端点流形上Poisson半群的双权不等式
The Two Weight Inequality for Poisson Semigroup on Manifold with Ends
摘要：
设$M = \mathbb R^m \sharp \mathcal R^n$是具有两个端点$\mathbb R^m \sharp \mathcal R^n$，$m > n \ge 3$的非二重流形.设$Δ$是$L^2(M)$上的Laplace-Beltrami算子，它是非负自伴的.则$Δ$及其平方根$\sqrtΔ$分别生成$L^2(M)$上的半群$e^{-tΔ}$和$e^{-t\sqrtΔ}$。给出了Poisson半群$e^{-t\sqrtΔ}$的两个权不等式在此情形下成立的检验条件.特别地，我们证明了对于测度$μ$在$M_{+}:=M\times (0,\infty)$上，$σ$在$M$上：$$ \|\mathsf{P}_σ(f)\|_{L^2(M_{+};μ)} \lesssim \|f\|_{L^2(M;σ)}, $$与$\mathsf{P}_σ(f)(x,t):= \int_M \mathsf{P}_t(x,y)f(y) \,dσ(y)$当且仅当检验条件对Poisson半群及其伴随成立.此外，在这些测试条件下，运营商的范数被证明是等价的最佳常数。
英文摘要：
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 $Δ$ be the Laplace--Beltrami operator which is non-negative self-adjoint on $L^2(M)$. Then $Δ$ and its square root $\sqrtΔ$ generate the semigroups $e^{-tΔ}$ and $e^{-t\sqrtΔ}$ on $L^2(M)$, respectively. We give testing conditions for the two weight inequality for the Poisson semigroup $e^{-t\sqrtΔ}$ to hold in this setting. In particular, we prove that for a measure $μ$ on $M_{+}:=M\times (0,\infty)$ and $σ$ on $M$: $$ \|\mathsf{P}_σ(f)\|_{L^2(M_{+};μ)} \lesssim \|f\|_{L^2(M;σ)}, $$ with $\mathsf{P}_σ(f)(x,t):= \int_M \mathsf{P}_t(x,y)f(y) \,dσ(y)$ 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.
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