Sharp weighted bounds for one-sided maximal operators

Abstract

In this note we prove some results about the best constants for the boundedness of the one-sided Hardy–Littlewood maximal operator in \(L^p(\mu )\), where \(\mu \) is a locally finite Borel measure, that in the two-sided weights have been obtained by Buckley (Trans Am Math Soc 340(1):253–272, 1993) and more recently by Hytonen and Perez (Anal PDE 6(4):777–818, 2013) and Hytonen et al. (J Funct Anal 263(12):3883–3899, 2012). To prove Bucley’s theorem for one-sided maximal operators, we follow the ideas of Lerner (Proc Am Math Soc 136(8):2829–2833, 2008). To obtain a better estimate in terms of mixed constants we follow the steps in Hytonen and Perez (Anal PDE 6(4):777–818, 2013) and Hytonen et al. (J Funct Anal 263(12):3883–3899, 2012) i.e., (a) getting a sharp estimate for the constant for the weak type type, in terms of the one-sided \(A_p\) constant, (b) obtaining a sharp reverse Holder inequality and (c) using Marcinkiewicz interpolation theorem. Our proofs of these facts are different from those in Hytonen and Perez (Anal PDE 6(4):777–818, 2013) and Hytonen et al. (J Funct Anal 263(12):3883–3899, 2012) and apply to more general measures.