The Two Weight Inequality for Poisson Semigroup on Manifold with Ends

TL;DR

This paper establishes testing conditions for the two weight inequality of the Poisson semigroup on a non-doubling manifold with ends, extending classical results to a more complex geometric setting.

Contribution

It provides necessary and sufficient testing conditions for the two weight inequality of the Poisson semigroup on a non-doubling manifold with ends, a novel extension of classical harmonic analysis results.

Findings

Characterization of two weight inequality via testing conditions

Equivalence of operator norm and testing constant

Application to non-doubling manifolds with ends

Abstract

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.