Function theory on the Neile parabola

We give a formula for the Caratheodory distance on the Neile parabola, the variety {z^2=w^3} restricted to the bidisk; thus making it the first variety with a singularity to have its Caratheodory distance explicitly computed. In addition, we relate this to a mixed Caratheodory-Pick interpolation problem for which known interpolation theorems do not apply. Finally, we prove a bounded holomorphic function extension result from the Neile parabola to the bidisk.Comment: 19 pages. Minor error corrected in theorem 4.

A Schwarz lemma on the polydisk

There is a known generalization of the classical Schwarz lemma to holomorphic functions from the polydisk to the disk. In this paper, we characterize those functions which satisfy equality everywhere in this generalized inequality: they are the transfer function of an n+1 by n+1 symmetric unitary, and in particular, are rational, inner, and belong to the Schur-Agler class of the polydisk. We also present some sufficient conditions for a function to be of this type.Comment: 12 pages. See also http://www.math.wustl.edu/~geknes

Algebraic pairs of isometries

We consider pairs of commuting isometries that are annihilated by a polynomial. We show that the polynomial must be inner toral, which is a geometric condition on its zero set. We show that cyclic pairs of commuting isometries are nearly unitarily equivalent if they are annihilated by the same minimal polynomial

Unions of Lebesgue spaces and A1 majorants

We study two questions. When does a function belong to the union of Lebesgue spaces, and when does a function have an A1 majorant? We provide a systematic study of these questions and show that they are fundamentally related. We show that the union ofLwp(ℝn)spaces withw∈Apis equal to the union of all Banach function spaces for which the Hardy–Littlewood maximal function is bounded on the space itself and its associate space

Algebraic pairs of isometries

We consider pairs of commuting isometries that are annihilated by a polynomial. We show that the polynomial must be inner toral, which is a geometric condition on its zero set. We show that cyclic pairs of commuting isometries are nearly unitarily equivalent if they are annihilated by the same minimal polynomial

Wolff's Theorem on ideals for matrices

We extend Wolff's theorem on ideals in the space H^∞ to the case involving matrices. Our work is based off the results of Andersson (1989) and Trent and Zhang (2007). At the end we show how the estimates in our theorem can be improved and how the theorem can be extended to other spaces, along with some other results. (Published By University of Alabama Libraries