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

Elliptic Methods for Solving the Linearized Field Equations of Causal Variational Principles