  We analyze electromagnetic scattering of TM polarized waves from a
diffraction grating consisting of a periodic, anisotropic, and possibly
negative-index dielectric material. Such scattering problems are important for
the modelization of, e.g., light propagation in nano-optical components and
metamaterials. The periodic scattering problem can be reformulated as a
strongly singular volume integral equation, a technique that attracts
continuous interest in the engineering community, but rarely received rigorous
theoretic treatment. In this paper we prove new (generalized) G\r{a}rding
inequalities in weighted and unweighted Sobolev spaces for the strongly
singular integral equation. These inequalities also hold for materials for
which the real part takes negative values inside the diffraction grating,
independently of the value of the imaginary part.
