I'm wondering what is known about the relation between time and memory for polynomial-time algorithms (which are necessarily also polynomial-space). In particular, I would like to learn what is known about less time vs. less memory, in the following sense:

Is it known whether (say) any language $L$ recognized by a polynomial-time algorithm $A$ which is polynomial-space $O(n^s)$ ($s>2$ say), can also be recognized by a polynomial-time algorithm $B$ which is polynomial-space $O(n^{s/2})$? Presumably any $B$ with less memory would need significantly more time, but can we restrict this extra time to polynomial-time?

So, recapping: is there for poly-time algorithms, some general way to compensate for less memory, in polynomial time? Answers and references greatly appreciated.

**update**: Dan Brumleve and Timothy Chow have given valuable insights and references in the comments and in this chat. It seems that time-space trade-offs like the one in this question are still a bridge too far.