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 (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.

We show that certain vanishing properties defining closed subspaces of generalized Morrey spaces are preserved under the action of various classical operators of harmonic analysis, such as maximal operators, singular-type operators, Hardy operators, and fractional integral operators. Those vanishing subspaces were recently used to deal with the delicate problem on the description of the closure of nice functions in Morrey norm.

It is proved that both oscillatory integral operators and fractional oscil-latory integral operators are bounded on weighted Morrey spaces. The corresponding commutators generated by BM O functions are also considered.

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