Bulletin of the
Korean Mathematical Society BKMS

In this paper we study Szeg\"{o} kernel projections for Hardy spaces in quaternionic Clifford analysis. At first we introduce the matrix Szeg\"{o} projection operator for the Hardy space of quaternionic Hermitean monogenic functions by the characterization of the matrix Hilbert transform in the quaternionic Clifford analysis. Then we establish the Kerzman-Stein formula which closely connects the matrix Szeg\"{o} projection operator with the Hardy projection operator onto the Hardy space, and we get the matrix Szeg\"{o} projection operator in terms of the Hardy projection operator and its adjoint. At last, we construct the explicit matrix Szeg\"o kernel function for the Hardy space on the sphere as an example, and get the solution to a Diriclet boundary value problem for matrix functions.

Keywords: Szeg\"{o} projections, quaternionic Clifford analysis, Hardy space, matrix function

MSC numbers: Primary 30G35, 15A66, 30C40, 31A25, 31B10

Supported by: This work was financially supported by National Natural Science Foundation of China (11601525, 12071485), Natural Science Foundation of Hunan Province (2020JJ4105).