  We consider subspaces of Morrey spaces defined in terms of various vanishing
properties of functions. Such subspaces were recently used to describe the
closure of $C_0^\infty(\mathbb{R}^n)$ in Morrey norm. We show that these
subspaces are invariant with respect to some classical operators of harmonic
analysis, such as the Hardy-Littlewood maximal operator, singular type
operators and Hardy operators. We also show that the vanishing properties
defining those subspaces are preserved under the action of Riesz potential
operators and fractional maximal operators.
