Sorting in sub-quadratic time and logspace simultaneously
The following paper proves that any comparison-based sorting algorithm must satisfy $TS = \Omega(n^2)$, where $T$ is the running time and $S$ is the amount of read-write space used.

Allan Borodin, Michael J. Fischer, David G. Kirkpatrick, Nancy A. Lynch, Martin Tompa.  A Time-Space Tradeoff for Sorting on Non-Oblivious Machines.  Journal of Computer and System Sciences, vol 22 pp.351--364, 1981.

In particular, if $S = O(\log n)$, then $T = \Omega(n^2/\log n)$, i.e., if you have only logarithmic space, then the running time has to be nearly quadratic.