We establish convolution inequalities for Besov spaces B_p,q^s and Triebel–Lizorkin spaces F_p,q^s . As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces A^s_p,q, A ∈ {B, F}. Our results apply to a wide class of convolution semigroups including the Gauß–Weierstraß semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian (−∆)^m, and we can derive various caloric smoothing estimates.