On some singular integral operatorsclose to the Hilbert transform

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