Complex Neumann type boundary problem and decomposition of Lebesgue spaces. (Q1433917)

In a series of papers the author has attained orthogonal decompositions of the Sobolev spaces \(W^m_2(G)\) with respect to the subspace of analytic and monogenic functions in the case of functions of one complex variable and in Clifford analysis, respectively. The orthogonal complement of this subspace is determined as the subspace \(W^{m+1}_{2,0}(G)\) consisting of functions in \(W^{m+1}_2(G)\) with vanishing boundary values. In the present paper this decomposition is proved for functions of several complex variables. The orthogonal complement of the subspace of analytic functions in \(L_2(G)\) is shown to be \(\text{div}_z\) \(\mathring D^1_2(G)\) with \(\text{div}_z\vec u =\sum^n_{\nu=1}\partial_{z_\nu} u_\nu\) \(\mathring D^1_2(G)=\{\vec u : G\to \mathbb{C}^n| \text{div}_z\vec u\in L_2(G),\vec u,\vec\mu_z) = 0\) on \(\partial G\}\) with the complex normal vector \(\vec \nu_z\) on \(\partial G\). The determination of this orthogonal complement is related to the complex Neumann problem for the Poisson equation. The solvability condition for this problem is shown to be equivalent to the Poincaré inequality in the case of a poly-disc.