We study the free Banach lattice $$\textrm{FBL}^{(p,\infty )}[E]$$
FBL
(
p
,
∞
)
[
E
]
with upper p-estimates generated by a Banach space E. Using a classical result of Pisier on factorization through $$L^{p,\infty }(\mu )$$
L
p
,
∞
(
μ
)
together with a finite dimensional reduction, it is shown that the spaces $$\ell ^{p,\infty }(n)$$
ℓ
p
,
∞
(
n
)
witness the universal property of $$\textrm{FBL}^{(p,\infty )}[E]$$
FBL
(
p
,
∞
)
[
E
]
isomorphically. As a consequence, we obtain a functional representation for $$\textrm{FBL}^{(p,\infty )}[E]$$
FBL
(
p
,
∞
)
[
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 $$\textrm{FBL}^{(p,\infty )}[E]$$
FBL
(
p
,
∞
)
[
E
]
. Notably, we prove that the norm for $$\textrm{FBL}^{(p,\infty )}[E]$$
FBL
(
p
,
∞
)
[
E
]
cannot be isometrically witnessed by $$L^{p,\infty }(\mu )$$
L
p
,
∞
(
μ
)
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 $$\ell ^p$$
ℓ
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 $$\ell ^{p,\infty }$$
ℓ
p
,
∞
is not injective in the class of Banach lattices with upper p-estimates, elucidating one of many difficulties arising in the study of $$\textrm{FBL}^{(p,\infty )}[E]$$
FBL
(
p
,
∞
)
[
E
]
.