  First microscopic theory for electron-phonon energy exchange in Anderson
insulators is developed. The major contribution to the cooling power as a
function of electron temperature is shown to be directly related to the
correlation function of the local density of electron states at small energy
difference argument. In Anderson insulators not far from localization
transition, this correlation function is strongly enhanced by wave-function's
multi-fractality and, additionally, by the presence of Mott's resonant pairs of
localized states. The theory we develop explains huge enhancement of the
cooling power observed in insulating Indium Oxide films as compared to
predictions of the theory previously developed for disordered metals. Our
results open the way to predict the conditions appropriate for the observation
of Many Body Localization transition those presence in electronic insulators
was advocated in the seminal paper by Basko, Aleiner and Altshuler (2006) but
have not been convincingly demonstrated yet.
