Quaternionic Hankel operators and approximation by slice regular functions

In this paper we study Hankel operators in the quaternionic setting. In particular we prove that they can be exploited to measure the $L^{\infty}$ distance of a slice $L^{\infty}$ function from the space of bounded slice regular functions.Comment: 19 page

From Hankel operators to Carleson measures in a quaternionic variable

We introduce and study Hankel operators defined on the Hardy space of regular functions of a quaternionic variable. Theorems analogous to those of Nehari anc C. Fefferman are proved.Comment: 19 page

The Mittag-Leffler Theorem for regular functions of a quaternionic variable

We prove a version of the classical Mittag-Leffler Theorem for regular functions over quaternions. Our result relies upon an appropriate notion of principal part, that is inspired by the recent definition of spherical analyticity.Comment: 10 page

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 $2$-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.Comment: 20 page

The orthogonal projection on slice functions on the quaternionic sphere

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.Comment: 6 page

A direct approach to quaternionic manifolds

The recent definition of slice regular function of several quaternionic variables suggests a new notion of quaternionic manifold. We give the definition of quaternionic regular manifold, as a space locally modeled on $\mathbb{H}^n$, in a slice regular sense. We exhibit some significant classes of examples, including manifolds which carry a quaternionic affine structure.Comment: 13 page