Weak weighted inequalities for a dyadic one-sided maximal function in $\mathbb {R}^{n}$

Abstract:

In this note we introduce a dyadic one-sided maximal function defined as \[ M^{+,d}f(x)=\sup _{Q\;dyadic\text {:}x\in Q}\frac {1}{\left | Q\right | } \int _{Q^{+}}\left | f\right | ,\] where $Q^{+}$ is a certain cube associated with the dyadic cube $Q$ and $f\in L_{loc}^{1}\left ( \mathbb {R}^{n}\right )$ . We characterize the pair of weights $\left ( w,v\right )$ for which the maximal operator $M^{+,d}$ applies $L^{p}\left ( v\right )$ into weak-$L^{p}\left ( w\right )$ for $1\leq p<\infty$.