What is know in general about the maps $\Omega^rS^n\rightarrow\Omega^sS^m$ between loop spaces of Spheres, or, perhaps to phrase it better, the groups $[\Omega^rS^n,\Omega^sS^m]$ for various values of $r$, $s$, $n$, $m$.
There are some obvious cases such as $r=s=0$, or $r=s=1$ in which case the James splitting gives a lot of useful information. Rationalization also goes a long way towards solving the problem.
I've gathered it's a pretty difficult problem, but are there any special cases that are known? The simplest non-trivial case would probably be $[\Omega^2S^n,\Omega^2S^m]$.