The sharp maximal function on spaces of generalized homogeneous type (Q1371030)

The paper deals with harmonic analysis on spaces of generalized type. The spaces considered were defined by \textit{A. Carbery}, \textit{J. Vance}, \textit{S. Wainger} and \textit{J. Wright} [ibid. 132, 119-140 (1995)] stimulated by the studies of the Hilbert transform and maximal functions associated to the curves on certain nilpotent Lie groups. An analogue of the Fefferman-Stein sharp maximal function is introduced, the equivalence between such a function and the maximal function of Hardy-Littlewood type in the \(L_p\)-norm is proved, and a John-Nirenberg type inequality is established. On the basis of the results obtained the corresponding BMO spaces and Hardy spaces are discussed and applications to singular integral operators are given.