We study the free Banach lattice \(\textrm{FBL}^{(p,\infty )}[E]\) 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 \(\textrm{FBL}^{(p,\infty )}[E]\) isomorphically. As a consequence, we obtain a functional representation for \(\textrm{FBL}^{(p,\infty )}[E]\) , answering a question from Oikhberg et al. [Free Banach lattices. J Eur Math Soc (JEMS), 2024]. 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 \(\textrm{FBL}^{(p,\infty )}[E]\) . Notably, we prove that the norm for \(\textrm{FBL}^{(p,\infty )}[E]\) 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 \(\textrm{FBL}^{(p,\infty )}[E]\) .
