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

We consider a class of non-doubling manifolds $\mathcal{M}$ that are the connected 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 of Hassell and the second author \cite{hs2019}, a particular decomposition of the resolvent operators $(\mathrm{\Delta }+k^2{)}^{-M}$, for $M\in {\mathbb{N}}^{\ast }$, will be used to demonstrate that the vertical square function operator $$Sf(x):={\left({\int }_0^{\mathrm{\infty }}\mid t\mathrm{\nabla }(I+t^2\mathrm{\Delta }{)}^{-M}f(x)\mid 2\frac{dt}{t}\right)}^{\frac{1}{2}}$$ 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 $\left\Vert f \right\Vert_{p} \lesssim \left\Vert S f\right\Vert_{p}$holdsfor$p \in (n_{min}',n_{min})$andthat$S$isunboundedfor$p \geq n_{min}$provided$2 M < n_{min}$. Similarly, for $M>1$, this method of proof will also be used to ascertain that the horizontal square function operator $$sf(x):={\left({\int }_0^{\mathrm{\infty }}{\mid t^2\mathrm{\Delta }(I+t^2\mathrm{\Delta }{)}^{-M}f(x)\mid }^2\frac{dt}{t}\right)}^{\frac{1}{2}}$$ is bounded on $L^p(\mathcal{M})$ for all $1<p<\mathrm{\infty }$ and weak-type $(1,1)$. Hence, for $p\ge n_{min}$, the vertical and horizontal square function operators are not equivalent and their corresponding Hardy spaces $H^p$ do not coincide.