Note on Nikodym maximal in the variable Lebesgue spaces

Abstract

In this paper we consider the k‐plane Nikodym maximal estimates in the variable Lebesgue spaces Lp(·)false(Rnfalse). We first formulate the problem about the boundedness of the k‐plane Nikodym maximal and show that the maximal estimate in Lpfalse(Rnfalse) is equivalent to that in Lp(·)false(Rnfalse) for pfalse(·false)∈LH0∩N∞. So, the optimal Nikodym maximal estimate in Lp(·)false(R2false) follows from Cordoba's estimate.