We give an overview of the behavior of the classical Hilbert Transform H seen as an operator on Lp(R) and on weak-Lp(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 Lp(R) → Lp(R), for 1 < p < ∞.

Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from $H_p\left(ℝ\right)$ to $L_p\left(ℝ\right)$ (1/(α+1) < p < ∞) and is of weak type (1,1), where $H_p\left(ℝ\right)$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function $⨍\in L_1\left(ℝ\right)$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on $H_p\left(ℝ\right)$ whenever 1/(α+1) < p < ∞. Thus, in case $⨍\in H_p\left(ℝ\right)$, the Riesz means converge...