We introduce and characterize a new class of subspaces \(V_p(m)\) of \(L^1(m)\) associated with a vector measure m, consisting of functions whose associated measure has finite p-variation. We establish that these subspaces are maximal with respect to operators acting on Banach function spaces with a lower p-estimate, meaning they are the largest subspaces of integrable functions that retain this property. Furthermore, we demonstrate that these spaces facilitate the study of restrictions of operators to subspaces where the lower p-estimate is preserved. We prove that if a vector measure induces a (p, 1)-summing integration operator, then it has finite p-variation, and the space \(L^1(m)\) coincides with the subspace of integrable functions satisfying a lower p-estimate \(V_p(m)\) . As an application, we show that the optimal domain of positive operators on (p, 1)-concave Banach lattices, operators on Banach spaces of cotype p, and operators on 2-concave Banach lattices necessarily possess a lower p-estimate or lower 2-estimate, respectively.
