Nathan Wagner will give a talk at 03/05/2024.

In these two talks, we will discuss the dyadic Hilbert transform, which is a useful model of its continuous counterpart and the prototypical example of a so-called “Haar-shift”. After discussing some background and motivation in the Lebesgue measure case, we will turn to the situation where the $L^2$ Haar functions are defined with respect to a locally finite Borel measure $\mu$, which may not satisfy the dyadic doubling condition. In this case, Lopez-Sanchez, Martell, and Parcet identified a weak regularity condition on the measure $\mu$ which characterizes weak-type and $L^p$ estimates for the dyadic Hilbert transform. I then will discuss joint work with Jose Conde-Alonso and Jill Pipher, where we obtain a domination of the dyadic Hilbert transform by a modified sparse form. Sparse domination is a common feature of modern harmonic analysis research, and often allows for quantitatively sharp weighted estimates. As an application, we characterize the class of weights where the dyadic Hilbert transform and related operators are bounded. A surprising novelty is that the usual (dyadic) Muckenhoupt $A_2$ condition is necessary, but no longer sufficient in the non-doubling setting, and our modified weight condition reflects the “complexity” of the underlying Haar shift.