L is as usual the complexity class DSPACE($\log n$), of languages decidable using a deterministic Turing machine using logarithmic workspace.
Is L closed under linear-time reductions?
It is possible that for some pair of languages Q and R, every linear-time reduction from Q to R requires more than logarithmic space. Up to linear space could be used while still remaining within linear time. So it isn't immediately obvious to me that L must be closed under linear-time reductions.
On the other hand, it is not clear what a counterexample would look like. One wants a language Q that can be linear-time many-one reduced to a language in L (for instance, to undirected graph st-connectivity), yet so that Q is not in L.