ON THE TOËPLITZ CORONA PROBLEM

The aim of this note is to characterize the vectors g = (g1,...,gk) of bounded holomorphic functions in the unit ball or in the unit polydisk of Cn such that the Corona is true for them in terms of the H2 Corona for measures on the boundary.

Let D be a bounded domain in C n , the Corona problem is: given functions g 1 , . . ., g N holomorphic and bounded in D such that: find f 1 , . . ., f N still holomorphic and bounded in D such that in D. This was solved for D = D, the unit disk in C by L. Carleson [8] and it is still open for n > 1 for the basic domains namely the unit ball B n and the unit polydisk D n .We shall link this question to a question on Toëplitz operators via the H p (µ) Corona.

Notations
We are interested by the basic domains, the unit ball in C n , D = B n , in fact any bounded convex domain with smooth boundary D, or the unit polydisc D = D n .
If D = D n we set bD = T n , the distinguished boundary; if D is a bounded convex domain with smooth boundary, bD = ∂D the topological boundary.
Recall that: Let M be the set of all probability measures on bD and for µ ∈ M and 1 ≤ p < ∞ let H p (µ) be the closure in L p (µ) of the holomorphic polynomials.
If µ ∈ M and f ∈ H ∞ (D) then, with the assumption that 0 ∈ D, for any r < 1, f r (z There is a subsequence of {f r , r < 1} which converges in (L 1 (µ), L ∞ (µ)) topology and uniformly on compact sets of Now suppose that the Corona problem is solvable, i.e.
we have, for any polynomial P : Let h ∈ H p (µ).Then there is a sequence {P k } k∈N of polynomials such that P k → h in H p (µ), hence: So if the Corona is true then the H p (µ) Corona is also true for any µ ∈ M: The aim of this paper is to show the converse. and we already know that: if D = B n , µ the Lebesgue's measure on ∂B n and 2 ≤ p < ∞, then CH p (µ) is true [2]; if D = D n , µ the Lebesgue's measure on T n and 1 ≤ p < ∞, then CH p (µ) is true [10], [11]; if D is strictly pseudo-convex, µ the Lebesgue's measure on ∂D and 2 ≤ p < ∞, then CH p (µ) is true [5]; if D is a bounded pseudo-convex domain with smooth boundary, µ the Lebesgue's measure on ∂D, then CH 2 (µ) is true [4].In the case n = 1, D = D the unit disc in C, µ the Lebesgue's measure on T, then CH 2 (µ) ⇒ CH ∞ (D) [12], by an operator method: the commutant lifting theorem of Nagy-Foias.
This means that the Corona theorem in one variable can be proved this way, hence there is some hope to prove a general version of the Corona theorem also by this way.

Proof of the theorem
We already seen that i) ⇒ ii); to prove that ii) ⇒ i) we shall use the minimax theorem of Von Neuman.The minimax theorem was already used by Berndtsson [6], [7] in order to get estimates on solutions of the ∂-equation; here the situation and the method are quite different.
We shall work with N = 2 in order to simplify notations.Because D is always convex containing 0, we may assume by dilation that the data g := (g 1 , g 2 ) are continuous up to the boundary, provided that the estimates do not depend on it.
Let Ω be an open set in D such that Ω ⊂ D, 0 ∈ Ω and let, for > 0, C be: this set is clearly convex in A(D) 2 .Let M be the set of probability measures on bD and for 0 < η ≤ 1 let M η = ηm + (1 − η)M, where m is the Lebesgue measure on bD; this is a convex weakly compact set.

Let us define N as
Then N is convex on C for µ fixed in M η and concave, in fact affine, and continuous on M for f fixed in C , hence we can apply the minimax theorem [9]: 2 ; by the very definition of H 2 (µ) there is a sequence which means that f ∈ C .We deduce that the left side of ( * ) is bounded by 1 δ 2 hence for any > 0, η > 0, γ > 0 there is a f ,η,γ ∈ C with sup µ∈Mη N (f ,η,γ , µ) ≤ 1 δ 2 + γ.Now let a ∈ D and ν a a representing measure for a supported by bD, then we have with µ because this is true for any a ∈ D we get Using Montel property we get that there is a f ∈ (H ∞ (D)) 2 bounded by 1 δ and such that g

Operator version
We shall give an operator version of the previous result strongly inspired by [3], but first we need some definitions.Let D be as before and µ ∈ M; for any function f in L ∞ (µ) define the Toëplitz operator T µ f on the Hilbert space H 2 (µ) by , where P µ is the orthogonal projection from L 2 (µ) on H 2 (µ).We can state: Corollary 3.1.Let D be a bounded convex domain containing 0 and with a smooth boundary or the unit polydisk D n of C n , n ≥ 1. Let: g 1 , . . ., g N ∈ H ∞ (D) and δ > 0. The following are equivalent: (ii) For all measures µ on bD, For D = D 2 , this was proved in [1]; they used a method specific to the bidisc which explicitly cannot work even for D 3 .
Proof: We shall prove that (ii) is equivalent to: (iii) For all measures µ on bD, and then we apply the theorem to be done.
(ii) ⇒ (iii) (same proof as in [1]): Let µ be a probability measure on bD and set G i := T µ gi ; by (ii) we get that the operator We can define: these are bounded operators on H 2 (µ) and clearly we get: Together with equation (1) this means precisely that the H 2 (µ) Corona is true, i.e.
(iii) ⇒ (ii): Let µ ∈ M, then by (iii) we have: and it has elements of norm less than 1 δ 2 h 2 2 ; S 0 is a subspace of the Hilbert space (H 2 (µ)) N hence there is a unique element k = (k 1 , . . ., k N ) in S h which is orthogonal to S 0 and hence of minimal norm.Then we get: k (2) From equation (3) we get: Using equation (2) we get: and the corollary.