The Hilbert transform has become increasingly popular over the years due to its wide ranging applications not only in mathematics, but also in many other applied areas. In a quest for more applications, studying various aspects of its two weight forms has been a subject of high interest as early as the 1970’s. Of special interest is the interface of the Hilbert transform with the notions of Carleson measures and the system of reproducing kernels in spaces of analytic functions. Though these notions have proved to be of fundamental importance and ubiquitous in the development of function theoretic spaces, their properties for many significant spaces, including the model subspace of the Hardy spaces H2 ;have not yet been well understood. The present thesis focuses on this interface and provides answers to several problems encompassing them.
The thesis consists of five chapters. The first chapter provides an up-to-date review of the relevant background literature. The remaining chapters contain results that have been published by, or intended for, international journals.
The work in chapter two covers the problems of unitarity, invertibility, boundedness, and surjective mapping properties of the two weight discrete Hilbert transforms, and a complete solution is obtained for the first one. Our solutions for the remaining problems are complete under a sparsity priori growth condition. Under such a condition, we describe bounded two weight Hilbert transforms in terms of a relatively simple A2 conditions. As a consequence, computable geometric criteria have been established for invertibility of such maps. Chapter two also provides all the basic underpinnings for the materials presented in Chapter three and Chapter four, where links have been established to interpolate all our results on the weighted transforms into statements about Carleson measures and systems of reproducing kernels in certain Hilbert spaces, of which de Branges spaces and model subspaces of H2; are prime examples. As an application, a connection to the Feichtinger conjecture, which is known to be equivalent to dozens of other conjectures including the famous Kadison–Singer problem, is pointed out and verified for certain classes of spaces.
Chapter five deals again with normalized reproducing kernel Riesz bases in model subspaces of H2 generated by the class of meromorphic inner functions. In this chapter, the approach to studying such bases digresses somewhat from the methods used in the preceding chapters. Here, we study the normalized kernel bases from an equality of spaces perspective. It is known that such bases can be described in terms of equality of spaces whenever the kernels are associated with points all from the real line. When the points are from the upper half-plane, it is now proved that the analogous conditions may still be sufficient while failing to be necessary.