  This paper studies random fields on the unit sphere. Traditionally, isotropic
Gaussian random fields are considered as the underlying statistical model of
the cosmic microwave background (CMB) data. This paper discusses the
generalized multifractional Brownian motion and its pointwise H\"older exponent
on the sphere. The multifractional approach is used to investigate the CMB data
from the Planck mission. These data consist of CMB radiation measurements at
narrow angles of the sky sphere. The obtained results suggest that the
estimated H\"older exponents for different CMB regions do change from location
to location. Therefore, CMB data are multifractional. Then the developed
methodology is used to suggest two approaches for detecting regions with
anomalies in cleaned CMB maps.
