Hyperholomorphic Bergman spaces and Bergman operators associated with domains in \(\mathbb C^2\) (Q941078)

The origins of quaternionic analysis (i.e. function theory for quaternions) date back to the work of Rudolf Fueter and his students. Much later it was realized that the theory can be successfully applied to the study of boundary value problems for Poisson's equation with Dirichlet boundary conditions, yielding an explicit representation formula for the solutions involving integral operators. One key ingredient here is the so-called Bergman (ortho)projector, associating to every square-integrable quaternion-valued function its holomorphic part (in the sense of a null solution of Dirac's operator). The article at hand considers so-called \(\theta\)-hyperholomorphic functions (\(\theta\) stands for an arbitrary angle), which can be regarded as a generalization of the notion of holomorphic functions in two complex variables. The authors show that \(\theta\)-hyperholomorphic functions transform covariantly under Möbius transformations up to a special constant, thus allowing to identify \(\theta\)-hyperholomorphic function spaces on conformally related domains. Furthermore they define the Bergman kernel (i.e. the kernel of the Bergman projector), reveal its properties and give explicit formulas for the Bergman kernel of the Ball and the upper half-space. They also prove an orthogonal decomposition of the space of square-integrable quaternionic functions analogous to the decomposition mentioned above.