  We use the well-posedness of transmission problems on classes of two-sided
Sobolev extension domains to give variational definitions for (boundary) layer
potential operators and Neumann-Poincar{\'e} operators. These classes of
domains contain Lipschitz domains, and also domains with fractal boundaries.
Although our variational formulation does not involve any measures on the
boundary, we recover the classical results in smooth domains by considering the
surface measure on the boundary. We discuss properties of these operators and
generalize basic results in imaging beyond the Lipschitz case.
