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 (Rⁿⁱ,δ)×(M_i,g), where M_i 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 M is not a doubling space. We completely describe the range of exponents p for which the Riesz transform on M is a bounded operator on L^p(M). 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 p∈(1,min_in_i), 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 R^N, and of Carron concerning the Riesz transform on connected sums.