Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operator in higher dimensions

Abstract

We give a quantitative characterization of the pairs of weights (w, v) for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak (p, p) type inequality for 1 ⩽ p < ∞. More precisely, given any measurable set E0, the estimatew({x∈ℝn:M+,d(XE0)(x)>t})⩽C[(w,v)]Ap+,d(ℛ)ptpv(E0) holds if and only if the pair (w,v) belongs to Ap+,d(ℛ), that is,|E||Q|⩽[(w,v)]Ap+,d(ℛ)(v(E)w(Q))1/p for every dyadic cube Q and every measurable set E ⊂ Q+. The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the non-dyadic case in ℝ2 by following the main ideas in L. Forzani, F. J. Martín-Reyes, S. Ombrosi (2011).