The class of $2 \times 2$ nonlinear hyperbolic systems with one genuinely nonlinear field and one linearly degenerate field are considered. Existence of global weak solutions for small initial data in fractional $BV$ spaces $BV^s$ is proved. The exponent $s$ is related to the usual fractional Sobolev derivative. Riemann invariants $w$ and $z$ corresponding respectively to the genuinely nonlinear component and to the linearly degenerate component play different key roles in this work. We obtain the existence of a global weak solution provided that the initial data written in Riemann coordinates $(w_0 , z_0)$ are small in $BV^s \times L^\infty , 1/3 ≤ s < 1$. The restriction on the exponent $s$ is due to a fundamental result of P.D. Lax, the variation of the Riemann invariant $z$ on the Lax shock curve depends in a cubic way of the variation of the other Riemann invariant $w$.