Essential norm estimates for multilinear singular and fractional integrals

We derive lower two-weight estimates for the essential norm (measure of noncompactness) for multilinear Hilbert and Riesz transforms, and Riesz potential operators in Banach function lattices. As a corollary we have appropriate results in weighted Lebesgue spaces. From these statements we conclude that there is no $(m+1)$-tuple of weights $(v,w_1,\dots ,w_m)$ for which these operators are compact from $L_{w_1}^{p_1}\times \cdots \times L_{w_m}^{p_m}$ to $L_v^q$.