In this paper the formalism of the electromagnetic inverse source theory is used to investigate radiation enhancement due to antenna substrates. Particular attention is given to sources that are confined within a spherical volume and are embedded within two nested spheres of arbitrary materials. Emphasis is given to the special case when the two nested spheres are made up of materials with oppositely signed constitutive parameters. The analysis comprises forward, or radiation, characterization for a given configuration as well as inverse-theoretic characterization. The forward characterization is focused on the singular-value spectrum of the linear source-to-field mapping relevant to each configuration while the inverse-theoretic characterization is performed via the so-called "minimum-energy" sources capable of generating a prescribed exterior field. The derived formulation is based on constrained optimization and multipole theory. Importantly, it is non-antenna-specific. Thus, this formulation enables fair comparison of different substrate configurations by comparing optimal radiation in each configuration (i.e., the "best" in each one), as governed by a formally tractable source-energy cost function that is physically motivated by Ohmic loss control. The derived theory is accompanied by numerical results illustrating the effects on radiation enhancement of particular substrate designs.