In this paper we study maximal sharp functions associated with arbitrary differential bases. The definition of these functions
goes back to the papers by F. John (1965), and by C. Fefferman and E. M. Stein (1972), where the classical bases consisting
of cubic intervals were considered.
We obtain conditions imposed on the basis, under which inequalities, known earlier in the case of a basis of cubes, are valid
for the considered maximal functions. The main results are formulated in terms of nonincreasing rearrangements. In the capacity
of applications, we obtain estimates of the rearrangements of subadditive operators acting in BMO. In particular, the estimate
for the Hilbert transform, obtained earlier by C. Bennett and K. Rudnick, follows.