Compact positive multilinear operators on Banach lattices

Let 1 < p₁, …, pₙ < ∞, 1≤ q < ∞ be such that ∑ᵢ₌₁ⁿ 1/pᵢ < 1/q and let μ₁, …, μₙ, ν be arbitrary measures. Generalizing known linear and multilinear results, we prove that all positive n-linear operators from p₁ × ⋯ × pₙ to Lq(ν) and from Lp₁(μ₁) × ⋯ × Lp₁(μₙ) to q are compact. This result, along with other related ones concerning free Banach lattices, shall emerge as consequences of some facts we prove about M-weakly compact multilinear operators on Banach lattices.