TWO OPEN PROBLEMS
BRETT D. WICK
Abstract. Here are two open problems related to the lectures given at the CIMPA Summer
School on “Real and Complex Analysis with Applications to Other Sciences”.
Problem 1: Stable Rank for Multiplier Algebras
Let D denote the Dirichlet space, namely the set of functions f that are holomorphic and
such that
Z
2
2
kf kD = |f (0)| + |f 0 (z)|2 dA(z) < ∞.
D
Let MD denote the collections of pointwise multipliers of the Dirichlet space. These are the
collection of functions ϕ such that ϕf ∈ D for all f ∈ D. We place a norm on this space in
the following way,
kϕkMD = inf {C : kϕf kD ≤ Ckf kD
∀f ∈ D} .
It is easy to see that MD = H ∞ ∩ X , where X is the collection of functions that generate
Carleson measures for the Dirichlet space D. One places a norm on the space X in the
following way
Z
2 0
2
2
2
2
2
2
|f (z)| |ϕ (z)| dA(z) ≤ C kf kD ∀f ∈ D .
kϕkX = |ϕ(0)| + inf C :
D
One easily sees that MD is an algebra. It is well known that the Corona problem for the
algebra MD holds. This can be found in the works of Tolokonnikov, Xiao, and Trent, [4, 5].
Each of these authors gave a different proof of this fact. The question we are interested in
is the following:
Question 0.1. Suppose that f1 , f2 ∈ MD are such that
0 < δ < |f1 (z)| + |f2 (z)| ≤ 1
∀z ∈ D.
There do there exist g1 , g2 , g1−1 ∈ MD such that kg1 kMD + kg2 kMD + kg1−1 kMD ≤ C(δ) and
1 = f1 (z)g1 (z) + f2 (z)g2 (z)
∀z ∈ D?
This problem is asking if the Bass stable rank of MD is one. It is well known that the
Bass stable rank of H ∞ is one. See the work of Treil, [3].
Key words and phrases. Stable Rank, Besov Sobolev Spaces, Corona Problem, Bilinear Forms, Carleson
Measures.
Research supported in part by a National Science Foundation DMS grant.
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BRETT D. WICK
Problem 2: Blinear forms on Spaces of Analytic Functions
Recall that a Hankel operator on the Hardy space H 2 is given by Hϕ (f ) = P− (ϕf ). A
restatement of Nehari’s Theorem is the boundedness of the following bilinear form
Sb (f, g) = hf g, biH 2 ,
namely, kSb k ≈ k|b0 (z)|2 (1−|z|2 )dA(z)kCM (H 2 ) . Where CM (H 2 ) is the collection of Carleson
measures for H 2 , and the norm of the Carleson measure (either given by the embedding or
by the equivalent geometric condition).
Note that D is a Hilbert space with inner product given by
Z
hf, giD = f (0)g(0) + f 0 (z)g 0 (z)dA(z).
D
Define an operator Tb : D × D → C by
Z
Tb (f, g) = hf g, biD = f (0)g(0)b(0) +
(f 0 (z)g(z) + f (z)g 0 (z)) b0 (z)dA(z).
D
It is known that kTb kD×D→C ≈ kbkX . See the paper [1]. Now, for 0 ≤ σ ≤
Z
2
2
kf kBσ2 = |f (0)| + |f 0 (z)|2 (1 − |z|2 )2σ dA(z) < ∞.
1
2
define
D
= D and B 21 = H 2 (with equivalence of norms). Define the
Then for σ = 0 we have that
2
space Xσ to be
Z
2
2
2
2 0
2
2 2σ
2
2
kϕkX = |ϕ(0)| + inf C :
|f (z)| |ϕ (z)| (1 − |z| ) dA(z) ≤ C kf kBσ2 ∀f ∈ D .
B02
D
Then we have X 1 = BM O and X0 = X from the problem above. Based on what happens for
2
the case of the Dirichlet space and for the Hardy space, the following problem is proposed:
Question 0.2. Define
Z
Hb (f, g) = f (0)g(0)b(0) +
(f 0 (z)g(z) + f (z)g 0 (z)) b0 (z)(1 − |z|2 )2σ dA(z).
D
Show that kHb kBσ2 ×Bσ2 →C ≈ kbkXσ .
Note that one of these inequalities is very easy to prove, the challenge is in the other direction. This other direction will require one to thoroughly understand the Carleson measures
for the space Bσ2 . See the paper by Stegenga [2] to start understanding these objects.
References
[1] Nicola Arcozzi, Richard Rochberg, Eric Sawyer, and Brett D. Wick, Bilinear forms on the Dirichlet space,
Anal. PDE 3 (2010), no. 1, 21–47. ↑2
[2] David A. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), no. 1, 113–139. ↑2
[3] S. Treil, The stable rank of the algebra H ∞ equals 1, J. Funct. Anal. 109 (1992), no. 1, 130–154. ↑1
[4] Tavan T. Trent, A corona theorem for multipliers on Dirichlet space, Integral Equations Operator Theory
49 (2004), no. 1, 123–139. ↑1
[5] Jie Xiao, The ∂-problem for multipliers of the Sobolev space, Manuscripta Math. 97 (1998), no. 2, 217–232.
↑1
TWO OPEN PROBLEMS
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Brett D. Wick, School of Mathematics, Georgia Institute of Technology, 686 Cherry
Street, Atlanta, GA USA 30332-0160
E-mail address: [email protected]