Riesz transforms on a class of non-doubling manifolds II

We consider a class of manifolds $\mathcal{M}$ obtained by taking the connected sum of a finite number of $N$-dimensional Riemannian manifolds of the form $(\mathbb{R}^{n_i}, \delta) \times (\mathcal{M}_i, g)$, where $\mathcal{M}_i$ 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 $\mathcal{M}$ is not a doubling space. In the first paper in this series, by the first and third authors, we considered the case where each $n_i$ is least $3$. In the present paper, we assume that one of the $n_i$ is equal to $2$, which is a special and particularly interesting case. Our approach is to construct the low energy resolvent and determine the asymptotics of the resolvent kernel as the energy tends to zero. We show that the resolvent kernel $(\Delta + k^2)^{-1}$ on $\mathcal{M}$ has an expansion in powers of $1/\log (1/k)$ as $k \to 0$, which is significantly different from the case where all $n_i$ are at least 3, in which case the expansion is in powers of $k$. We express the Riesz transform in terms of the resolvent to show that it is bounded on $L^p(\mathcal{M})$ for $1 < p \leq 2$, and unbounded for all $p > 2$.