Giulia Sarfatti

A Bloch-Landau Theorem for Slice Regular Functions

The Bloch-Landau Theorem is one of the basic results in the geometric theory of holomorphic functions. It establishes that the image of the open unit disc \(\mathbb{D}\) under a holomorphic function f (such that f(0)=0 and f′(0)=1) always contains an open disc with radius larger than a universal constant. In this paper we prove a Bloch-Landau type Theorem for slice regular functions over the skew field ℍ of quaternions. If f is a regular function on the open unit ball \(\mathbb{B}\subset\mathbb{H}\), then for every \(w \in \mathbb{B}\) we define the regular translation \(\tilde{f}_{w}\) of f. The peculiarities of the non commutative setting lead to the following statement: there exists a universal open set contained in the image of \(\mathbb{B}\) through some regular translation \(\tilde{f}_{w}\) of any slice regular function \(f: \mathbb{B}\to\mathbb{H}\) (such that f(0)=0 and ∂ C f(0)=1). For technical reasons, we introduce a new norm on the space of regular functions on open balls centred at the origin, equivalent to the uniform norm, and we investigate its properties.

The Bohr Theorem for slice regular functions

In this paper we prove the Bohr Theorem for slice regular functions. Following the historical path that led to the proof of the classical Bohr Theorem, we also extend the Borel-Caratheodory Theorem to the new setting.

Landau–Toeplitz theorems for slice regular functions over quaternions

The theory of slice regular functions of a quaternionic variable extends the notion of holomorphic function to the quaternionic setting. This theory, already rich of results, is sometimes surprisingly different from the theory of holomorphic functions of a complex variable. However, several fundamental results in the two environments are similar, even if their proofs for the case of quaternions need new technical tools. In this paper we prove the Landau-Toeplitz Theorem for slice regular functions, in a formulation that involves an appropriate notion of regular

From Hankel Operators to Carleson Measures in a Quaternionic Variable

2

The orthogonal projection on slice functions on the quaternionic sphere

-diameter. We then show that the Landau-Toeplitz inequalities hold in the case of the regular

Quaternionic Hankel operators and approximation by slice regular functions

n

Some Hilbert spaces related with the Dirichlet space

-diameter, for all

Shift invariant subspaces of slice L^2 functions

n\geq 2

Invariant Metrics for the Quaternionic Hardy Space

. Finally, a

Quaternionic Hardy spaces

3