THE CORONA THEOREM FOR THE MULTIPLIER ALGEBRAS ON WEIGHTED DIRICHLET SPACES by BERHANU TEKLE KIDANE A DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of The University of Alabama TUSCALOOSA, ALABAMA 2009 Copyright Berhanu Kidane 2009 ALL RIGHTS RESERVED ABSTRACT In this dissertation we give a proof of “The Corona Theorem for Infinitely Many Functions for the Multiplier Algebras on Weighted Dirichlet Spaces”, and we obtain explicit estimates on the size of the solution. We denote the open unit disc of the complex plane by D, and for α ∈ (0, 1) we denote by Dα the Weighted Dirichlet Spaces of all holomorphic functions on D, and by kfk 0r kfk the weighted Dirichlet norm. Dα α To prove the Dα Corona Theorem and find bounds for the solution we use a similar technique employed by Trent [Tr 2] in his proof of the corona theorem for Dirichlet space. The first challenge is to find a norm that is expressed on the boundary ∂D of D. In this dissertation we first establish such a norm. The new norm basically enables us to extend multipliers on the weighted Dirichlet spaces to multipliers on a related weighted harmonic Dirichlet spaces. Finally using Schur’s Theorem and Cauchy’s Transform we are able to prove the Dα corona theorem and give bounds for the solution. ii ACKNOWLEDGMENTS I am especially grateful to my family who have helped me in more ways than I can name. From my family, a very special thanks goes to my brother Mesfin. It would have been a very difficult task to begin and complete my studies without my brother’s constant support and encouragement. Above all I would like to give my heartfelt thanks to my advisor Dr. Trent. This research would be an impossible task without my advisor’s support and professional guidance. A very special thanks goes to the dissertation committee members for their thoughtful comments and valuable suggestions. Finally, my gratitude goes to staff, graduate students, and faculty members of the Department of Mathematics of The University of Alabama, for all the support I received through the course of the years. iii CONTENTS ABSTRACT ......................................................................i ACKNOWLEDGMENTS . ii 1. INTRODUCTION . 1 1.1. Historical Background . 1 2. REPRODUCING KERENEL HILBERT SPACES . 5 2.1. Definitions and Properties . 5 2.2. Examples . 8 2.3. Characterization of Reproducing Kernels . .11 3. MULTIPLIER ALGEBRAS . 13 4. WEIGHTED DIRICHLET SPACES . .15 4.1. Algebra of Operators On Dα ............................................16 5. THE CORONA THEOREM . 20 5.1. Carleson’s Corona Theorem . .20 5.2. H∞(D) Corona Theorem . .22 5.3. The Dirichlet Corona Theorem . 24 iv 5.4. Norm Estimates . 25 5.5. The Weighted Corona Theorem . 44 5.6. The Cauchy Transform . 56 5.7. Estimates for Kb ........................................................69 6. FUTURE RESEARCH . .90 REFERENCES . 91 v 1. INTRODUCTION 1.1. Historical Background In Mathematics the word Theorem is a major statement that is proved; we can say the word Theorem is synonymous to the word “Main Result”. Theorems tend to be given names, The Fundamental Theorem of Calculus, Fermat’s Little Theorem, The Corona Theorem and etc. In contrast, a Conjecture is a statement that has not been proven. A conjecture is not the same as a theorem. Famous conjectures tend to be given names too: The Poincare hypothesis, The Riemann hypothesis etc. When we come to the Corona problem, it was the Japanese Mathematician Kakutani who first stated the conjecture in the beginning of the 1940’s. The word, “Corona ” denotes the set of maximal ideals in H∞(D) which are not in the W ∗ (weak star) closure of the maximal ideals in the unit disk D. Carleson proved that the corona is empty. Lennart Carleson has made many fundamental contributions to Harmonic Analysis, Complex Analysis, dynamical system, and other areas of mathematics. Standing out among them is his 1962 solution of the famous Corona Problem. The corona theorem for finitely many functions is referred as the Carlson Corona Theorem or the H∞(D) Corona Theorem. Carleson’s proof of the corona theorem is very much illustrative (for example; in his proof Carleson introduced what is now 1 know as Carleson’s measure, a fundamental tool in complex and harmonic analysis), but at the same time highly complicated. The need to come up with a simpler proof and immediate generalization of the Carleson’s Corona Theorem (say for infinitely many functions) are perhaps the first steps to take followed by a multitude set of corona problems on different spaces. Hormander introduced the ∂-technique for the corona problem and Wolff simplified Carleson’s proof of the corona theorem using the ∂-technique. Rosenblum [R] and Tolokonnikov [T] independently proved the Carleson’s Corona Theorem for the infinite version. Fuhrman and Vasyunin proved the Matrix version of the H∞(D) Corona Theorem. The Hp (1 ≤ p ≤ ∞) corona Theorem for a finite number of bounded analytic functions on polydisks was established independently by Li [Li] and Lin [L], the infinite version was done by Trent [Tr3] The Corona theorem on Dirichlet space for finite set of functions is done by Tolokonnikov; while the corresponding theorem for infinitely many functions from the multiplier algebra of the Dirichlet space is due to Trent[Tr2]. In this thesis, we extend Trent’s result on Dirichlet space to weighted Dirichlet spaces for the weight α ∈ (0, 1). Our proof is done in a similar fashion to that of what Trent did for the Dirichlet space. 2 The main problem is, we don’t know how to prove the weighted corona theorem and give the estimates based on known norms. That is, based on norms that come directly from the definition of the weighted Dirichlet spaces. Let’s say a little more about this. By definition the norm for f ∈ Dα is given by ∞ P α 2 1/2 kfkα = ( (n + 1) |an| ) n=0 For α = 1, D is the Dirichlet space, and ∞ 2 P 2 kfkD = (n + 1) |an| (1.1.1) n=0 Or equivalently the norm can be given by 2 2 R 0 2 kfk = kfk 2 + |f (z) | dA(z) (1.1.2) D H (D) D or π π Z Z it iθ 2 2 2 f(e ) − f(e ) kfk = kfk 2 + dσ(t)dσ(θ) (1.1.3) D H (D) |eit − eiθ|2 −π −π In the proof of the Dirichlet Corona Theorem (D Corona Theorem) for Dirichlet space, T. Trent used the norm in (1.1.3). Directly proving the D Corona Theorem using the norms (1.1.1) and (1.1.2) is still an open problem. Since the technique employed by Trent [Tr2] can easily be extended to the weighted case, our major task is to find a norm for the weighted Dirichlet spaces that is similar to the norm given in (1.1.3). 3 In this thesis, for the weight α ∈ (0, 1), we succeed in finding such a norm, which is most crucial for our work. The next important thing to do is to give bounds for the solution; our approach for this case is different from what is given in [Tr2]. We use Schur’s theorem and the Cauchy transform and some other techniques to get the estimates. 4 2. REPRODUCING KERNEL HILBERT SPACES In this section we explain the idea of a reproducing kernel Hilbert space in general and on the unit disk. We state some facts and give some standard examples of reproducing kernel Hilbert spaces. Good references for this sections are N. K. Nikolski [Ni], J. Agler and J.E. McCarthy [AM], and V. I. Paulsen [P]. 2.1. Definitions and Properties Let R and C denotes the set of real numbers and complex numbers respectively. We will use F to denote either R or C. Given a non - empty set E, F(E, F) denotes the set of all functions from E to F. F(E, F) with the usual operations of addition and scalar multiplication is a vector space over F. Definition 2.1.1. Given a set E we will say that H ≡ H(E) is a reproducing kernel Hilbert space (RKHS) on E over F if (i) H is a subspace of F(E, F) (ii) H is endowed with an inner product h , i which makes H into a Hilbert space 5 (iii) for every x ∈ E, the linear evaluational functional Lx : H → F defined by Lx(f) = f(x), is bounded. If H is a RKHS on E, then by Riesz Representation Theorem every bounded linear functional is given by the inner product with a unique vector in H; thus for every y ∈ E, there exists a unique vector, ky ∈ H, such that for every f ∈ H, f(y) = hf, kyi. Definition 2.1.2. The function ky is called the Reproducing Kernel for the point y. The 2-variable function K(x, y), defined by K(x, y) = ky(x) is called the Reproducing Kernel for H. Note that K : E × E → F, and so K(., y) is in H. The following properties are immediate consequences of the definition of RKHS. Property 2.1.3. The reproducing kernel of a Hilbert space is unique. Property 2.1.4. Suppose H is a RKHS on E with reproducing kernel K(x, y) then (i) K(x, y) = ky(x) = hky, kxi (ii) K(x, x) ≥ 0 (iii) K(x, y) = K(y, x) (iv) | K(x, y) |2 ≤ K(x, x)K(y, y). ∞ Suppose H is a RKHS on E with reproducing kernel K(x, y).