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 $(\Delta + k^{ 2})^{ -M}$, for $M \in \mathbb{ N}^{ *}$, will be 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 $\left\Vert f \right\Vert_{ p} \lesssim \left\Vert S f\right\Vert_{ p}$ holds for $p \in (n_{ min}',n_{ min})$ and that $S$ is unbounded for $p \geq n_{ min}$ provided $2 M < n_{ min}$. Similarly, for $M > 1$, this method of proof will also be used to ascertain that the horizontal square function operator $$ sf(x) := \left(\int^{ \infty}_{ 0} \left|t^{ 2}\Delta (I + t^{ 2} \Delta)^{ -M}f(x)\right|^{ 2} \, \frac{ dt}{ t}\right)^{ \frac{ 1}{ 2}} $$ is bounded 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 function operators are not equivalent and their corresponding Hardy spaces $H^p$ do not coincide.
Categories: math.AP math.FA