Necessary and sufficient condition for a VMO function

The purpose of this paper is to prove a necessary and sufficient condition for a vanishing mean oscillation function. To obtain this result we use an extension to Morrey spaces of an inequality by Fefferman and Stein concerning the maximal and the sharp maximal function, the fractional maximal function used by Muckenhoupt and Wheeden and the spaces generated by blocks well studied by Shirai, Komori and Mizuhara.