A geometric proof of Bourgain's \(L^2\) estimate of maximal operators along analytic vector fields (Q2012940)

Let \(\Omega \in {\mathbb R}^2\) be a bounded open set and \(v: \Omega' \to S^1\) be a unit vector field defined on a neighborhood \(\Omega'\) of the closure \(\overline{\Omega}\). For a fixed small positive number \(\varepsilon_0>0\), define the maximal operator associated with the vector field \(v\) truncated at the scale \(\varepsilon_0\) by \[ M_{v, \varepsilon_0}f(x) := \sup_{\varepsilon < \varepsilon_0} \frac{1}{2\varepsilon} \int_{-\varepsilon}^{\varepsilon} | f(x+tv(x))| \, dt. \] \textit{J. Bourgain} [Lond. Math. Soc. Lect. Note Ser. 137, 111--132 (1989; Zbl 0692.42006)]. proved that if \(v\) is real analytic on \(\Omega'\) then \(M_{v, \varepsilon_0}\) is bounded on \(L^2(\Omega)\). The \(L^p\) bounds and its singular integral variant \[ H_{v, \varepsilon_0}f(x) := \sup_{\varepsilon < \varepsilon_0} \int_{-\varepsilon}^{\varepsilon} | f(x+tv(x))| \, \frac{dt}{t} \] were obtained by \textit{E. M. Stein} and \textit{B. Street} [Adv. Math. 229, No. 4, 2210--2238 (2012; Zbl 1242.42010)] via a very different method. The author gives a geometric proof of Bourgain's results by using tools developed by \textit{M. T. Lacey} and \textit{X. Li} [Trans. Am. Math. Soc. 358, No. 9, 4099--4117 (2006; Zbl 1095.42010); Mem. Am. Math. Soc. 965, i-viii, 72 p. (2010; Zbl 1190.42005)].