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 Hytönen and Pérez (Anal PDE 6(4):777–818, 2013) and Hytönen 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 Hytönen and Pérez (Anal PDE 6(4):777–818, 2013) and Hytönen 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 Hölder inequality and (c) using Marcinkiewicz interpolation theorem. Our proofs of these facts are different from those in Hytönen and Pérez (Anal PDE 6(4):777–818, 2013) and Hytönen et al. (J Funct Anal 263(12):3883–3899, 2012) and apply to more general measures.