Generalized Ladner's Theorem Ladner's Theorem states that if P ≠ NP, then there is an infinite hierarchy of complexity classes strictly containing P and strictly contained in NP.  The proof uses the completeness of SAT under many-one reductions in NP.  The hierarchy contains complexity classes constructed by a kind of diagonalization, each containing some language to which the languages in the lower classes are not many-one reducible.

This motivates my question:


  Let C be a complexity class, and let D be a complexity class that strictly contains C.  If D contains languages that are complete for some notion of reduction, does there exist an infinite hierarchy of complexity classes between C and D, with respect to the reduction?


More specifically, I would like to know if there are results known for D = P and C = LOGCFL or C = NC, for an appropriate notion of reduction.



Ladner's paper already includes Theorem 7 for space-bounded classes C, as Kaveh pointed out in an answer.  In its strongest form this says: if NL ≠ NP then there is an infinite sequence of languages between NL and NP, of strictly increasing hardness.  This is slightly more general than the usual version (Theorem 1), which is conditional on P ≠ NP.  However, Ladner's paper only considers D = NP.
The answer to your question is "yes" for a wide variety of classes and reductions, including logspace reductions and the classes you mentioned, as is proved in these papers:


  H. Vollmer.  The gap-language technique revisited.  Computer Science Logic, Lecture Notes in Computer Science Vol. 533, pages 389-399, 1990.
  
  K. Regan and H. Vollmer.  Gap-languages and log-time complexity classes.  Theoretical Computer Science, 188(1-2):101-116, 1997.


(You can download gzipped postscript files of these papers here.)

The proofs follow the basic principle of Uwe Schöning's extension of Ladner's theorem:


  Uwe Schöning.  A uniform approach to obtain diagonal sets in complexity classes.  Theoretical Computer Science 18(1):95-103, 1982.


Schöning's proof has always been my favorite proof of Ladner's theorem -- it's both simple and general.