Let ( M , d ) be a metric space, equipped with a Borel measure μ satisfying suitable compatibility conditions. An amalgam A p q ( M ) is a space which looks locally like L p ( M ) but globally like L q ( M ) . We consider the case where the measure μ ( B ( x , ρ ) of the ball B ( x , ρ ) with centre x and radius ρ behaves like a polynomial in ρ , and consider the mapping properties between amalgams of kernel operators where the kernel ker K ( x , y ) behaves like d ( x , y ) - a when d ( x , y ) ≤ 1 and like d ( x , y ) - b when d ( x , y ) ≥ 1 . As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems for Laplace–Beltrami operators on Riemannian manifolds and for certain subelliptic operators on Lie groups of polynomial growth.