Abstract:
In this paper, we give the sufficient conditions for the compactness of sets in generalized
Morrey spaces M
w(·)
p . This result is an analogue of the well-known Fréchet–Kolmogorov theorem on
the compactness of a set in Lebesgue spaces Lp, p > 0. As an application, we prove the compactness
of the commutator of the Riesz potential [b, Iα] in generalized Morrey spaces, where b ∈ VMO
(VMO(Rn
) denote the BMO-closure of C
∞
0
(Rn
)). We prove auxiliary statements regarding the
connection between the norm of average functions and the norm of the difference of functions in the
generalized Morrey spaces. Such results are also of independent interest.
