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We consider a class of non-doubling manifolds $\mathcal{M}$ that are theconnected sum of a finite number of $N$-dimensional manifolds of the form$\mathbb{R}^{n_{i}} \times \mathcal{M}_{i}$. Following on from the work ofHassell and the second author \cite{hs2019}, a particular decomposition of theresolvent operators $(\Delta + k^{2})^{-M}$, for $M \in \mathbb{N}^{*}$, willbe used to demonstrate that the vertical square function operator $$ Sf(x) :=\left( \int^{\infty}_{0} \left|t \nabla (I + t^{2}  \Delta)^{-M}f(x)\right|{2} \frac{dt}{t}\right)^{\frac{1}{2}} $$ is bounded on$L^{p}(\mathcal{M})$ for $1 < p < n_{min} = \min_{i} n_{i}$ and weak-type$(1,1)$. In addition, it will be proved that the reverse inequality $\left\Vertf \right\Vert_{p} \lesssim \left\Vert S f\right\Vert_{p}$ holds for $p \in(n_{min}',n_{min})$ and that $S$ is unbounded for $p \geq n_{min}$ provided $2M < n_{min}$.  Similarly, for $M > 1$, this method of proof will also be used to ascertainthat the horizontal square function operator $$ sf(x) :=\left(\int^{\infty}_{0} \left|t^{2}\Delta (I + t^{2}  \Delta)^{-M}f(x)\right|^{2} \, \frac{dt}{t}\right)^{\frac{1}{2}} $$ isbounded on $L^{p}(\mathcal{M})$ for all $1 < p < \infty$ and weak-type $(1,1)$.Hence, for $p \geq n_{min}$, the vertical and horizontal square functionoperators are not equivalent and their corresponding Hardy spaces $H^p$ do notcoincide.

Vertical and horizontal Square Functions on a Class of Non-Doubling Manifolds