Octonionic Kerzman-Stein operators

Abstract: In this paper we consider generalized Hardy spaces in the octonionic setting associated to arbitrary Lipschitz domains where the unit normal field exists almost everywhere. First we discuss some basic properties and explain structural differences to the associative Clifford analysis setting. The non-associativity requires special attention in the definition of an appropriate inner product and hence in the definition of a generalized Szeg\"o projection. Whenever we want to apply classical theorems from reproducing kernel Hilbert spaces we first need to switch to the consideration of real-valued inner products where the Riesz representation theorem holds. Then we introduce a generalization of the dual Cauchy transform for octonionic monogenic functions which represents the adjoint transform with respect to the real-valued inner product $\langle \cdot, \cdot \rangle_0$ together with an associated octonionic Kerzman-Stein operator and related kernel functions. Also in the octonionic setting, the Kerzman-Stein operator that we introduce turns out to be a compact operator. A motivation behind this approach is to find an approximative method to compute the Szeg\"o projection of octonionic monogenic functions offering a possibility to tackle BVP in the octonions without the explicit knowledge of the octonionic Szeg\"o kernel which is extremely difficult to determine in general. We also discuss the particular cases of the octonionic unit ball and the half-space. Finally, we relate our octonionic Kerzman-Stein operator to the Hilbert transform and particularly to the Hilbert-Riesz transform in the half-space case.