We develop the theory of Fourier multipliers acting on L p ( γ ) {L_p}(\gamma ) where γ \gamma is a Lipschitz curve of the form γ = { x + i g ( x ) } \gamma = \{ x + ig(x)\} with ‖ g ‖ ∞ > ∞ \left \| g\right \| _\infty > \infty and ‖ g ′ ‖ ∞ > ∞ \left \| g\prime \right \| _\infty > \infty . The aim is to better understand convolution singular integrals B B defined naturally on such curves by \[ B u ( z ) = p.v. ∫ γ φ ( z − ζ ) u ( ζ ) d ζ Bu(z) = {\text {p.v.}}\int _\gamma {\varphi (z - \zeta )u(\zeta )d\zeta } \] for almost all z ∈ γ z \in \gamma .