Maximal operators associated to families of flat curves in the plane
Let \(C\) be a smooth compact curve in the plane, with the property that the curvature vanishes at most of order \(m- 2\), \(m\geq 2\). The author shows that the maximal operator defined by \[ Mf(x)= \sup_{t> 0} |\int f(x- ty) d\sigma(y)| \] is bounded on \(L^p(\mathbb{R}^2)\) if and only if \(p> m\). This is a version of a result due to \textit{J. Bourgain} [J. Anal. Math. 47, 69-85 (1986; Zbl 0626.42012)] who proved the case \(m= 2\). In his proof the author relies on estimates of \textit{G. Mockenhaupt}, the reviewer and \textit{Ch. D. Sogge} [Ann. Math., II. Ser. 136, No. 1, 207-218 (1992; Zbl 0759.42016); J. Am. Math. Soc. 6, No. 1, 65-130 (1993; Zbl 0776.58037)] and suitable scaling arguments. He also discusses a variable coefficient analogue where the associated canonical relation is folding of order \(k\), with two-sided folds, as defined by \textit{D. H. Phong} and \textit{E. M. Stein} [Int. Math. Res. Not. 1991, No. 4, 49-60 (1991; Zbl 0761.46033)].
scientific article; zbMATH DE number 727838
scientific article