Marletta, G. Maximal functions with mitigating factors in the plane. (English) Zbl 0947.42013 J. Lond. Math. Soc., II. Ser. 59, No. 2, 647-656 (1999). Suppose that \(\gamma:[0,1]\to{\mathbb R}^{2}\) is a curve with curvature \(\kappa\), not passing through the origin, and consider the maximal operator \(M_{\sigma}\), where \(\sigma\geq 0\), given by \[ M_{\sigma}f(x) = \sup_{t>0}\int^{1}_{0}|f(x-t\gamma(s))|\kappa(s)^{\sigma} ds. \] When \(\kappa\) never vanishes, then a theorem of Bourgain implies that \(M_{\sigma}\) is bounded on \(L^{p}({\mathbb R}^{2})\) if \(p>2\). In this paper it is shown that for curves with a flat point satisfying certain technical assumptions, \(M_{\sigma}\) is \(L^{p}\)-bounded provided that \(p>\max\{ \sigma^{-1},2\}\). Reviewer: Michael Cowling (Sydney) Cited in 2 Documents MSC: 42B25 Maximal functions, Littlewood-Paley theory Keywords:maximal functions; curvature PDFBibTeX XMLCite \textit{G. Marletta}, J. Lond. Math. Soc., II. Ser. 59, No. 2, 647--656 (1999; Zbl 0947.42013) Full Text: DOI