Lower bounds for the dyadic Hilbert transform

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.