  We investigate the interaction between the electrons of a two-dimensional
metal and the acoustic phonons of an underlying piezoelectric substrate.
Fundamental inequalities can be obtained from general energy arguments. As a
result, phonon mediated attraction can be proven to never overcome electron
Coulomb repulsion, at least for long phonon wavelengths. We study the influence
of these phonons on the possible pairing instabilities of a two-dimensional
electron gas such as graphene.
