 < p < ∞ and m ≥ 1. We define a “tree semi-norm” ∗ k·k σ by Bp,m(Bn)
1 p Z σ p ∗ X 2 m kf k σ = 1 − |z| D f (z) dλn (z) . Bp,m(Bn) cα Bd (cα,C2) α∈Tn B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 120 / 131 Sketch of Proofs Pointwise Multipliers
Lemma 35 We have m−1 ∗ X j kf k σ + ∇ f (0) ≈ kf k σ . Bp,m(Bn) Bp,m(Bn) j=0 ∞ σ n Let ϕ ∈ H (Bn) ∩ Bp (Bn). If m > p − σ and 0 ≤ σ < ∞, then ϕ is a σ pointwise multiplier on Bp (Bn) if and only if
m+σ p 2 m 1 − |z| ∇ ϕ (z) dλn (z) (24)
σ n is a Bp (Bn)-Carleson measure on Bn. If m > 2 p − σ and n 0 ≤ σ < p + 1, then (24) can be replaced by σ p 2 m 1 − |z| D ϕ (z) dλn (z) .
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 121 / 131 Sketch of Proofs The Real Variable Besov Space
Definition 36 We denote by X m the vector of all differential operators of the form X1X2 ··· Xm where each Xi is either the identity operator I, the operator 2 D, or the operator 1 − |z| R. We calculate the products X1X2 ··· Xm by 2 composing Da and 1 − |a| R and then setting a = z at the end. Note 2 that Da and 1 − |a| R commute since the first is an antiholomorphic derivative and the coefficient z in R = z · ∇ is holomorphic.
Definition 37
We define the norm k·k σ for f smooth on the ball n by Λp,m(Bn) B
1 Z σ p p 2 m kf k σ ≡ 1 − |z| X f (z) dλn (z) . Λp,m(Bn) Bn
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 122 / 131 Sketch of Proofs Estimates for the Differentiation Operators
Lemma 38
m p !m α ∂ 4 (w, z) m (z − w) F (w) ≤ C D F (w) m = |α| ∂w α 1 − |w|2
n 2 1 o |Dz 4 (w, z)| ≤ C 1 − |z| 4 (w, z) 2 + 4 (w, z) q 2 2 1 − |z| R 4 (w, z) ≤ C 1 − |z| 4 (w, z)
! m n o 1 − |z|2 2 Dm (1 − wz)k ≤ C |1 − wz|k z |1 − wz| !m m n o 1 − |z|2 1 − |z|2 Rm (1 − wz)k ≤ C |1 − wz|k |1 − wz| B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 123 / 131 Sketch of Proofs The Main Estimates
σ From the Koszul complex we must show that g ∈ Bp (Bn) where
1 2 g = Ω0h − Λf Γ0 = Ω1h − Λ C0,0 Ω2h − Λ Γ3 0 f n,s1 1 f 1 = Ω1h − Λ C0,0 Ω2h + Λ C0,0 Λ Γ3 0 f n,s1 1 f n,s1 f 1 = Ω1h − Λ C0,0 Ω2h + Λ C0,0 Λ C0,1 Ω3h − Λ Γ4 0 f n,s1 1 f n,s1 f n,s2 2 f 2 . . = Ω1h − Λ C0,0 Ω2h + Λ C0,0 Λ C0,1 Ω3h − Λ C0,0 Λ C0,1 Λ C0,2 Ω4h − · · · 0 f n,s1 1 f n,s1 f n,s2 2 f n,s1 f n,s2 f n,s3 3 + (−1)n Λ C0,0 ··· Λ C0,n−1Ωn+1h f n,s1 f n,sn n ≡ F 0 + F 1 + ··· + F n n = X F µ µ=0
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 124 / 131 Sketch of Proofs The Main Estimates
The goal is to establish
kgk σ ≤ Cδ,n,N,p khk σ , Bp (Bn) Λp (Bn) which we accomplish by showing that
µ kF k σ ≤ C khk σ , 0 ≤ µ ≤ n, B ( n) δ,n,N,p Λ ( n) p,m1 B p,mµ B
for a choice of integers mµ satisfying n − σ < m < m < ··· < m < ··· < m . p 1 2 ` n
Recall that we defined both of the norms kF k σ and kF k σ Bp,mµ (Bn) Λp,mµ (Bn) for smooth functions F in the ball Bn.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 125 / 131 Sketch of Proofs The Main Estimates
The norms k·k σ will now be used to estimate the composition of Λp,m(Bn) Charpentier solution operators in each function
F µ = Λ C0,0 ··· Λ C0,µ−1Ωµ+1h f n,s1 f n,sµ µ as follows. We will use the facts that
Z σ p p 2 m khk σ ≈ 1 − |z| X h (z) dλn (z) , Bp (Bn) Bn σ p p p 2 m kgk ≈ kgk + 1 − |z| X g (z) dλn (z) , MBσ( ) ∞ σ p Bn Bp (Bn)−Carleson
n n for 0 ≤ σ < p + 1 and m > 2 p − σ .
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 126 / 131 Sketch of Proofs Schur’s Lemma on Besov-Sobolev Spaces
Lemma 39
Let a, b, c, t ∈ R. Then the operator
2a 2b p c Z 1 − |z| 1 − |w| 4 (w, z) Ta,b,c f (z) = n+1+a+b+c f (w) dV (w) Bn |1 − wz|
t p 2 is bounded on L Bn; 1 − |w| dV (w) if and only if c > −2n and
−pa < t + 1 < p (b + 1) .
We will use this Lemma for appropriate choices of a, b, c. This, plus some more computations, shows that the it is possible to obtain estimates to ∂ σ in the space Λp,m (Bn).
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 127 / 131 Sketch of Proofs Obtaining the Estimates
We will accomplish this by showing
µ kF k σ ≤ Cn,σ,p,δ khk σ , 0 ≤ µ ≤ n. Bp (Bn) Bp (Bn)
Then we can then estimate by
µ µ+1 kF k σ ≤ T1 ◦ T2 ◦ · · · ◦ TµΩ h . Bp (Bn) µ σ Bp (Bn)
Here Tµ = Taµ,bµ,cµ for appropriate aµ, bµ, cµ. So, we have
µ µ+1 kF k σ ≤ Cn,σ,p,δ Ω h . Bp (Bn) µ σ Bp (Bn) µ+1 Finally, show that for Ωµ arising from the Koszul complex, then
µ+1 Ω h ≤ Cn,σ,p,δ khk σ . µ σ Bp (Bn) Bp (Bn)
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 128 / 131 Sketch of Proofs Multilinear Estimates and Embeddings
To control this last term we use the following lemma: Lemma 40
n Suppose that 1 < p < ∞, 0 ≤ σ < ∞,M ≥ 1, m > 2 p − σ and M+1 α = (α , ..., α ) ∈ with |α| = m. For g , ..., g ∈ M σ σ 0 M Z+ 1 M Bp (Bn)→Bp (Bn) σ and h ∈ Bp (Bn) we have
Z pσ 2 α1 p αM p α0 p 1 − |z| |(X g1)(z)| · · · |(X gM )(z)| |(X h)(z)| dλn (z) Bn M p ≤ C Y M khkp . n,M,σ,p gj σ σ Bσ( ) B ( n)→B ( n) p Bn j=1 p B p B
µ+1 This follows then since the term Ωµ h is essentially what appears in the left hand side above.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 129 / 131 Sketch of Proofs Open Problems and Future Directions
∞ 1 Does the algebra H (Bn) of bounded analytic functions on the ball have a Corona in its maximal ideal space? 2 Does the Corona Theorem for the multiplier algebra of the 1 2 n Drury-Arveson space B2 (Bn) extend to more general domains in C ? 3 Can we prove a Corona Theorem for any algebra in higher dimensions that is not the multiplier algebra of a Hilbert space with the complete 1 n Nevanlinna-Pick property? Any 2 < σ ≤ 2 would be extremely interesting. 4 Can one prove the equivalence between a “weakened” version of the 1 n Baby Corona Problem and the Corona Problem when 2 < σ < 2 ? This would be useful to approach the above problem.
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 130 / 131 Conclusion
Thank You!
B. D. Wick (Georgia Tech) The Corona Problem MSRI 2011 131 / 131