Lower bounds for the dyadic Hilbert transform

TL;DR

This paper investigates lower bounds for the dyadic Hilbert transform, establishing conditions under which such bounds exist and providing new insights that relate to the classical Hilbert transform.

Contribution

It derives new lower bounds for the dyadic Hilbert transform depending on the relative position of intervals, extending understanding of its behavior.

Findings

Bounds exist when one interval is contained in the other

Additional constraints are needed when intervals are disjoint or partially overlapping

A new bound involving the mean of the function over the interval is established

Abstract

In this paper, we seek lower bounds of the dyadic Hilbert transform (Haar shift) of the form $\left\Vert S f\right\Vert_{L^2(K)}\geq C(I,K)\left\Vert f\right\Vert_{L^2(I)}$ where $I$ and $K$ are two dyadic intervals and $f$ supported in $I$. If $I\subset K$ such bound exist while in the other cases $K\subsetneq I$ and $K\cap I=\emptyset$ such bounds are only available under additional constraints on the derivative of $f$. In the later case, we establish a bound of the form $\left\Vert S f\right\Vert_{L^2(K)}\geq C(I,K)|\left\langle f\right\rangle_I|$ where $\left\langle f\right\rangle_I$ is the mean of $f$ over $I$. This sheds new light on the similar problem for the usual Hilbert transform that we exploit.