The T1 theorem for the Hilbert transform fails when p ≠ 2

Given 1 < p < ∞, p ≠ 2, we show that the T1 theorem for the Hilbert transform fails for Lp, despite holding for p = 2. More precisely, we construct a pair of locally finite positive Borel measures (σ, ω) that satisfy the two-tailed Ap condition, and satisfy both of the Lp-testing conditions for the Hilbert transform H, yet Hσ: Lp(σ) ↛ Lp(ω). In the opposite direction, the T1 theorem for the Hilbert transform for p = 2 was proved a decade ago in the two part paper [LaSaShUr3] and [Lac].