The H p -Corona Theorem for the Polydisc

Lin, Kai-Ching

doi:10.2307/2154627

Cited by 9 publications

(9 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First, the exact constant occurs in Theorem B, so finding the optimal estimates in the corona theorem resides in the proof of Theorem A. Second, if the scalar corona problem is similarly posed over, say the bidisk, then for a finite set {f j } m j=1 of bounded analytic functions on D × D, Theorem A holds for a constant depending on m and e. See Lin [11] and Li [10]. Therefore the corona problem for the bidisk reduces to the question of whether Theorem B, an operator corona theorem, holds for H 2 (D × D).…”

Section: B(e)mentioning

confidence: 99%

A New Estimate for the Vector Valued Corona Problem

Trent

2002

Journal of Functional Analysis

View full text Add to dashboard Cite

We improve an estimate of Fuhrmann and Vasyunin in the vector valued corona theorem. © 2002 Elsevier Science (USA) Key Words: Hilbert space; vector valued corona theorem; ideals of operators; Carleson; Tolokonnikov; Vasyunin.In this paper we will give an improved estimate in the vector valued corona theorem for the unit disk. Our proof will be split into two parts: first, we will resurrect the operator corona theorem from the 1970s, and then we will use a Hilbert space argument, based on a version of Wolff's ideas. Since we use Hilbert space methods, the Riesz representation theorem replaces the usual "-techniques.In 1962 B(e, m) for all z ¥ D. This paper [5] spawned numerous investigations in many directions, but we will mention only those most closely related to our work. For the scalar case above, Hormander introduced "-methods, culminating in Wolff's surprising proof of the corona theorem. (See [8].) Rosenblum [14] and,

show abstract

Section: B(e)mentioning

confidence: 99%

A New Estimate for the Vector Valued Corona Problem

Trent

2002

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

“…Amar proved in [5] when k = 2 and when D is the ball of C n that if φ ∈ H p (D) then there exist f 1 , f 2 ∈ H p (D) such that f 1 g 1 + f 2 g 2 = φ. This result was generalized by Lin in [26] for any k in the polydisc of C n and by Andersson and Carlsson in [8][9][10] for any k and for D a general strictly pseudoconvex domain of C n .…”

Section: Introductionmentioning

confidence: 82%

A Berndtsson–Andersson operator solving $${\overline{\partial}}$$ -equation with W α -estimates on convex domains of finite type

Alexandre

2010

Math. Z.

View full text Add to dashboard Cite

The Carleson measures were first introduced by Carleson in order to solve the corona problem for the disc in C. The notion of Carleson measure can be generalized to any homogeneous space and were also used in the context of the corona problem in C n for example by Varopoulos, Amar and Andersson and Carlsson. One of the steps to solve the H p corona problem in a pseudoconvex domain is to solve the ∂ equation for a form μ satisfying a Carleson condition and get norm estimates of the solution in term of the Carleson norm of μ. The main goal of this paper is to consider this question in the case of convex domains of finite type and to get estimates linked to the multitype of the domain.

show abstract

“…An equivalent way of phrasing this theorem is that the point evaluation functionals Over the years a lot of effort was put into proving an analogue of this celebrated theorem in several variables, and some results were obtained [9,85,88,116,117,120]; see also the recent survey [55]. However, the most natural several variables analogues of Theorem 49, …”

Section: The Curvature As Indexmentioning

confidence: 99%