Hilbert transforms and the Cauchy integral in euclidean space

We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being unique, but under suitable boundary conditions we prove existence and uniqueness of conjugates. The proof also yields invertibility results for a new class of generalized double layer potential operators on Lipschitz surfaces and boundedness of related Hilbert transforms.Comment: Some minor corrections mad

Hölder norm estimate for a Hilbert transform in Hermitian Clifford analysis

A Hilbert transform for Holder continuous circulant (2 x 2) matrix functions, on the d-summable (or fractal) boundary I" of a Jordan domain Omega in a"e(2n) , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the Holder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the Holder exponents, the diameter of I" and a specific d-sum (d > d) of the Whitney decomposition of Omega. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary

The Calderon projection over C* algebras

We construct the Calderon projection on the space of Cauchy datas for a twisted Dirac operator in the Mischenko--Fomenko pseudodifferential calculus for operators acting on bundles of finitely generated $C^*$--Hilbert modules on a compact manifold with boundary. In particular an invertible double is constructed generalizing the classical result

Riemann-Hilbert problems for poly-Hardy space on the unit ball

In this paper, we focus on a Riemann–Hilbert boundary value problem (BVP) with a constant coefficients for the poly-Hardy space on the real unit ball in higher dimensions. We first discuss the boundary behaviour of functions in the poly-Hardy class. Then we construct the Schwarz kernel and the higher order Schwarz operator to study Riemann–Hilbert BVPs over the unit ball for the poly- Hardy class. Finally, we obtain explicit integral expressions for their solutions. As a special case, monogenic signals as elements in the Hardy space over the unit sphere will be reconstructed in the case of boundary data given in terms of functions having values in a Clifford subalgebra. Such monogenic signals represent the generalization of analytic signals as elements of the Hardy space over the unit circle of the complex plane

Hardy spaces of solutions of generalized Riesz and Moisil-Teodorescu systems

Hardy spaces of solutions of generalized Riesz and generalized Moisil-Teodorescu systems in half space Rm+1,+ , and of their non-tangential L2-boundary values in Rm are characterized