  The existence theory is developed for solutions of the inhomogeneous
linearized field equations for causal variational principles. These equations
are formulated weakly with an integral operator which is shown to be bounded
and symmetric on a Hilbert space endowed with a suitably adapted weighted
$L^2$-scalar product. Guided by the procedure in the theory of linear elliptic
partial differential equations, we use the spectral calculus to define
Sobolev-type Hilbert spaces and invert the linearized field operator as an
operator between such function spaces. The uniqueness of the resulting weak
solutions is analyzed. Our constructions are illustrated in simple explicit
examples. The connection to the causal action principle for static causal
fermion systems is explained.
