  The galaxy cluster MS 0735.6+7421 hosts two large X-ray cavities, filled with
radio emission, where a decrease of the Sunyaev-Zel'dovich (SZ) effect has been
detected, without establishing if its origin is thermal (from a gas with very
high temperature) or non-thermal. In this paper we study how thermal and
non-thermal contributions to the SZ effect in the cavities are related; in
fact, Coulomb interactions with the thermal gas modify the spectrum of low
energy non-thermal electrons, which dominate the non-thermal SZ effect; as a
consequence, the intensity of the non-thermal SZ effect is stronger for lower
density of the thermal gas inside the cavity. We calculate the non-thermal SZ
effect in the cavities as a function of the thermal density, and compare the SZ
effects produced by thermal and non-thermal components, and with the one from
the external Intra Cluster Medium (ICM), searching for the best frequency range
where it is possible to disentangle the different contributions. We find that
for temperatures inside the cavities higher than $\sim1500$ keV the non-thermal
SZ effect is expected to dominate on the thermal one, particularly at high
frequencies ($\nu>500$ GHz), where it can also be a non-negligible fraction of
the SZ effect from the external ICM. We also discuss the possible sources of
astrophysical bias (as kinetic SZ effect and foreground emission from Galactic
dust) and possible ways to address them, as well as necessary improvements in
the modeling of the properties of cavities and the ICM.
