  In this article, we consider a class of strictly hyperbolic triangular
systems involving a transport equation. Such systems are known to create
measure solutions for the initial value problem. Adding a stronger
transversality assumption on the fields, we are able to obtain solutions in
$L^\infty$ under optimal fractional $BV$ regularity of the initial data. Our
results show that the critical fractional regularity is $s=1/3$. We also
construct an initial data that is not in $BV^{1/3}$ but for which a blow-up in
$L^\infty$ occurs, proving the optimality of our results.
