  In this paper, we consider $2 \times 2$ hyperbolic systems of conservation
laws in one space dimension with characteristic fields satisfying a condition
that encompasses genuine nonlinearity and linear degeneracy as well as
intermediate cases, namely, with standard notations, $r_i\cdot \nabla \lambda_i
\geq 0$.
  We prove the existence of entropy solutions in the fractional $BV$ spaces
${W}_p(\mathbb{R})$ of functions of bounded $p$-variation, $p \in
[1,\frac{3}{2}]$, for small initial data.
