Remarks on the Hilbert transform and on some families of multiplier operators related to it

We give an overview of the behavior of the classical Hilbert Transform $H$ seen as an operator on $L^p(\mathbb{R})$ and on weak-$L^p(\mathbb{R})$, then we consider other operators related to $H$. In particular, we discuss various versions of Discrete Hilbert Transform and Fourier multipliers periodized in frequency, giving some partial results and stating some conjectures about their sharp bounds $L^p(\mathbb{R})\rightarrow L^p(\mathbb{R})$, for $1 < p < \infty$.