A note on the limiting weak-type behavior for maximal operators

Abstract:

We study the following open question raised by Janakiraman in (2006): for $f\in L^1 (\mathbb {R}^n )\cap L^{\infty } (\mathbb {R}^n )$ and $\lambda > 0$, what is the limiting behavior of \[ \left [m\left (\{x \in \mathbb {R}^n :M(|f|^p )(x)>\lambda \}\right )\right ]^{1/p} \] as $p\to \infty$? In this note, we give a complete answer to this question.