A Geometric Proof of Bourgain’s $$L^2$$ L 2 Estimate of Maximal Operators Along Analytic Vector Fields

Abstract

Bourgain (A remark on the maximal function associated to an analytic vector field. Analysis at Urbana, Cambridge University Press, Cambridge, 1989) proved that the maximal operator associated with an analytic vector field is bounded on \(L^2\). In the present paper, we give a geometric proof of Bourgain’s result by using the tools developed by Lacey and Li in (Trans Am Math Soc 358(9):4099–4117, 2006) and (Mem Am Math Soc 205 (965):viii+72, 2010).