Riesz transforms on a class of non-doubling manifolds

Abstract

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