A geometric proof of Bourgain’s \(L^2\) estimate of maximal operators along analytic vector fields. (English) Zbl 1377.42019
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
\[
M_{v, \varepsilon_0}f(x) := \sup_{\varepsilon < \varepsilon_0} \frac{1}{2\varepsilon} \int_{-\varepsilon}^{\varepsilon} | f(x+tv(x))| \, dt.
\]
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 E. M. Stein and 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 M. T. Lacey and 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)].
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 E. M. Stein and 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 M. T. Lacey and 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)].
Reviewer: Yasuo Komori-Furuya (Kanagawa)
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
References:
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| [3] | Bourgain, J.: A remark on the maximal function associated to an analytic vector field. Analysis at Urbana, vol. I (Urbana, IL, 1986-1987), pp. 111-132, London Mathematical Society Lecture Note Series, 137, Cambridge University Press, Cambridge (1989) · Zbl 1285.42014 |
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| [6] | Lacey, M., Li, X.: Maximal theorems for the directional Hilbert transform on the plane. Trans. Am. Math. Soc. 358(9), 4099-4117 (2006) · Zbl 1095.42010 · doi:10.1090/S0002-9947-06-03869-4 |
| [7] | Lacey, M., Li, X.: On a conjecture of E.M. Stein on the Hilbert transform on vector fields. Mem. Am. Math. Soc. 205 (965), viii+72 (2010). ISBN: 978-0-8218-4540-0 · Zbl 1190.42005 |
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