We consider operators of the form $\left(\Omega f\right)\left(y\right)={ʃ}_{-\infty }^{\infty }\Omega (y,u)f\left(u\right)du$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h\in L^{\infty }$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space ${Ḃ}_1^{0,1}$ (= B) into itself. In particular, all operators with $h\left(y\right)=e^{{i\left|y\right|}^a}$, a > 0, a ≠ 1, map B into itself.

Let m: ℝ → ℝ be a function of bounded variation. We prove the $L^p\left(ℝ\right)$-boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by $T_{m}f\left(x\right)=p.v.\int k(x-y)m(x+y)f\left(y\right)dy$ where $k\left(x\right)={\sum }_{j\in \mathbb{Z}}{2}^j{\phi }_j\left(2^{j}x\right)$ for a family of functions ${{\phi }_j}_{j\in \mathbb{Z}}$ satisfying conditions (1.1)-(1.3) given below.

Using classical results on conjugate functions, we give very short proofs of theorems of Erdös–Turán and Blatt concerning the angular distribution of the roots of polynomials. Then we study some examples.

The Stein-Weiss theorem that the distribution function of the Hilbert transform of the characteristic function of E depends only on the measure of E is generalized to the ergodic Hilbert transform.

We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1,0)}{(}^2)$, $H^{(0,1)}{(}^2)$, or $H^{(1,1)}{(}^2)$. We prove that the maximal Fejér operator is bounded from $H^{(1,0)}{(}^2)$ or $H^{(0,1)}{(}^2)$ into weak-$L^1{(}^2)$, and also bounded from $H^{(1,1)}{(}^2)$ into $L^1{(}^2)$. These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces $L^{1}lo{g}^{+}L{(}^2)$, $L^1{\left(lo{g}^{+}L\right)}^2{(}^2)$, and $L^{\mu }{(}^2)$ with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate two conjectures....

Let $Tf\left(x\right)=p.v.{ʃ}_{ℝ¹}{e}^{iP(x-y)}f\left(y\right)/(x-y)dy$, where P is a real polynomial on ℝ. It is proved that T is bounded on the weighted H¹(wdx) space with w ∈ A₁.

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 &lt; p &lt; ∞.