Riesz potentials and amalgams

Let (M, d) be a metric space, equipped with a Borel measure mu 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 mu(B(x, rho) of the ball B(x, rho) with centre x and radius rho behaves like a polynomial in rho, 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) less than or equal to 1 and like d(x, y)(-b) when d(x, y) greater than or equal to 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.