Higher spin fields on smooth domains. (English) Zbl 1009.30029

Brackx, F. (ed.) et al., Clifford analysis and its applications. Proceedings of the NATO advanced research workshop, Prague, Czech Republic, October 30-November 3, 2000. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 25, 389-398 (2001).
This paper deals with homogeneous monogenic polynomials of degree \(k\), which are just the spherical monogenics of degree \(k\) denoted by \({\mathcal M}_k\).
As the key operator the Rarita-Schwinger operator is introduced which act in \({\mathcal M}_k\). This operator is one of the components of a decomposition of the Dirac operator. By the help of the Fischer inner product \((\cdot,\cdot)_u\) on the boundary of a \(C^\infty\)-submanifold-with-boundary \(\overline\Omega\) in \(\mathbb{R}^n\) the space \(L_2(\partial\Omega|{\mathcal M}_k)\) is defined by the \(\mathbb{C}_n\)-inner product \[ \langle f,g\rangle_{\partial\Omega}= \int_{\partial\Omega} (f(x,u), g(x,u))_u ds(x). \] The fundamental solution of \(R_k(\partial_x)\) is constructed basing on the reproducing kernel \(K_k(u,v)\) for inner spherical monogenics of degree \(k\). The following factorization is obtained \[ R_k(\partial_x) D_{2k+1}= \Delta^{k+1} \] with a differential operator of order \(2k+1\), which can be calculated. Basic integral relations (Stokes theorem, Cauchy-Pompeiu theorem, Cauchy integral formula) are deduced. It is proved that with \(R_k(\Omega):=\text{ker} R_k(\partial_x)\) the corresponding Cauchy transform \(C_k\) maps \(C^\infty(\partial\Omega)\) onto \(R^\infty_K(\Omega):=R_K(\Omega)\cap C^\infty(\overline\Omega)\). Several decompositions of \(L_2(\partial\Omega,{\mathcal M}_K)\) are studied, Kerzman-Stein formulae are considered. Finally, the self-adjointness of the Cauchy transform is geometric characterized. This paper is clearly written and important in this field.
For the entire collection see [Zbl 0977.00025].


30G35 Functions of hypercomplex variables and generalized variables