### Some orthogonal decompositions of Sobolev spaces and applications

Two kinds of orthogonal decompositions of the Sobolev space W̊₂¹ and hence also of $W{\u2082}^{-1}$ for bounded domains are given. They originate from a decomposition of W̊₂¹ into the orthogonal sum of the subspace of the ${\Delta}^{k}$-solenoidal functions, k ≥ 1, and its explicitly given orthogonal complement. This decomposition is developed in the real as well as in the complex case. For the solenoidal subspace (k = 0) the decomposition appears in a little different form. In the second kind decomposition the ${\Delta}^{k}$-solenoidal...