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

Abstract

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 $(\hat{I}”+k^2{)}^{-M}$, for $M\in {\mathbb{N}}^{\ast }$, will be used to demonstrate that the vertical square function operator $$Sf(x):={\left({\int }_0^{\infty }\left|t\nabla (I+t^2\hat{I}”{)}^{-M}f(x)\right|2\frac{dt}{t}\right)}^{\frac{1}{2}}$$ 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 ${\Vert f\Vert }_p\lesssim {\Vert Sf\Vert }_p$ holds for $p\in (n_{min}^{\prime },n_{min})$ and that $S$ is unbounded for $p\ge n_{min}$ provided $2M<n_{min}$. Similarly, for $M>1$, this method of proof will also be used to ascertain that the horizontal square function operator $$sf(x):={\left({\int }_0^{\infty }{\left|t^2\hat{I}”(I+t^2\hat{I}”{)}^{-M}f(x)\right|}^2\frac{dt}{t}\right)}^{\frac{1}{2}}$$ is bounded on $L^p(\mathcal{M})$ for all $1<p<\infty$ and weak-type $(1,1)$. Hence, for $p\ge n_{min}$, the vertical and horizontal square function operators are not equivalent and their corresponding Hardy spaces $H^p$ do not coincide.