Boundedness of strong maximal functions with respect to non-doubling measures

Abstract

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The main purpose of this paper is to establish a boundedness result for strong maximal functions with respect to certain non-doubling measures in ....... More precisely, let ...... be a product measure which is not necessarily doubling in ...... (only assuming ...... is doubling on ...... for ......), and let [omega] be a nonnegative and locally integral function such that ...... is in ...... uniformly in ...... for each ......, let ......, ......, and ...... be the strong maximal function defined by ...... where ...... is the collection of rectangles with sides parallel to the coordinate axes in ....... Then we show that ...... is bounded on ...... for ....... This extends an earlier result of Fefferman (Am. J. Math. 103:33-40, 1981 ) who established the ...... boundedness when ...... is the Lebesgue measure on ...... and ...... is doubling with respect to rectangles in ......, [omega] satisfies a uniform ...... condition in each of the variables except one.

Moreover, we also establish some boundedness result for the Cordoba maximal functions (Córdoba A. in Harmonic Analysis in Euclidean Spaces, pp. 29-50, 1978 ) associated with the Córdoba-Zygmund dilation in ...... with respect to some non-doubling measures. This generalizes the result of Fefferman-Pipher (Am. J. Math. 119:337-369, 1997 ).