1. Statement of result

Let H^{∞}(𝔻) be the Banach algebra of bounded analytic functions of the disk, with the usual L^{∞} norm. We define the spectrum σ(H^{∞}(𝔻)) of H^{∞}(𝔻) to be the space of all Banach algebra homomorphisms λ : H^{∞}(𝔻) → ℂ from H^{∞}(𝔻) to ℂ, thus λ is complexlinear, preserves products, and maps 1 to 1. We remark that such homomorphisms are automatically contractions:

|λ(f)| ≤ ∥f∥_{H^{∞}(𝔻)}.

This is an easy consequence of the observation that f−z is invertible in H^{∞}(𝔻) whenever |z| > ∥f∥_{H^{∞}(𝔻)}. In particular, λ maps B to \overline{𝔻}, where B is the unit ball in H^{∞}(𝔻). Thus σ(H^{∞}(𝔻)) can be viewed as a closed and bounded subset of \overline{𝔻}^{B}, using the product topology. By Tychonoff’s theorem we thus see that σ(H^{∞}(𝔻)) is compact.

Every point z ∈ 𝔻 induces a homomorphism λ_{z} : H^{∞}(𝔻) → ℂ defined by λ_{z}(f) := f(z). It is easy to see that this map z ↦ λ_{z} is continuous and injective, and so we can view 𝔻 as being a subset of H^{∞}(𝔻). Note that it is not all of H^{∞}(𝔻), since this would imply that 𝔻 is compact, which is absurd.

In these notes we prove

corona-qual Theorem 1.1 (Corona theorem, qualitative version). 𝔻 is dense in σ(H^{∞}(𝔻)).

This theorem was first proven by Carleson ^{\underline{carleson}} [1]; somewhat later Wolff ^{\underline{wolff}} [3] gave a shorter proof, which is the one we present here. The main differences are that we give things more of a real analysis slant than a complex analysis one, for instance foregoing the factorisation theory of H^{p}. We also use Uchiyama’s ^{\underline{uchiyama}} [4] constructive decomposition of BMO to avoid use of the Hahn-Banach theorem (and thus give a fully constructive proof).

2. First reduction - quantitative formulation

We first reduce the above qualitative assertion to its quantitative counterpart.

corona-quant Theorem 2.1 (Corona theorem, quantitative version). Let f_{1}, …, f_{k} ∈ H^{∞}(𝔻) be such that inf_{z∈𝔻} ∑_{j=1}^{k} |f_{j}(z)|^{2} > 0. Then there exists g_{1}, …, g_{k} ∈ H^{∞}(𝔻) such that f_{1}g_{1} + … + f_{k}g_{k} = 1.

Let us show the equivalence of the two formulations of the Corona theorem.

Proof of Theorem ^{\underline{corona-qual}} 1.1 assuming Theorem ^{\underline{corona-quant}} 2.1. Suppose for contradiction that 𝔻 was not dense in σ(H^{∞}(𝔻)). Then, by the definition of product topology, there exists λ ∈ σ(H^{∞}(𝔻)), f_{1}, …, f_{k} ∈ B, and ε > 0 such that

sup_{1≤j≤k} |λ(f_{j}) − λ_{z}(f_{j})| ≥ ε

for all z ∈ 𝔻. Writing f̃_{j} := f_{j} − λ(f_{j}), we thus have λ(f_{j}) = 0 and

sup_{1≤j≤k} |λ_{z}(f̃_{j})| ≥ ε

for all z ∈ 𝔻, and hence

inf_{z∈𝔻} ∑_{j=1}^{k} |f̃_{j}(z)|^{2} > 0.

We apply Theorem ^{\underline{corona-quant}} 2.1 and obtain g_{1}, …, g_{k} such that f_{1}g_{1} + … + f_{k}g_{k} = 1. Applying λ to both sides we conclude 0 = 1, which is absurd. □

Proof of Theorem ^{\underline{corona-quant}} 2.1 assuming Theorem ^{\underline{corona-qual}} 1.1. Suppose for contradiction that we could find f_{1}, …, f_{k} ∈ H^{∞}(𝔻) such that inf_{z∈𝔻} ∑_{j=1}^{k} |f_{j}(z)|^{2} > 0, but such that f_{1}g_{1} + … + f_{k}g_{k} ≠ 1 for all g_{1}, …, g_{k} ∈ H^{∞}(𝔻). In other words, the ideal of H^{∞}(𝔻) generated by f_{1}, …, f_{k} avoids 1. It must then also avoid the open ball of radius 1 around 1 in H^{∞}, since all such elements are invertible. In particular, the closure of this ideal still avoids 1. A standard application of Zorn’s lemma^{1} thus shows that f_{1}, …, f_{k} are contained in a maximal ideal which avoids 1, which then easily induces a homomorphism λ ∈ H^{∞}(𝔻) which annihilates f_{1}, …, f_{k}. However by hypothesis we see that sup_{1≤j≤k} |λ(f_{j})−λ_{z}(f_{j})| is uniformly bounded away from zero for all z ∈ 𝔻, thus λ lies in the exterior of 𝔻, a contradiction. □

3. Second reduction: \overline{∂} problem

Now we perform some algebraic manipulations to reduce things to a \overline{∂} problem. Fix f_{1}, …, f_{k} as in Theorem ^{\underline{corona-quant}} 2.1 and extending smoothly to \overline{𝔻}. Recall that we have the differential operators

∂ := \frac{1}{2}(∂_{x} − i∂_{y}); \overline{∂} := \frac{1}{2}(∂_{x} + i∂_{y}).

Recall that a function f : 𝔻 → ℂ lies in H^{∞}(𝔻) if it is C^{1} on 𝔻, obeys the Cauchy-Riemann equation \overline{∂}f = 0, and lim sup_{t→1^{−}} ∥f∥_{L^{∞}(𝕋_{t})} < ∞, where where 𝕋_{t} is the circle of radius t. Thus our objective is to find C^{1}(𝔻) functions g_{1}, …, g_{k} solving the equations and bounds

∑_{j=1}^{k} f_{j}g_{j} = 0  (3.1) hgo
\overline{∂}g_{j} = 0 for all 1 ≤ j ≤ k  (3.2) hnum
lim sup_{t→1^{−}} ∥g_{j}∥_{L^{∞}(𝕋_{t})} < ∞ for all 1 ≤ j ≤ k.  (3.3) ho

Now from the hypothesis inf_{z∈𝔻} ∑_{j=1}^{k} |f_{j}(z)|^{2} > 0 and the smoothness of the f_{j}, we can find a function ϕ ∈ C^{∞}(𝔻) such that

lim sup_{t→1^{−}} ∥ϕ∥_{L^{∞}(𝕋_{t})} < ∞

and

ϕ ∑_{j} f_{j}\overline{f_{j}} = 1.  (3.4) phij

This gives us a solution to (^{\underline{hgo}} 3.1) and (^{\underline{ho}} 3.3) but not to (^{\underline{hnum}} 3.2):

∑_{j=1}^{k} f_{j}(\overline{f_{j}}ϕ) = 1.

1This is of course non-constructive, but this direction of the equivalence is not needed to prove the Corona theorem.

This suggests using the ansatz

g_{j} := \overline{f_{j}}ϕ + ∑_{l=1}^{k} ω_{jl}f_{l}

for some antisymmetric functions ω_{jl}, since this will automatically satisfy (^{\underline{hgo}} 3.1). To satisfy (^{\underline{ho}} 3.3) it will suffice to have

lim sup_{t→1^{−}} ∥ω_{jl}∥_{L^{∞}(𝕋_{t})} < ∞

for all j, l.

We now recast (^{\underline{hnum}} 3.2) in terms of the ω_{jl}. Since \overline{∂}f_{l} = 0, we arrive at

∑_{l=1}^{k}(\overline{∂}ω_{jl})f_{l} = −\overline{∂}(\overline{f_{j}}ϕ)

or in vector notation

\overline{∂}Ωf = −\overline{(}\overline{f}ϕ)

where Ω is the matrix of the ω_{jl} and f is the column vector of the f_{j}. From (^{\underline{phij}} 3.4) we have

ϕ\overline{f}^{T} f = 1  (3.5) phif

and so one possible candidate for \overline{∂}Ω is −\overline{(}\overline{f}ϕ)ϕf^{T}. But this is not anti-symmetric. On the other hand, if we apply \overline{∂} to (^{\underline{phif}} 3.5) we obtain

\overline{∂}(ϕ\overline{f})^{T} f = 0

and so

fϕ\overline{∂}(ϕ\overline{f})^{T} f = 0.

In particular we see that an anti-symmetric candidate for \overline{∂}Ω is

fϕ\overline{∂}(ϕ\overline{f})^{T} − \overline{(}\overline{f}ϕ)ϕf^{T} .

Thus it will suffice to solve the problem

Ω = −Ω^{T}
\overline{∂}Ω = fϕ\overline{∂}(ϕ\overline{f})^{T} − \overline{(}\overline{f}ϕ)ϕf^{T}
lim sup_{t→1^{−}} ∥Ω∥_{L^{∞}(𝕋_{t})} < ∞.

Actually it will suffice to solve the de-symmetrised problem

\overline{∂}Ω = fϕ\overline{∂}(ϕ\overline{f})^{T}
lim sup_{t→1^{−}} ∥Ω∥_{L^{∞}(𝕋_{t})} < ∞

since the matrix-valued function Ω − Ω^{T} will then solve the previous problem.

4. Third reduction: Carleson measures

The claim will now follow immediately from the following two lemmas and the principle of superposition.

Lemma 4.1 (Carleson measure estimates). Let F := fϕ\overline{∂}(ϕ\overline{f})^{T}. Then

sup_{I} \frac{1}{|I|} ∫_{Q(I)} |F (z)|^{2} log \frac{1}{|z|} dxdy < ∞  (4.1) fuzz

and

sup_{I} \frac{1}{|I|} ∫_{Q(I)} |∂F (z)| log \frac{1}{|z|} dxdy < ∞  (4.2) fuzz-2

where the supremum is over all arcs I, and where Q(I) is the sector {z ∈ 𝔻\{0} : z/|z| ∈ I; |z| ≥ 1 − |I|}.

solvo Lemma 4.2 (Wolff criterion for \overline{∂}-solvability). Let F : \overline{𝔻} → ℂ be a C^{1} function such that

∫_{Q(I)} |F (z)|^{2} log \frac{1}{|z|} dxdy ≤ |I|

and

∫_{Q(I)} |∂F (z)| log \frac{1}{|z|} dxdy ≤ |I|

for all arcs I. Then there exists a function u ∈ C^{1}(𝔻) such that \overline{∂}u = F and lim sup_{t→1^{−}} ∥u∥_{L^{∞}(𝕋_{t})} ≤ C for some absolute constant C.

5. Proof of Carleson measure estimate

We first prove (^{\underline{fuzz}} 4.1). Since f and ϕ are already bounded, it suffices to show that

sup_{I} \frac{1}{|I|} ∫_{Q(I)} |\overline{∂}(ϕ\overline{f})|^{2} log \frac{1}{|z|} dxdy < ∞.

From the chain rule and the bounds on ϕ, f we quickly see that

|\overline{∂}(ϕ\overline{f})| ≤ C_{ϕ,f}|∂f|

for some finite constant C_{ϕ,f} depending on ϕ, f, so we end up wishing to show

sup_{I} \frac{1}{|I|} ∫_{Q(I)} |∂f|^{2} log \frac{1}{|z|} dxdy < ∞.

But the left-hand side is essentially the square of the BMO(𝕋) norm of the holomorphic vector-valued function f, which is bounded by the L^{∞} norm of f, and the claim follows.

Now we prove (^{\underline{fuzz-2}} 4.2). Applying the Leibnitz and chain rules, and recalling that ∂ annihilates f and \overline{∂} annihilates \overline{f}, one eventually obtains the estimate

|∂F | ≤ C′_{ϕ,f}|∂f|^{2}

and the claim follows as before.

6. Proof of solvability criterion

We now establish solvability. Observe that given any solution to the equation \overline{∂}u = F, one can obtain a new solution by adding or subtracting an H^{∞}(𝔻) function. Now recall the Fefferman-Stein theorem ^{\underline{fefferman-stein}} [2] that any BMO(𝕋) function can be split into the sum of an L^{∞}(𝕋) function and the Hilbert transform of an L^{∞}(𝕋) function, with the norms of both factors comparable to the BMO(𝕋) norm of the original. (This theorem was later given a constructive proof by Uchiyama.) Because of this, we see that instead of requiring u|_{𝕋_{t}} to be uniformly bounded in L^{∞}, it will suffice for u|_{𝕋_{t}} to be uniformly bounded in BMO(𝕋_{t}). In fact, we can go further: it suffices for the Riesz projection P_{−}u|_{𝕋_{t}} to functions with anti-holomorphic boundary values to be in BMO(𝕋_{t}) for 1/2 < t < 1. Indeed, if this were the case, one can easily use the Fefferman-Stein theorem obtain for each 1/2 < t < 1 a decomposition u = u_{t} + v_{t} on the disk 𝔻_{t} := {|z| < t} where u_{t} was uniformly bounded in L^{∞}(𝕋_{t′}) for all t′ < t and v_{t} was in H^{∞}(𝔻_{t}). Then we have \overline{∂}u_{t} = F on 𝔻_{t}; taking t → 1 and using Montel’s theorem on normal families to extract a locally uniform convergent subsequence we obtain the claim.

Note that the Riesz projection P_{−}u|_{𝕋_{t}} is in fact independent of the particular choice of u. Thus, let us select an arbitrary smooth solution to \overline{∂}u = F on 𝔻_{t} (e.g. by restricting to the disk of radius (1 + t)/2, where everything is smooth, and using the Bargmann transform). By Fefferman-Stein duality ^{\underline{fefferman-stein}} [2] , we have

∥P_{−}u|_{𝕋_{t}}∥_{BMO(𝕋_{t})} ≤ C sup{| ∫_{𝕋_{t}}(P_{−}u)h| : ∥h∥_{H^{1}(𝕋_{t})} = 1}

where the integration is respect to arc-length measure. Using the P_{−} (whose adjoint P_{+} is bounded on H^{1}) we may assume h is in fact in complex H^{1}, so that it extends to a function on ℍ^{1}(𝔻_{t}). Then we can drop the P_{−}. From Green’s formula we have

| ∫_{𝕋_{t}} uh| ∼ | ∫_{𝔻_{t}} Δ(uh) log \frac{t}{|z|} dxdy|.

We expand

Δ(uh) = ∂\overline{∂}(uh) = ∂(F h) = (∂F)h + F (∂h).

If we decompose h into H^{1} atoms and use the Carleson measure hypotheses on F, we quickly obtain the desired bounds.

Remark 6.1. An inspection of the proof shows that one can in fact obtain a quantitative version of the corona theorem: the H^{∞}(𝔻) norms of the g_{j} in Theorem ^{\underline{corona-quant}} 2.1 are in fact bounded by a polynomial combination of the H^{∞}(𝔻) norms of the f_{j}, and of the quantity inf_{z∈𝔻} ∑_{j=1}^{k} |f_{j}(z)|^{2} > 0.

References

carleson [1] L. Carleson, Interpolations by bounded analytic functions and the Corona problem, Ann. of Math. (2) 76 (1962), 547–559. 148 (1982), 215–241.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555, USA.

  E-mail address: tao@math.ucla.edu

fefferman-stein [2] C. Fefferman, E. M. Stein, H^{p} spaces of several variables, Acta Math. 129 (1972), 137–193

wolff [3] T. W. Gamelin, Wolff’s proof of the Corona Theorem, Israel J. Math., 37 (1980), 113-119.

uchiyama [4] A. Uchiyama, A constructive proof of the Fefferman-Stein decomposition of BMO(ℝ^{n}), Acta Math.