A New Inequality For The Hilbert Transform

Sakin Demir

Published 2020-09-12Version 1

Suppose that $\{a_j\}\in l^1$ has finite support. Then we prove that there is a constant $C$ such that $$\sum_{n=1}^\infty\sharp\left\{k\in\mathbb{Z}:\left| \sum_{i=-n}^n \! \raise{1ex}\hbox{${}'$} \frac{a_{k+i}}{i} \right| > \lambda\right\} \leq \frac{C}{\lambda}\sum_{i=-\infty}^\infty |a_i|$$ for all $\lambda>0$. We show as a corollary that one can use a transference argument to have an analogue result for the ergodic Hilbert transform.