On a olvability condition for systems with an injective symbol in terms of iterations of double layer potentials

We prove existence of the $H^p(D)$-limit for iterations of the double layer potentials constructed from a Hodge parametrix on a smooth compact manifold $X$ (here $D$ is an open connected subset in $X$). The limit is the orthogonal projection of the Sobolev space $H^p(D)$ onto the closed subspace of $H^p(D)$-solutions of some elliptic operator $P$ of order $p\geq 1$. Using this result, we obtain a formula for Sobolev solutions to the equation $Pu=f$ in $D$ if such exist. Solutions are given in the form of series whose summands are iterations of the double layer potentials. We also construct a similar expansion for the Neumann $P$-problem in $D$.