  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 $, $1/3 \leq s<1$. 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 $w$.
