Riesz transforms on a class of non-doubling manifolds

We consider a class of manifolds M obtained by taking the connected sum of a finite number of N-dimensional Riemannian manifolds of the form (Formula presented.) where (Formula presented.) is a compact manifold, with the product metric. The case of greatest interest is when the Euclidean dimensions n_i are not all equal. This means that the ends have different ‘asymptotic dimension’, and implies that the Riemannian manifold (Formula presented.) is not a doubling space. We completely describe the range of exponents p for which the Riesz transform on (Formula presented.) is a bounded operator on (Formula presented.) Namely, under the assumption that each n_i is at least 3, we show that Riesz transform is of weak type (1, 1), is continuous on L^p for all (Formula presented.) and is unbounded on L^p otherwise. This generalizes results of the first-named author with Carron and Coulhon devoted to the doubling case of the connected sum of several copies of Euclidean space (Formula presented.) and of Carron concerning the Riesz transform on connected sums.