Maximal operators along piecewise linear curves near $L^1$ Neal Bez 42B25maximal operatorpiecewise linear curveweak type estimate 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 $\sigma\in(0,\frac{1}{2})$. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3606 10.1512/iumj.2009.58.3606 en Indiana Univ. Math. J. 58 (2009) 1639 - 1658 state-of-the-art mathematics http://iumj.org/access/