Maximal functions with respect to differential bases measuring mean oscillation

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Abstract

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.

Original languageEnglish
Pages (from-to)41-58
Number of pages18
JournalAnalysis Mathematica
Volume24
Issue number1
DOIs
StatePublished - 1998
Externally publishedYes

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