  In this paper the averaged weak (AWEC) and averaged null (ANEC) energy
conditions, together with uncertainty principle-type restrictions on negative
energy (``quantum inequalities''), are examined in the context of evaporating
black hole backgrounds in both two and four dimensions. In particular,
integrals over only half-geodesics are studied. We determine the regions of the
spacetime in which the averaged energy conditions are violated. In all cases
where these conditions fail, there appear to be quantum inequalities which
bound the magnitude and extent of the negative energy, and hence the degree of
the violation. The possible relevance of these results for the validity of
singularity theorems in evaporating black hole spacetimes is discussed.
