We study inequalities of the type j R Rd Φ(K*f )| ≲ ∥f ∥p L1(Rd ) , where the kernel K is homogeneous of order α - d and possibly vector-valued, the function Φ is positively p-homogeneous, and p = d/(d - α). Under mild regularity assumptions on K and Φ, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions f . © 2024 Scuola Normale Superiore. All rights reserved.