2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/107469In a prior work [Hilbert transform along smooth families of lines math.CA/0310345] the authors introduced a variant of the Kakeya maximal function associated with Lipschitz maps from the plane into the unit circle. In this paper, we improve the known estimates for this maximal operator--and raise the conjecture that the bounds established are optimal.12 pages. The L^2 estimate for the maximal function in this paper is correct. A claimed L^p inequality, for 1<p<2, had an incomplete proofClassical Analysis and ODEs42text