Let $d(\textbf{w},p)$, $1\leq p<\infty$, denote the Lorentz sequence space, where $\textbf{w}=(w_n)_{n=1}^\infty\in c_0\setminus\ell_1$ is a normalized decreasing weight.

Is there very much known about the complemented subspaces of $d(\textbf{w},p)$? In general (i.e., without any restrictions on $\textbf{w}$ or $p$), I can only find two: $\ell_p$ and $d(\textbf{w},p)$ itself.

**Question 1.** What complemented subspaces of $d(\textbf{w},p)$, besides $\ell_p$ and itself, are already known?

There has been found a third distinct complemented subspace in case $\textbf{w}$ satisfies a certain (NUC) condition, namely that $\inf_k(\sum_{i=1}^{2k}w_n)/(\sum_{i=1}^kw_n)=1$. In this case $d(\textbf{w},p)$ contains a 1-complemented subspace isomorphic to $\oplus_p(\ell_\infty^n)$.

But can we find other complemented subspaces in the general case?

The obvious thing to try first is to look for constant-coefficient block basic sequences. If the length of the blocks is bounded then they span another copy of $d(\textbf{w},p)$. However, if their lengths tend to infinity, then their span will be distinct from $d(\textbf{w},p)$. In this case, the problem is to show that the resulting space is not isomorphic to $\ell_p$.

This is pretty easy when $p=1$ or $p=2$. In these cases, since $\ell_1$ and $\ell_2$ each admit a unique unconditional basis, it's sufficient to make sure that the constant-coefficient blocks in $d(\textbf{w},p)$ aren't equivalent to those bases. This can be done by taking constant-coefficient block sequences $(d_i^{(k)})_{i=1}^{N_k}$ of fixed length $k$, choosing $N_k$ sufficiently large that they fail increasingly badly to dominate $\ell_p^{N_k}$. Then glue those sequences together for all $k\in\mathbb{N}$, pushing them out far enough so that they're disjoint.

I suspect that this will work for all $1\leq p<\infty$, but proving it is not as straightforward as the above. However, if it could be shown that $\ell_p$ does not contain uniformly complemented copies of $\text{span}(d_n)_{n=1}^N$, where $(d_n)_{n=1}^\infty$ is the basis for a Lorentz sequence space, then that would do the trick. Thus:

**Question 2.** Does $\ell_p$ contain uniformly complemented copies of $\text{span}(d_n)_{n=1}^N$?

Thanks guys!