Geodesic
The space $\mathbf {L_1(L_p)}$ is primary for 1 p ∞
Forum of Mathematics, Sigma, Tome 10 (2022)

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The classical Banach space $L_1(L_p)$ consists of measurable scalar functions f on the unit square for which

$ \begin{align*}\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx \infty.\end{align*} $

We show that $L_1(L_p) (1 p \infty )$ is primary, meaning that whenever $L_1(L_p) = E\oplus F$, where E and F are closed subspaces of $L_1(L_p)$, then either E or F is isomorphic to $L_1(L_p)$. More generally, we show that $L_1(X)$ is primary for a large class of rearrangement-invariant Banach function spaces. 
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     title = {The space $\mathbf {L_1(L_p)}$ is primary for 1 < p < \ensuremath{\infty}},
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Richard Lechner; Pavlos Motakis; Paul F.X. Müller; Thomas Schlumprecht. The space $\mathbf {L_1(L_p)}$ is primary for 1 < p < ∞. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.25

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