IUMJ

Title: Maximal operators along piecewise linear curves near $L^1$

Authors: Neal Bez

Issue: Volume 58 (2009), Issue 4, 1639-1658

Abstract:

For certain piecewise linear plane curves $\Gamma$ and convex functions $\Phi$ we address the question of whether the maximal operator along $\Gamma$ is of weak type $\Phi(L)$. For example, when $\Gamma$ is the graph of the continuous map $\gamma$ for which $\gamma(2^k)=2^{2k}$ and $\gamma$ is linear on $[2^k,2^{k+1}]$, $k\in\mathbb{Z}$, then the maximal operator along $\Gamma$ is not of weak type $L(\log L)^{\sigma}$ for any $\sigma\in(0,\frac{1}{2})$.