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Let $M = \mathbb R^m \sharp \mathcal R^n$ be a non-doubling manifold with twoends $\mathbb R^m \sharp \mathcal R^n$, $m > n \ge 3$. Let $\Delta$ be theLaplace--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 givetesting conditions for the two weight inequality for the Poisson semigroup$e^{-t\sqrt{\Delta}}$ to hold in this setting. In particular, we prove that fora 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 $\mathsf{P}_\sigma(f)(x,t):= \int_M \mathsf{P}_t(x,y)f(y) \,d\sigma(y)$ ifand 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 bestconstant in these testing conditions.

The Two Weight Inequality for Poisson Semigroup on Manifold with Ends