Bellman函数与Bakry定理中的线性无量纲估计
Bellman function and linear dimension-free estimates in a theorem of Bakry
摘要：
通过使用显式Bellman函数，我们证明了与具有Bakry-Emery曲率从下方有界的加权黎曼流形$(M,μ_ϕ)$相关的拉普拉斯算子的双线性嵌入定理。作用于$L^p(M,μ_ϕ)$和$L^q(T^*M,μ_ϕ)$的笛卡儿积，$1/p+1/q=1$的嵌入，涉及独立于流形的维数和在$p$中线性的估计。因此，我们得到了相应的移位里斯变换的$L^p$范数的线性无量纲估计。我们所有的证明都是分析的。
英文摘要：
By using an explicit Bellman function, we prove a bilinear embedding theorem for the Laplacian associated with a weighted Riemannian manifold $(M,μ_ϕ)$ having the Bakry-Emery curvature bounded from below. The embedding, acting on the cartesian product of $L^p(M,μ_ϕ)$ and $L^q(T^*M,μ_ϕ)$, $1/p+1/q=1$, involves estimates which are independent of the dimension of the manifold and linear in $p$. As a consequence we obtain linear dimension-free estimates of the $L^p$ norms of the corresponding shifted Riesz transform. All our proofs are analytic.
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