  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 $p > 2$.
