Slice Hyperholomorphic Bloch, Besov and Dirichlet Spaces

Gentili G., Stoppato C., Struppa D.C.

Colombo F., Kraußhar R.S., Sabadini I.

In this paper we consider the symmetry behavior of slice monogenic functions under Möbius transformations. We describe the group under which slice monogenic functions are taken into slice monogenic functions. We prove a transformation formula for composing slice monogenic functions with Möbius transformations and describe their conformal invariance. Finally, we explain two construction methods to obtain automorphic forms in the framework of this function class. We round off by presenting a precise algebraic characterization of the subset of slice monogenic linear fractional transformations within the set of general Möbius transformations.

Gal S.G., Sabadini I.

In this paper we continue our study on the density of the set of quaternionic polynomials in function spaces of slice regular functions on the unit ball by considering the case of the Bloch and Besov spaces of the first and of the second kind. Among the results we prove, we show some constructive methods based on the Taylor expansion and on the convolution polynomials. We also provide quantitative estimates in terms of higher order moduli of smoothness and of the best approximation quantity. As a byproduct, we obtain two new results for complex Bloch and Besov spaces.

Avetisyan K., Gürlebeck K.

Alpay D., Colombo F., Sabadini I.

El-Sayed Ahmed A.

In this paper, we obtain some characterizations for general Besov-type spaces by employing certain power series satisfying certain growth conditions in lieu of the weight function in Clifford analysis. The obtained results extend and generalize the corresponding results which are given in [6, 23].

El-Fallah O., Kellay K., Mashreghi J., Ransford T.

The Dirichlet space is one of the three fundamental Hilbert spaces of holomorphic functions on the unit disk. It boasts a rich and beautiful theory, yet at the same time remains a source of challenging open problems and a subject of active mathematical research. This book is the first systematic account of the Dirichlet space, assembling results previously only found in scattered research articles, and improving upon many of the proofs. Topics treated include: the Douglas and Carleson formulas for the Dirichlet integral, reproducing kernels, boundary behaviour and capacity, zero sets and uniqueness sets, multipliers, interpolation, Carleson measures, composition operators, local Dirichlet spaces, shift-invariant subspaces, and cyclicity. Special features include a self-contained treatment of capacity, including the strong-type inequality. The book will be valuable to researchers in function theory, and with over 100 exercises it is also suitable for self-study by graduate students.

Bisi C., Stoppato C.

The regular fractional transformations of the extended quaternionic space have been recently introduced as variants of the classical linear fractional transformations. These variants have the advantage of being included in the class of slice regular functions, introduced by Gentili and Struppa in 2006, so that they can be studied with the useful tools available in this theory. We first consider their general properties, then focus on the regular Möbius transformations of the quaternionic unit ball $\mathbb{B}$ , comparing the latter with their classical analogs. In particular we study the relation between the regular Möbius transformations and the Poincaré metric of $\mathbb{B}$ , which is preserved by the classical Möbius transformations. Furthermore, we announce a result that is a quaternionic analog of the Schwarz-Pick lemma.

Stoppato C.

Quaternionic Moebius transformations have been investigated for more than 100 years and their properties have been characterized in detail. In recent years G. Gentili and D. C. Struppa introduced a new notion of regular function of a quaternionic variable, which is developing into a quite rich theory. Several properties of regular quaternionic functions are analogous to those of holomorphic functions of one complex variable, although the diversity of the non-commutative setting introduces new phenomena. Unfortunately, the (classical) quaternionic Moebius transformations are not regular. However, in this paper we are able to construct a different class of Moebius-type transformations that are indeed regular. This construction requires several steps: we first find an analog to the Casorati-Weierstrass theorem and use it to prove that the group $${Aut(\mathbb{H})}$$ of biregular functions on $${\mathbb{H}}$$ coincides with the group of regular affine transformations. We then show that each regular injective function from $${\widehat{\mathbb{H}} = \mathbb{H}\cup \{\infty\}}$$ to itself belongs to a special class of transformations, called regular fractional transformations. Among these, we focus on the ones which map the unit ball $${\mathbb{B} = \{q \in \mathbb{H} : |q| < 1 \}}$$ onto itself, called regular Moebius transformations. We study their basic properties and we are able to characterize them as the only regular bijections from $${\mathbb{B}}$$ to itself.

Bernstein S., Gürlebeck K., Reséndis O. L.F., Tovar S. L.M.

Reséndis O. L.F., Tovar S. L.M.

In this paper we give a generalization of the Zhao F(p, q, s)-spaces by using operators instead of functions. In this way we unify and simplify several important results about the classic spaces D p , $${\mathcal{Q}}_{p}$$ ,Bα, etc.

El-Sayed Ahmed A., Gürlebeck K., Reséndis L.F., Tovar S. L.M.

In this article we give the definition of Bp, q spaces of hyperholomorphic functions. Then, we characterize hypercomplex Bloch space by these Bp, q spaces. One of the main results is a general Besov-type characterization for quaternionic Bloch functions that generalizes a Stroethoff theorem. Furthermore, some important basic properties of these Bp, q spaces are also considered.

Reséndis O.L., Tovar S.L.

In this paper we obtain a Besov-type characterization for quaternione Bloch functions that generalizes a Stroethoff theorem.

Gürlebeck K., Malonek H.R.

We consider a scale of weighted spaces a quaternion-valued functions of three real variables. This scale generalises the idea of Qp-spaces in complex function theory. The goal of this paper is to prove that the inclusions of spaces from the scale are strict inclusion. As a tool we prove some properties of special monogenic polynomials which have an importance in their own right independently of their use in the scale of Qp-spaces.

Wu Z.

We study commutators between multiplication by a function, called the symbol, and Riesz transformations on the Besov spaces. We characterize symbols, by potential capacity, for which the associated commutators are bounded. Clifford analysis plays a key role in our approach.