Let C be a complexity class, and let L be a language such that PC ⊆ PL. Then it’s natural (and easy to prove) that L is C-hard with respect to Cook reductions (polytime Turing reductions).
This fact is not (known to be) true for Karp (polytime many-one) reductions: for instance, we have PNP ⊆ PUNSAT, but UNSAT is not NP-hard under Karp reductions (unless NP = coNP). On the other hand, most of the times we use Karp reductions to order problems by hardness, since they seem to provide a finer structure.
Are there some “natural” or “interesting” sufficient conditions for PC ⊆ PL to imply that L is C-hard under Karp reductions?
If that implication held for all C and L, then the two notions of reducibility would coincide (see Tsuyoshi Ito’s comment below); I would be interested in knowing about any interesting class of problems where this happens, if any is known.