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Maximal functions with mitigating factors in the plane. (English) Zbl 0947.42013

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 \(p>\max\{ \sigma^{-1},2\}\).

MSC:

42B25 Maximal functions, Littlewood-Paley theory
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