The Two Weight Inequality for Poisson Semigroup on Manifold with Ends

Let $M = \mathbb R^m \sharp \mathcal R^n$ be a non-doubling manifold with two ends $\mathbb R^m \sharp \mathcal R^n$, $m > n \ge 3$. Let $\Delta$ be the Laplace--Beltrami operator which is non-negative self-adjoint on $L^2(M)$. Then $\Delta$ and its square root $\sqrt{\Delta}$ generate the semigroups $e^{-t\Delta}$ and $e^{-t\sqrt{\Delta}}$ on $L^2(M)$, respectively. We give testing conditions for the two weight inequality for the Poisson semigroup $e^{-t\sqrt{\Delta}}$ to hold in this setting. In particular, we prove that for a measure $\mu$ on $M_{+}:=M\times (0,\infty)$ and $\sigma$ on $M$: $$\|\mathsf{P}_\sigma(f)\|_{L^2(M_{+};\mu)} \lesssim \|f\|_{L^2(M;\sigma)},$$ with $\mathsf{P}_\sigma(f)(x,t):= \int_M \mathsf{P}_t(x,y)f(y) \,d\sigma(y)$ if and only if testing conditions hold for the Poisson semigroup and its adjoint. Further, the norm of the operator is shown to be equivalent to the best constant in these testing conditions.