arXiv subjects
Publications and source records attributed to Giulia Sarfatti.
We do a preliminary study of the reproducing kernel Hilbert space having as kernel $k^d$, where $d$ is a positive integer and $k$ is the reproducing kernel of the analytic Dirichlet space.
The theory of slice regular functions of a quaternionic variable, introduced in 2006 by Gentili and Struppa, extends the notion of holomorphic function to the quaternionic setting. This fast growing theory is already rich of many results and has interesting applications. In this setting, the present paper is devoted to introduce and study the quaternionic counterparts of Hardy spaces of holomorphic functions of one complex variable. The basic properties of the theory of quaternionic Hardy spaces are investigated, and in particular a Poisson-type representation formula, the notions of outer function, singular function and inner function are given. A quaternionic (partial) counterpart of the classical $H^p$-factorization theorem is proved. This last result assumes a particularly interesting formulation for a large subclass of slice regular functions, where it is obtained in terms of an outer function, a singular function and a quaternionic Blaschke product.
We study the $L^p$ norm of the orthogonal projection from the space of quaternion valued $L^2$ functions to the closed subspace of slice $L^2$ 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-Carathéodory Theorem to the new setting.
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 $\mathbb{H}$ 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 $\partial_C f(0)=1$). 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 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 $2$-diameter. We then show that the Landau-Toeplitz inequalities hold in the case of the regular $n$-diameter, for all $n\geq 2$. Finally, a $3$-diameter version of the Landau-Toeplitz Theorem is proved using the notion of slice $3$-diameter.
Riemannian metrics on the unit ball of the quaternions, which are naturally associated with the reproducing kernel Hilbert spaces. We study the metric arising from the Hardy space in detail. We show that, in contrast to the one-complex variable case, no Riemannian metric is invariant under regular self-maps of the quaternionic ball.