The endpoint Fefferman-Stein inequality for the strong maximal function with respect to nondoubling measure

Let dµ(x1, . . . , xn) = dµ1(x1) · · · dµn(xn) be a product measure which is not necessarily doubling in Rn (only assuming dµi is doubling on R for i = 2, . . . , n), and Mn be the strong maximal function defined by { Mathematical Formulas } where R is the collection of rectangles with sides parallel to the coordinate axes in Rn, and ω, ν are two nonnegative functions. We give a sufficient condition on ω, ν for which the operator Mn is bounded from L(1 + (log+)n−1)(νdµ) to L1, (ωdµ). By interpolation, Mn is bounded from Lp(νdµ) to Lp(ωdµ), 1 p ∞.