Compact positive multilinear operators on Banach lattices

Let $1<p_1,\dots ,p_n<\mathrm{\infty },1\le q<\mathrm{\infty }$ be such that ${\sum }_{i=1}^n\frac{1}{p_i}<\frac{1}{q}$ and let ${\mu }_1,\dots ,{\mu }_n,\nu$ be arbitrary measures. Generalizing known linear and multilinear results, we prove that all positive $n$-linear operators from ${\mathrm{\ell }}_{p_1}\times \cdots \times {\mathrm{\ell }}_{p_n}$ to $L_q(\nu )$ and from $L_{p_1}({\mu }_1)\times \cdots \times L_{p_1}({\mu }_n)$ to ${\mathrm{\ell }}_q$ 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.