  The R\'enyi function plays an important role in the analysis of multifractal
random fields. For random fields on the sphere, there are three models in the
literature where the R\'enyi function is known explicitly. The theoretical part
of the article presents multifractal random fields on the sphere and develops
specific models where the R\'enyi function can be computed explicitly. Then
these results are applied to the Cosmic Microwave Background Radiation data
collected from the Planck mission. The main statistical model used to describe
these data in the literature is isotropic Gaussian fields. We present numerical
multifractality studies and methodology based on simulating random fields,
computing the R\'enyi function and the multifractal spectrum for different
scenarios and actual CMB data.
