  The modern study of singular integral operators on curves in the plane began
in the 1970's. Since then, there has been a vast array of work done on the
boundedness of singular integral operators defined on lower dimensional sets in
Euclidean spaces. In recent years, mathematicians have attempted to push these
results into a more general metric setting particularly in the case of singular
integral operators defined on curves and graphs in Carnot groups.
  Suppose $X = Y^*$ for a separable Banach space $Y$. Any separable metric
space can be isometrically embedded in such a Banach space via the Kuratowski
embedding. Suppose $\Gamma = \gamma([a,b])$ is a curve in $X$ whose
$w^*$-derivative is H\"{o}lder continuous and bounded away from 0. 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 is $L^p$-bounded along $\Gamma$.
  We also prove a version of David's ``good lambda'' theorem for upper regular
measures on doubling metric spaces.
