It is known that $L \subsetneq PH$? Is it known whether $Logspace$ is strictly contained in the polynomial time hierarchy ?
Are there oracles relative to which these classes are equal / distinct ?
This is equivalent to $LOGSPACE≠NP$ (which is obviously open). The proof of that equivalence relativizes (at least under the usual oracle models).
And there are oracles making $LOGSPACE = NP$ (the PSPACE-complete TQBF works) and making them not equal (the oracle separating P from NP works).