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We solve time-harmonic Maxwell's equations in anisotropic, spatiallyhomogeneous media in intersections of $L^p$-spaces. The material laws aretime-independent. The analysis requires Fourier restriction-extension estimatesfor perturbations of Fresnel's wave surface. This surface can be decomposedinto finitely many components of the following three types: smooth surfaceswith non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvaturevanishing along one-dimensional submanifolds, but without flat points, andsurfaces with conical singularities. Our estimates are based on newBochner-Riesz estimates with negative index for non-elliptic surfaces.

Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner-Riesz estimates with negative index for non-elliptic surfaces