The Dirichlet space: A Survey

In this paper we survey many results on the Dirichlet space of analytic functions. Our focus is more on the classical Dirichlet space on the disc and not the potential generalizations to other domains or several variables. Additionally, we focus mainly on certain function theoretic properties of the Dirichlet space and omit covering the interesting connections between this space and operator theory. The results discussed in this survey show what is known about the Dirichlet space and compares it with the related results for the Hardy space.Comment: 35 pages, typoes corrected, some open problems adde

Supporting the active learning of collaborative database browsing techniques

We describe the implications of a study of database browsing behaviour for the development of a system to support more effective browsing. In particular we consider the importance of collaborative working, both in learning browsing skills and in co‐operating on a shared information‐retrieval task. From our study, we believe that an interface to support collaboration should promote the awareness of the activities of others, better visualization of the information data structures being browsed, and effective communication of the browsing process

The Corona Problem for Kernel Multiplier Algebras

We prove an alternate Toeplitz corona theorem for the algebras of pointwise kernel multipliers of Besov-Sobolev spaces on the unit ball in $\mathbb{C}^{n}$, and for the algebra of bounded analytic functions on certain strictly pseudoconvex domains and polydiscs in higher dimensions as well. This alternate Toeplitz corona theorem extends to more general Hilbert function spaces where it does not require the complete Pick property. Instead, the kernel functions $k_{x}\left(y\right)$ of certain Hilbert function spaces $\mathcal{H}$ are assumed to be invertible multipliers on $\mathcal{H}$, and then we continue a research thread begun by Agler and McCarthy in 1999, and continued by Amar in 2003, and most recently by Trent and Wick in 2009. In dimension $n=1$ we prove the corona theorem for the kernel multiplier algebras of Besov-Sobolev Banach spaces in the unit disk, extending the result for Hilbert spaces $H^\infty\cap Q_p$ by A. Nicolau and J. Xiao.Comment: v1: 34 pages. v2: 34 pages, typos corrected. v3: 35 pages, typos corrected, presentation improved. v4 35 pages, typos corrected and referee comments included. v5 35 pages, additional reference added and remark to prior related work include

The Hyt\"onen-Vuorinen L^{p} conjecture for the Hilbert transform when (4/3)<p<4 and the measures share no point masses

In the case (4/3)<p<4, and assuming a pair of locally finite positive Borel measures on the real line have no common point masses, we prove two conjectures of T. Hyt\"onen and E. Vuorinen from 2018 on two weight testing theorems for the Hilbert transform on weighted L^{p} spaces. Namely, the two weight norm inequality holds (1) if and only if the global quadratic interval testing conditions hold, (2) if and only if the local quadratic interval testing, the quadratic Muckenhoupt, and the quadratic weak boundedness conditions all hold. We also give a slight improvement of the second conjecture in this setting by replacing the quadratic Muckenhoupt conditions with two smaller conditions.Comment: 104 page