  We develop a new heat kernel method that is suited for a systematic study of
the renormalization group flow in Horava gravity (and in Lifshitz field
theories in general). This method maintains covariance at all stages of the
calculation, which is achieved by introducing a generalized Fourier transform
covariant with respect to the nonrelativistic background spacetime. As a first
test, we apply this method to compute the anisotropic Weyl anomaly for a
(2+1)-dimensional scalar field theory around a z=2 Lifshitz point and
corroborate the previously found result. We then proceed to general scalar
operators and evaluate their one-loop effective action. The covariant heat
kernel method that we develop also directly applies to operators with spin
structures in arbitrary dimensions.
