Plemelj projections in discrete quaternionic analysis

Abstract

The discrete quaternionic analysis has been placed recently in the same framework as its continuous counterpart. To discover more similarities, we study the solvability of the Dirichlet problem for discrete Cauchy–Fueter system. A serious difficulty arises due to the absence of the notion of the non-tangential limit on lattices. To overcome it, we introduce the two layered structure of discrete boundaries. It turns out that the restriction of the discrete Cauchy–Bitsadze integral on the inner layer of the discrete boundary plays the same role as the non-tangential limit of its continuous counterpart from the inside of the domain, and the analogous result holds true for the outer layer. Moreover, the discrete Sokhotski–Plemelj formula is established over an arbitrary bounded boundary. This allows us to give a criterion for the solvability of the Dirichlet problem via discrete Plemelj projections. More remarkably, we can show that the scaling limits of discrete Plemelj projections are the continuous ones. Our crucial tool is the asymptotic expansion of the fundamental solution of the discrete Cauchy–Fueter operator, instead of the classical method by V. Thomee.