Reproducing kernel functions of solutions to polynomial Dirac equations in the annulus of the unit ball in Rn and applications to boundary value problems

Let D:=∑i=1n∂∂xiei be the Dirac operator in Rn and let P(X)=amXm+⋯+a1X1+a0 be a polynomial with complex coefficients. Differential equations of the form P(D)f=0 are called polynomial Dirac equations. In this paper we consider Hilbert spaces of Clifford algebra-valued functions that satisfy such a polynomial Dirac equation in annuli of the unit ball in Rn. We determine an explicit formula for the Bergman kernel for solutions of complex polynomial Dirac equations of any degree m in the annulus of radii μ and 1 where μ∈]0,1[. We further give formulas for the Szegö kernel for solutions to polynomial Dirac equations of degree m