  Suppose $X$ is the dual of a separable Banach space. The $w^*$ derivative
$\gamma'$ of a curve $\gamma:[a,b] \to X$ satisfies $\tfrac{d}{ds} \langle
\gamma(s),y\rangle|_{s=t} = \langle \gamma'(t),y\rangle$ for every $y \in Y$.
We prove that any convolution type singular integral operator associated with a
1-dimensional Calder\`{o}n-Zygmund kernel which is uniformly $L^2$-bounded on
lines must be bounded in $L^p$ along $\Gamma = \gamma([a,b])$ whenever $\Gamma$
is regular and $\gamma'$ is H\"{o}lder continuous.
