2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/218215We extend to all planar simply connected domains Makarov-type results about the Hausdorff dimension of $p$-harmonic measure pioneered by Lewis and Bennewitz in the context of quasidisks. The key to our analysis is a gradient estimate using the distance to the boundary and constants that only depend on $p$. This is achieved by studying the conformal map from the unit disk to the simply connected domain to construct good quasicurves from a point in the domain to the boundary.Complex VariablesAnalysis of PDEs35J25, 35J70text