The class of languages that can be certified in a small amount of space
(Note that X is not known to differ from NP.) ​ Yes - X is equal to PSPACE. ​ In fact, one gets that from just brute force, rather than needing Savitch's theorem.

You might also be interested in NL, which is only known to satisfy

L ​ $\subseteq$ ​ NL ​ $\subseteq$ ​ coNLOGLOGTIME-uniform SAC1 ​ ​ .

(Note that, to get a certificate characterization of NL, we probably must

"restrict the verifier to have one-way access to the certificate".)