# complemented $\ell_p$ subspaces in $\ell_p$ sums of spaces

Note: By "subspace" I always mean an infinite-dimensional closed subspace.

Notation.

Let us write $$\oplus_p\ell_q^n:=\left(\bigoplus_{n=1}^\infty\ell_q^n\right)_{\ell_p}\;\;\;\text{ and }\;\;\;\oplus_0\ell_q^n:=\left(\bigoplus_{n=1}^\infty\ell_q^n\right)_{c_0}.$$ If $X$ is a Banach space, we will also write $$\ell_p(X)=\left\{(x_n)_{n=1}^\infty\in X^\mathbb{N}:\sum_{n=1}^\infty\|x_n\|_X^p<\infty\right\},$$ endowed with the norm $$\|(x_n)_{n=1}^\infty\|_{\ell_p(X)}:=\left(\sum_{n=1}^\infty\|x_n\|_X^p\right)^{1/p}.$$ We define $c_0(X)$ similarly.

Question 1a. Let $1\leq p<\infty$ and $1\leq q\leq\infty$. Does every subspace of $\oplus_p\ell_q^n$ admit a complemented subspace isomorphic to $\ell_p$? More precisely, if $Y$ is a susbpace of $\oplus_p\ell_q^n$, does there exist a subspace $Z$ of $Y$ which is isomorphic to $\ell_p$ and complemented in $\oplus_p\ell_q^n$?

Question 1b. Does every subspace of $\oplus_0\ell_q^n$ admit a complemented subspace isomorphic to $c_0$?

Question 2a. Let $1\leq p\neq q<\infty$. Does every subspace of $\ell_p(\ell_q)$ admit a complemented subspace isomorphic to either $\ell_p$ or $\ell_q$?

Question 2b. Does every subspace of $\ell_p(c_0)$ admit a complemented subspace isomorphic to $\ell_p$ or $c_0$?

I believe the answers to these questions are already known, in which case references would be much appreciated. I suspect something even stronger is true, e.g. that every basic sequence admits a complemented basic subsequence equivalent to $\ell_p$ (or $\ell_q$), but all I really need is a complemented subspace.

I asked this is stackexchange because I thought it would be easy and already known, but maybe it is more appropriate for MO.

Answer to Q1b is particularly easy as this space embeds into $c_0$, hence it is saturated with complemented copies of $c_0$.