Bounded Calderon maps and signed jump identities

Assume the Axiom of Choice and Countable Choice, as in the single- and double-layer transmission theorems. For $\lambda \in H^{-1\mathrm{/}2}(\mathrm{\Gamma })$ and $\varphi \in H^{1\mathrm{/}2}(\mathrm{\Gamma })$, the operators in Calderon boundary operators and trace conventions are bounded as $V:H^{-1\mathrm{/}2}\to H^{1\mathrm{/}2},K:H^{1\mathrm{/}2}\to H^{1\mathrm{/}2},K^{\mathrm{\prime }}:H^{-1\mathrm{/}2}\to H^{-1\mathrm{/}2},W:H^{1\mathrm{/}2}\to H^{-1\mathrm{/}2}\mathrm{.}$ Their weak traces satisfy ${\gamma }^{-}S\lambda ={\gamma }^{+}S\lambda =V\lambda ,{\mathrm{\partial }}_n^{-}S\lambda =K^{\mathrm{\prime }}\lambda +\frac{1}{2}\lambda ,{\mathrm{\partial }}_n^{+}S\lambda =K^{\mathrm{\prime }}\lambda -\frac{1}{2}\lambda ,$ ${\gamma }^{-}D\varphi =K\varphi -\frac{1}{2}\varphi ,{\gamma }^{+}D\varphi =K\varphi +\frac{1}{2}\varphi ,{\mathrm{\partial }}_n^{-}D\varphi ={\mathrm{\partial }}_n^{+}D\varphi =-W\varphi \mathrm{.}$ All equalities are in the stated Sobolev trace or dual spaces and use the outward normal orientation fixed on the A page.

The double-layer transmission theorem identifies the weak double layer with its off-surface kernel integral for every global $H^{1\mathrm{/}2}(\mathrm{\Gamma })$ density, including compatible facewise smooth densities reaching polyhedral edges. For those facewise smooth densities, Finite-energy transmission pairs have the double-layer kernel representation across polyhedral edges also computes the face-interior kernel realization of $K$ and the signed half-jumps. The single-layer transmission theorem identifies the weak single layer with its kernel for bounded densities, including discontinuous P0 densities. The common double-layer conormal defines the hypersingular operator weakly; no ordinary absolutely convergent hypersingular boundary integral or pointwise half-jump at edges or vertices is asserted.