Characterization of BMO in terms of rearrangement-invariant Banach function spaces

The atomic decomposition of Hardy spaces by atoms defined by rearrangement-invariant Banach function spaces is proved in this paper. Using this decomposition, we obtain the characterizations of BMO and Lipschitz spaces by rearrangement-invariant Banach function spaces. We also provide the sharp function characterization of the rearrangement-invariant Banach function spaces.