Looking for examples of maps $\Omega^lS^{n+l}\to\Omega^kS^{m+k}$ with $l>k$ As the title says I am looking for examples of essential maps 
$\Omega^lS^{n+l}\to\Omega^kS^{m+k}$, with $l>k$, which may or may not be an iterated loop map. As an example, James fibration $S^n\to \Omega S^{n+1}\to \Omega S^{2n+1}$ provides a map $\Omega^2S^{2n+1}\to S^n$ which we may look at its iterated loops and get maps $\Omega^lS^{2n+1}\to\Omega^{l-2}S^n$.  
One source which I have thought of is James-Hopf maps, or maps which are somehow constructed in a combinatorial manner, but I don't see how to construct them. Let me add that the usual James-Hopf maps produce maps into infinite loop spaces, but I strictly want $0\leqslant k<l<+\infty$.
I will be grateful for any advise or hint on any helpful reference.
 A: Localize at a prime $p$. Then (this is from my comment above) there is a fiber sequence
$$J_{p^k−1}S^{2n} \to \Omega S^{2n+1} \to \Omega S^{2npk+1},$$
coming from the combinatorial James-Hopf map (see Proposition 5.2.2 of Neisendorfer's book). In particular, you obtain a map $\Omega^2 S^{2np^k+1} \to J_{p^k−1}S^{2n}$, so projecting onto the top cell of the partial James construction gives a $p$-local map $\Omega^2 S^{2np^k+1} \to S^{2n(p^k−1)}$. Similarly, the $p^k$th decomposition James-Hopf map gives a fiber sequence $$J_{p-1} S^{2n} \to \Omega S^{2n+1} \to \Omega S^{2np^k+1},$$
which gives maps $\Omega^2 S^{2np^k+1} \to S^{2n(p-1)}$.
I'll note that one can get a number of stable maps, using the Snaith splitting (see Ravenel's https://pdfs.semanticscholar.org/9747/cd5562647aeab35c02e06c0214c3fb1d99a3.pdf): there is a splitting
$$\Sigma^\infty \Omega^{m} S^{n} \simeq \bigvee_{k\geq 1} \Sigma^\infty D_{n,m}(k),$$
where $D_{n,m}(k) = S^{(n-m)k}\wedge_{\Sigma_k} C^+_{m,k}$; here, $C_{m,k}$ is the configuration space of subsets of size $k$ of $\mathbf{R}^m$, and $\Sigma_k$ acts on $S^{(n-m)k}$ by permuting the $k$ smash factors of $S^{n-m}$. We can project onto one of these factors, and then onto its top cell.
In order to answer your question, which lives in the unstable world, now needs to know if this desuspends in order to get unstable maps. For this, you should look at Cohen-May-Taylor's "Splitting of certain spaces CX"; there, they show that for $n<\infty$ and $k$ fixed, there is a map $\Sigma^N \Omega^m S^n\to \Sigma^N D_{n,m}(k)$ inducing Snaith's stable decomposition (these are also called James-Hopf maps; more generally, you can define them for any operad). In particular, we obtain maps $\Omega^m S^n \to \Omega^N \Sigma^N D_{n,m}(k)$. Composition with the projection to the top cell of $D_{n,m}(k)$ gives a map $\Omega^m S^n \to \Omega^N S^{N+?}$.
Let us now choose $n\geq m$. In order to get the condition in your question, we somehow need to ensure that $m>N$. This involves finding the minimum possible value for the integer $N$. I don't have time to work through this (so I don't know whether or not these resulting maps can possibly satisfy your condition, although I suspect they don't), but Cohen's "The unstable decomposition of $\Omega^2 \Sigma^2 X$ and its applications" has a very readable introduction, and refers to Cohen-Mahowald's "Unstable properties of $\Omega^n S^{n+k}$".
