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In this paper the averaged weak (AWEC) and averaged null (ANEC) energyconditions, together with uncertainty principle-type restrictions on negativeenergy (``quantum inequalities''), are examined in the context of evaporatingblack hole backgrounds in both two and four dimensions. In particular,integrals over only half-geodesics are studied. We determine the regions of thespacetime in which the averaged energy conditions are violated. In all caseswhere these conditions fail, there appear to be quantum inequalities whichbound the magnitude and extent of the negative energy, and hence the degree ofthe violation. The possible relevance of these results for the validity ofsingularity theorems in evaporating black hole spacetimes is discussed.Comment: Sections 2.1 and 2.2 have been revised and some erroneous statements  corrected. The main conclusions and the figures are unchanged. 27 pp, plain  Latex, 3 figures available upon reques

Averaged energy conditions and evaporating black holes