We prove resolvent estimates in $L^p$-spaces for time-harmonic Maxwell’s equations in two spatial dimensions and in three dimensions in the partially anisotropic case. In the two-dimensional case the estimates are sharp. We consider anisotropic permittivity and permeability, which are both taken to be time-independent and spatially homogeneous. For the proof we diagonalize time-harmonic Maxwell’s equations to equations involving Half-Laplacians. We apply these estimates to localize eigenvalues for perturbations by potentials and to derive a limiting absorption principle in intersections of $L^p$-spaces.