For hyperbolic systems of one-dimensional conservation laws, the theory of existence of global weak enropy solutions for the initial value problem is generally performed for small initial data in $BV$ or, rarely, in $L^\infty$. In this presentation, the intermediate spaces $BV^s$, 0 < s < 1, are used, $BV=BV^1$ and $L^\infty=BV^0$. The $2\times 2$ strictly hyperbolic systems with genuinely nonlinear or lineraly degenerate fields are considered, so there are three cases. The first case, a full genuinely nonlinear system, is already known since Glimm-Lax 1970 and Bianchini-Colombo-Monti 2010, for small $L^\infty$ initial data there is a smoothing in $BV$ like for the scalar case with an uniformly convex flux, Lax and Oleinik 1957. For the second case, a full linearly degenerate system, the existence holds in any $BV^s$. For the third case, the main part of the talk, one field is genuinely nonlinear and the other one is linearly degnerate, a critical fractional regularity $s=1/3$ appears. Optimality of $s=1/3$ is proven on a triangular system.