Hilbert transforms and maximal operators along planar vector fields

Abstract

In harmonic analysis, there is a conjecture (attributed to Zygmund) stating that the directional maximal operator along a Lipschitz planar vector field is weakly bounded on L^2. In this thesis, we present some recent progress towards this conjecture and its singular integral variant, which is that the directional Hilbert transform along a Lipschitz vector field is weakly bounded on L^2.<br /> In Chapter 1 we will first state these two conjectures and explain some partial progress that has been made. Afterwards we will state the main results of the present thesis.<br /> In Chapter 2 we will prove the L^2 boundedness of the directional Hilbert transform along planar measurable vector fields which are constant along suitable Lipschitz curves. Jones' beta numbers will play a crucial role when handling vector fields of the critical Lipschitz regularity.<br /> In Chapter 3 we will generalise the L^2 bounds in Chapter 2 to L^p for all p>3/2. To achieve this, we need to study a new paraproduct, which is indeed a one-parameter family of paraproducts, with each paraproduct living on one Lipschitz level curve of the vector field.<br /> In Chapter 4, by using the techniques presented in Chapter 2 and 3, we will provide a geometric proof of Bourgain's L^2 estimate of the maximal operator along analytic vector fields.