  In the context of gauge/gravity duality, we verify two types of gauge theory
low-energy theorems, the dilation Ward identities and the decoupling of heavy
flavor. First, we provide an analytic proof of non-trivial dilation Ward
identities for a theory holographically dual to a background with gluon
condensate (the self-dual Liu--Tseytlin background). In this way an important
class of low-energy theorems for correlators of different operators with the
trace of the energy-momentum tensor is established, which so far has been
studied in field theory only. Another low-energy relationship, the so-called
decoupling theorem, is numerically shown to hold universally in three
holographic models involving both the quark and the gluon condensate. We show
this by comparing the ratio of the quark and gluon condensates in three
different examples of gravity backgrounds with non-trivial dilaton flow. As a
by-product of our study, we also obtain gauge field condensate contributions to
meson transport coefficients.
