Corona Problem Research Articles

A Vector-valued Hp Corona Theorem on the Polydisk

For the corona problem on the bidisk, we find analytic solutions belonging to the Orlicz-type space \(\exp {\left( {L^{{\frac{1}{3}}} } \right)}.\) In addition, for 1 ≤ p < ∞, an \( \mathcal{H}^{p} {\left( {D^{2} } \right)}\) corona theorem is established. Similar techniques can be used for the polydisk.

The Corona Problem in Carleman Algebras on Stein Manifolds

Hörmander L 2 estimates for the \(\bar\partial\)-operator on Stein Manifolds are applied to the solution of the Corona problem in Carleman Algebras on Stein Manifolds.

The matrix-valued [formula omitted] corona problem in the disk and polydisk

In this paper, we consider the matrix-valued H p corona problem in the disk and polydisk. The result for the disk is rather well-known, and is usually obtained from the classical Carleson Corona Theorem by linear algebra. Our proof provides a streamlined way of obtaining this result and allows one to get a better estimate on the norm of the solution. In particular, we were able to improve the estimate found in the recent work of Trent in [J. Funct. Anal. 189 (2002) 267–282]. Note that, the solution of the H ∞ matrix corona problem in the disk can be easily obtained from the H 2 corona problem either by factorization, or by the Commutant Lifting Theorem. The H p corona problem in the polydisk was originally solved by Lin in [Bull. Sci. Math. 110(2) (1986) 69–84, Trans. Amer. Math. Soc. 341 (1994) 371–375]. The solution used Koszul complexes and was rather complicated because one had to consider higher order ∂ ¯ -equations. Our proof is more transparent and it improves upon Lin's result in several ways. First, we deal with the more general matrix corona problem. Second, we were able to show that the norm of the solution is independent of the number of generators. Additionally, we illustrate that the norm of the solution of the H 2 corona problem in the polydisk D n grows at most proportionally to n . Our approach is based on one that was originated by Andersson in [Math. Z. 201 (1989) 121–130]. In the disk it essentially depends on Green's Theorem and duality to obtain the estimate. In the polydisk we use Riesz projections to reduce the problem to the disk case.

On a generalized corona problem on the unit disc

Let g , f 1 , … , f n ∈ H ∞ g,f_ 1,\dots ,f_ n\in H^{\infty } . We give a sufficient condition on the size of a function g g in order for it to be in the ideal generated by f 1 , … , f n f_ 1,\dots ,f_ n . In particular, this improves Cegrell’s result on this problem.

Sobolev space estimates for solutions of the equation∂¯u=f on polycylinders

In trying to improve Weinstock′s results on approximation by holomorphic functions on certain product domains, we are led to estimates in Sobolev spaces for the ‐operator on polycylinders for (γ, q)‐forms. This generalizes our results for the same operator on polycylinders previously obtained, and can be applied to a number of other problems such as the Corona problem.

An operator corona theorem

In this paper some new positive results in the Operator Corona Problem are obtained in a rather general situation. The main result is that, under some additional assumptions about a bounded analytic operator-valued function F in the unit disc D, the condition F* (z)F(z) ≥ δ 2 I ∀ z ∈ D (δ > 0) implies that F has a bounded analytic left inverse. Typical additional assumptions are (any of the following): (1) The trace norms of defects I - F*(z)F(z) are uniformly (in z ∈ D) bounded. The identity operator I can be replaced by an arbitrary bounded operator here, and F* F can be changed to FF*; (2) The function F can be represented as F = F 0 + F 1 , where F 0 is a bounded analytic operator-valued function with a bounded analytic left in verse, and the Hilbert-Schmidt norms of operators F 1 (z) are uniformly (in z E D) bounded. It is now well-known that without any additional assumption, the condition F*F > δ 2 I is not sufficient for the existence of a bounded analytic left inverse. Another important result of the paper is the so-called Tolokonnikov's Lemma, which says that a bounded analytic operator-valued function has an analytic left inverse if and only if it can be represented as a part of an invertible bounded analytic function. This result was known for operator-valued functions such that the operators F(z) act from a finite-dimensional space, but the general case is new.

Diffraction by systems of finite strips

The talk is devoted to wave diffraction problems for the two-dimensional Helmholtz equation. Boundary-transmission problems for a union of intervals (i.e., strips of a negligible thickness) which lie on a system of one or several parallel lines are considered in a Sobolev space setting. These problems include the mixed Dirichlet/Neumann and impedance boundary and transmission problems. Such problems for a union of intervals are reduced to equivalent systems of Wiener–Hopf equations on Lebesgue spaces over the positive real semiaxis. The matrix symbols of corresponding Wiener–Hopf operators are semialmost periodic, that is, they are continuous on the real line and have, in general, different almost periodic behavior at semineighborhoods of infinity. The solvability of obtained systems of Wiener–Hopf equations is studied on the basis of factorization techniques for oscillating and sectorial matrix functions and a corona problem approach developed in M. A. Bastos, Yu. I. Karlovich, and A. F. dos Santos [Integr. Eq. Oper. Th. 42, 22–56 (2002)], A. Bottcher, Yu. I. Karlovich, and I. M. Spitkovsky [Convolution Operators and Factorization of Almost Periodic Matrix Functions (Birkhauser-Verlag, Basel, 2002)], and L. P. Castro and F.-O. Speck [Integr. Equat. Oper. Th. 37, 169–207 (2000)].

Hölder Estimates for the ∂¯-Operator on Polycylinders

Hölder estimates for the ∂¯-operator are obtained on polycylinders and applied to a number of problems-cohomology with bounds, the corona problem and approximation of Hölder functions by holomorphic functions.

ON THE LOWER BOUND ESTIMATES OF SECTIONS OF THE CANONICAL BUNDLES OVER A RIEMANN SURFACE

We give a lower bound estimate of the sum of the square norm of the sections of the pluricanonical bundles over a Riemann surface of genus greater than 2 and Gauss curvature -1. Such an estimate must depend on the injective radius of the Riemann surface. However, using this estimate, we give a uniform estimate of the corona problem on Riemann surface. Here "uniform" means that the estimate depends only on the genus of Rieman surface, not on the injective radius.

Wiener-Hopf factorization for a class of oscillatory symbols

Two classes of 2×2 matrix symbols involving oscillatory functions are considered, one of which consists of triangular matrices. An equivalence theorem is obtained, concerning the solution of Riemann-Hilbert problems associated with each of them. Conditions for existence of canonical generalized factorization are established, as well as boundedness conditions for the factors. Explicit formulas are derived for the factors, showing in particular that only one of the columns needs to be calculated. The results are applied to solving a corona problem.

Estimates of solutions of the $H^p$ and BMOA corona problem

We prove new sharper estimates of solutions to the \(H^p\)-corona problem in strictly pseudoconvex domains; in particular we show that the constant is independent of the number of generators. We also obtain sharper estimates for solutions to the BMOA corona problem. The proofs also lead to new results about the Taylor spectrum of analytic Toeplitz operators on \(H^p\) and BMOA.

Vanishing theorems on manifolds with piecewise strictly pseudodonvex boundaris

Let ω be a relatively compact subdomain of a complex manifold X with boundary ∂ ω strictly pseudoconvex and let there be a C2 strictly plurisubhannonic function defined in a neighborhood of the closure of ω in X. It is shown that the cohomology groups Hq (ωDp ) vanish for q ≥ 1, p ≥ 0, where Dp is the sheaf of germs of holomorphic p-forms on ω. This settles, in the negative, the question raised by I. Lieb concerning whether the vanishing of Hq (ωDp ) for q ≥ 1, p ≥. 0 implies that ω is Stein or not. When Ω has only a piecewise strictly pseudoconvex boundary, it also shows that Hq (ωVn) vanishes for q ≥ 1. This is applied to the Gleason problem, the weak corona problem and an approximation problem.

Backward Shift Invariant Operator Ranges

Results on first order Ext groups for Hilbert modules over the disk algebra are used to study certain backward shift invariant operator ranges, namely de Branges–Rovnyak spaces and a more general class called H(W;B) spaces. Necessary and sufficient conditions are given for the groups Ext1A(D)(H2C, H(W;B)) to vanish whereH2Cis thedualof the vector-valued Hardy module, H2C. One condition involves an extension problem for the Hankel operator with symbolB,ΓB, but viewed as a module map from H2Cinto H(W;B). The group Ext1A(D)(H2C, H(W;B))=(0) precisely whenΓBextends to a module map from L2Cinto H(W;B) and this in turn is equivalent to the injectivity of H(W;B) in the category of contractive HilbertA(D)-modules. This result applied to the de Branges–Rovnyak spaces yields a connection between the extension problem for the HankelΓB and the operator corona problem.

The Hp corona theorem in analytic polyhedra

TheHp corona problem is the following: Letg1, ...,gm be bounded holomorphic functions with 0<δ≤Σ‖gi‖. Can we, for anyHp function ϕ, findHp functionsu1, ...,um such that Σgiui=ϕ? It is known that the answer is affirmative in the polydisc, and the aim of this paper is to prove that it is in non-degenerate analytic polyhedra. To prove this, we construct a solution using a certain integral representation formula. TheHp estimate for the solution is then obtained by localization and some harmonic analysis results in the polydisc.

Uniform estimates with weights for the $$\bar \partial - equation$$

This paper concernsL∞-variants of Hormanders weightedL2-estimates for the\(\bar \partial - equation\). In particular, we discuss a conjecture concerning suchL∞-estimates which is related to the corona problem in the ball, and show a weaker version of this conjecture. The proof uses a refinedL2-estimate for the canonical solution to the\(\bar \partial - equation\). An alternative approach based on von Neumann’s Minimax theorem is also given.

Explicit solutions for the corona problem with Lipschitz data in the polydisc

This paper contains considerations of various versions of the classical corona problem on domains in complex n-dimensional space. Although we do not solve the H°° corona problem, we do obtain positive results in other topologies. We also provide explicit constructions for solutions.

Factorization of functions in subspaces of L^1 and applications to the corona problem

Factorization of functions in subspaces of L^1 and applications to the corona problem

A Note on Infinitesimal Generators of Semigroups on H2/φ H2

Let φ be an inner function on the unit disc D . Let A (φ) be the commutant of the compression of the standard unilateral shift onto H 2/φ H 2. Necessary and sufficient conditions for a complex number to be in the resolvent Set of the infinitesimal generator of a C 0 semigroup of operators in A (φ) are given. In this case, the resolvents of the infinitesimal generator are obtained by solving a corona problem for unbounded functions whose real part is bounded above.

On the Corona problem

We show that if f1, f2 are bounded holomorphic functions in the unit ball\(\mathbb{B}\) of ℂn such that\(\forall z \in \mathbb{B}\), ¦f1(z)¦2 + ¦f2(z)2¦2 ≥ δ2 >; 0, then any functionh in the Hardy space\(H^p (\mathbb{B})\),p < +∞ can be decomposed ash = f1h1+ f2h2 with\(h_i \in H^p (\mathbb{B})\). The Corona theorem in\(\mathbb{B}\) would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction.

Corona problem and flows

Using a version of Ambrose's theorem, we generalize the corona theorem to the setting of bounded analytic functions induced by flows. The structure of the maximal ideal space is also investigated; in particular, an open dense subset, just like the open unit disc in the classical case, is determined.