Banach lattices with upper $p$-estimates: free and injective objects

We study the free Banach lattice $FBL^{(p,\infty)}[E]$ with upper $p$-estimates generated by a Banach space $E$. Using a classical result of Pisier on factorization through $L^{p,\infty}(\mu)$ together with a finite dimensional reduction, it is shown that the spaces $\ell^{p,\infty}(n)$ witness the universal property of $FBL^{(p,\infty)}[E]$ isomorphically. As a consequence, we obtain a functional representation for $FBL^{(p,\infty)}[E]$. More generally, our proof allows us to identify the norm of any free Banach lattice over $E$ associated with a rearrangement invariant function space. After obtaining the above functional representation, we take the first steps towards analyzing the fine structure of $FBL^{(p,\infty)}[E]$. Notably, we prove that the norm for $FBL^{(p,\infty)}[E]$ cannot be isometrically witnessed by $L^{p,\infty}(\mu)$ and settle the question of characterizing when an embedding between Banach spaces extends to a lattice embedding between the corresponding free Banach lattices with upper $p$-estimates. To prove this latter result, we introduce a novel push-out argument, which when combined with the injectivity of $\ell^p$ allows us to give an alternative proof of the subspace problem for free $p$-convex Banach lattices. On the other hand, we prove that $\ell^{p,\infty}$ is not injective in the class of Banach lattices with upper $p$-estimates, elucidating one of many difficulties arising in the study of $FBL^{(p,\infty)}[E]$.