Multidimensional distributions and generalized Hilbert transforms in Clifford analysis

The classical Hilbert transform on the real line is a well–known singular integral operator with applications in the theoretical description of many devices and systems. To our knowledge, J. Horváth was the first to introduce a multidimensional vector valued generalization of the Hilbert transform in the framework of Clifford analysis. Clifford analysis is a higher dimensional function theory in the framework of a Clifford algebra which may be seen as a multidimensional generalization of the theory of holomorphic functions in one complex variable and – at the same time – as a refinement of harmonic analysis. In this doctoral thesis we study some specific families of multidimensional distributions in the framework of Euclidean Clifford analysis, meanwhile constructing several generalizations of the Clifford–Hilbert transform, their kernels belonging to one of those families of distributions (Part I). Next, we adopt the idea of an anisotropic (also called metric dependent or metrodynamical) Clifford setting, which offers the possibility of adjusting the co–ordinate system to preferential and not necessarily mutually orthogonal directions. In this area of Clifford analysis, we construct the so–called anisotropic Clifford–Hilbert transform (Part II). Finally, new higher dimensional Hilbert transforms are developed in the framework of Hermitean Clifford analysis, a recent and successful branch of Clifford analysis, offering a refinement of the Euclidean case (Part III)

Generalized multidimensional Hilbert transforms in Clifford analysis

Two specific generalizations of the multidimensional Hilbert transform in Clifford analysis are constructed. It is shown that though in each of these generalizations some traditional properties of the Hilbert transform are inevitably lost, new bounded singular operators emerge on Hilbert or Sobolev spaces of L2-functions

De veralgemeende (hyperbolische) tangentialis van tweede orde in Clifford-analyse

B

A matrix Hilbert transform in Hermitean Clifford analysis

AbstractOrthogonal Clifford analysis is a higher dimensional function theory offering both a generalization of complex analysis in the plane and a refinement of classical harmonic analysis. During the last years, Hermitean Clifford analysis has emerged as a new and successful branch of it, offering yet a refinement of the orthogonal case. Recently in [F. Brackx, B. De Knock, H. De Schepper, D. Peña Peña, F. Sommen, submitted for publication], a Hermitean Cauchy integral was constructed in the framework of circulant (2×2) matrix functions. In the present paper, a new Hermitean Hilbert transform is introduced, arising naturally as part of the non-tangential boundary limits of that Hermitean Cauchy integral. The resulting matrix operator is shown to satisfy properly adapted analogues of the characteristic properties of the Hilbert transform in classical analysis and orthogonal Clifford analysis

The Hermitean Hilbert-Dirac connection

Hermitean Clifford analysis is a recent branch of Clifford analysis, refining the Euclidean case; it focusses on the simultaneous null solutions, called Hermitean monogenic functions, of two complex Dirac operators which are invariant under the action of the unitary group. The specificity of the framework, introduced by means of a complex structure creating a Hermitean space, forces the underlying vector space to be even dimensional. Thus, any Hilbert convolution kernel in R-2n should originate from the non-tangential boundary limits of a corresponding Cauchy kernel in R2n+2. In this paper we show that the difficulties posed by this inevitable dimensional jump can be overcome by following a matrix approach. The resulting matrix Hermitean Hilbert transform also gives rise, through composition with the matrix Dirac operator, to a Hermitean Hilbert-Dirac convolution operator "factorizing" the Laplacian and being closely related to Riesz potentials