  In this paper, we study different types of weighted Besov and
Triebel-Lizorkin spaces with variable smoothness. The function spaces can be
defined by means of the Littlewood-Paley theory in the field of Fourier
analysis, while there are other norms arising in the theory of partial
differential equations such as Sobolev-Slobodeckij spaces. It is known that two
norms are equivalent when one considers constant regularity function spaces
without weights. We show that the equivalence still holds for variable
smoothness and weights, which is accomplished by making use of shifted maximal
functions, Peetre's maximal functions, and the reverse H\"older inequality.
Moreover, we obtain a weighted regularity estimate for time-fractional
evolution equations and a generalized Sobolev embedding theorem without
weights.
