On $\mathcal L$, $\mathcal{N\!L}$, $\mathcal L^2$, $\mathcal P$ and $\mathcal{N\!P}$
You can check the following paper:

Translational lemmas, polynomial time, and $ (\log n)^j$-space by Ronald V. Book (1976).

Figures 1 and 2 in the paper give the summary of what is known and what is unknown. 

I put Theorem 3.10 in the paper here:


$ DTIME(poly(n)) \neq DSPACE(poly(\log n)) $;
for every $ j \geq 1 $, $ DTIME(n^j) \neq DSPACE(poly(\log n)) $;
for every $ j,k \geq 1 $, $ DTIME(n^j) \neq DSPACE( (\log n)^k ) $.