  Ordinary fracture functions, describing hadrons production in the deep
inelastic scattering target fragmentation region, are generalized to account
for the production of hadrons in arbitrary number, thus offering a renewed
framework for dealing with QCD initial state radiation. We also propose a new
jet-like observable which measures beam remnants and low-$p_{\perp}$ scattering
fragments and derive its QCD evolution equations by using Jet Calculus.
Possible implications for semi-inclusive deep inelastic scattering and
hadron-hadron reactions are shortly discussed.
