Related papers: Slice regularity and harmonicity on Clifford algeb…
In this paper the certain 4-dimensional algebra in 4-dimensional pseudo-Riemannian space with signature (1, -1, -1, -1) is constructed. On the basis of this algebra the elements of the analysis, i.e. the theory of 4-dimensional functions of…
We study Cauchy problems associated to elliptic operators acting on vector-valued functions and coupled up to the first-order. We prove pointwise estimates for the spatial derivatives of the semigroup associated to these problems in the…
In this paper we introduce the Dirac and spin-Dirac operators associated to a connection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac operators associated with a Levi-Civita connection on a Riemannian (Lorentzian)…
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 $f'(0)=1$)…
The aim of this paper is to study some features of slice semi-regular functions $\mathcal{RM}(\Omega)$ on a circular domain $\Omega$ contained in the skew-symmetric algebra of quaternions $\mathbb{H}$ via the analysis of a family of linear…
Slice analysis is a generalization of the theory of holomorphic functions of one complex variable to quaternions. Among the new phenomena which appear in this context, there is the fact that the convergence domain of…
The concept of generalized partial-slice monogenic functions has been recently introduced to include the two theories of monogenic functions and of slice monogenic functions over Clifford algebras. The main purpose of this article is to…
We study global properties of quaternionic slice regular functions (also called s-regular) defined on symmetric slice domains. In particular, thanks to new techniques and points of view, we can characterize the property of being one-slice…
In this paper, we are concerned with the S-polyregularity the regular dot product of slice regular functions. We prove that the product of a slice regular function and right quaternionic polynomial function is a S-polyregular function and…
We introduce two classes of right quaternionic Hilbert spaces in the context of slice polyregular functions, generalizing the so-called slice and full hyperholomorphic Bargmann spaces. Their basic properties are discussed, the explicit…
This treatise investigates holomorphic functions defined on the space of bicomplex numbers introduced by Segre. The theory of these functions is associated with Fueter's theory of regular, quaternionic functions. The algebras of quaternions…
The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows to recover quantum mechanics as mechanics on a non-differentiable (fractal) space-time. The…
In this paper we develop the theory of Schauder estimates for the fractional harmonic oscillator $H^\sigma=(-\Delta+|x|^2)^\sigma$, $0<\sigma<1$. More precisely, a new class of smooth functions $C^{k,\alpha}_H$ is defined, in which we study…
We prove quadratic estimates for complex perturbations of Dirac-type operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral projections of the Hodge--Dirac operator on…
In this paper, we study the (complex) geometry of the set $S$ of the square roots of $-1$ in a real associative algebra $A$, showing that $S$ carries a natural complex structure, given by an embedding into the Grassmannian of…
In this paper, we introduce a pair of multiplication-like operations, $L_0$ and $L_1$, which derive $k$-regular functions from $(k+1)$-regular functions. The investigation of the inverse problem naturally leads to a deeper study of the…
The theory of quaternionic slice regular functions was introduced in 2006 and successfully developed for about a decade over symmetric slice domains, which appeared to be the natural setting for their study. Some recent articles paved the…
The theory of generalized partial-slice monogenic functions is considered as a syhthesis of the classical Clifford analysis and the theory of slice monogenic functions. In this paper, we investigate the Cauchy integral formula and the…
Rarita-Schwinger equation plays an important role in theoretical physics. Bure\v s et al. generalized it to arbitrary spin $k/2$ in 2002 in the context of Clifford algebras. In this article, we introduce the mean value property, Cauchy's…
In this article, the authors survey and review the studies of boundary value problems for regular functions in Clifford analysis, which include theoretical foundations and useful methods. Its theoretical bases consist of the generalized…