The Hilbert transform along the extended evil ladder curve

Let(g)\tilde] (t)\tilde \gamma (t) be an even function extended by the evil ladder (tri-adic Cantor function) γ(t). Denote the curveG(t) = (t, (g)\tilde] (t))\Gamma (t) = (t, \tilde \gamma (t)) (-1≤1≤1). It is shown that the Hilbert transform along this curve Γ is bounded onL ²(ℝ²).