In this note, we prove Cheng-Yau type local gradient estimate for harmonic functions on Alexandrov spaces with Ricci curvature bounded below. We adopt a refined version of Moser's iteration which is based on Zhang-Zhu's Bochner type formula in "Yau’s gradient estimates on Alexandrov spaces." Our result improves the previous one of Zhang-Zhu in the case of negative Ricci lower bound.
"A note on local gradient estimate on Alexandrov spaces." Tohoku Math. J. (2) 66 (2) 259 - 267, 2014. https://doi.org/10.2748/tmj/1404911862