Lower bounds for the dyadic Hilbert transform

Abstract

In this paper, we seek lower bounds of the dyadic Hilbert transform (Haar shift) of the form ${\Vert Sf\Vert }_{L^2(K)}\ge C(I,K){\Vert f\Vert }_{L^2(I)}$ 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 $K⊊I$ and $K\cap I=\varnothing$ such bounds are only available under additional constraints on the derivative of $f$. In the later case, we establish a bound of the form ${\Vert Sf\Vert }_{L^2(K)}\ge C(I,K)|{⟨f⟩}_I|$ where ${⟨f⟩}_I$ is the mean of $f$ over $I$. This sheds new light on the similar problem for the usual Hilbert transform that we exploit.

arXiv commentv5: Only changes to the abstract so symbols display properly, Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6, Université Paul Sabatier \_ Cellule Mathdoc 2017