论文标题
在较低的RICCI边界下的1形式上的热流。功能不平等，光谱理论和热内核
Heat flow on 1-forms under lower Ricci bounds. Functional inequalities, spectral theory, and heat kernel
论文作者
论文摘要
我们在$ \ mathrm {rcd}（rcd}（k，\ infty）$ space $（m，\ mathsf {m，\ mathsf {d d} $（m，mathrm {rcd} $（m，mathrm {rcd} $）上$ k \ in \ boldsymbol {\ mathrm {r}} $。 We show Hess-Schrader-Uhlenbrock's inequality and, if $(M,\mathsf{d},\mathfrak{m})$ is also an $\mathrm{RCD}^*(K,N)$ space, $N\in(1,\infty)$, Bakry-Ledoux's inequality for $（\ Mathsf {h} _t）_ {t \ geq 0} $ W.R.T. $ l^2（m）$上的热流$（\ Mathsf {p} _t）_ {t \ geq 0} $。也讨论了这些估计的可变版本。结合一项对数sobolev不等式的研究，以$ 1 $形式的不等式，以前的不等式产生了$（\ mathsf {h} _t）_ {t \ geq 0} $，$ l^p $ -properties $（\ mathsf {h} _t）_ { 然后，我们在其发电机的光谱（Hodge laplacian $ \ smash {\vecδ} $）的频谱之间建立了明确的包含，该杂物是负功能性laplacian $-Δ$的杂物和schrödingeroperator $-Δ+k $。在$ \ mathrm {rcd}^*（k，n）$案件中，我们证明了$ \ smash {\vecΔ^{ - 1}} $如果$ m $是紧凑的，并且$ l^p $ -spectrum of $ \ smash smash smash {\ smash {\vecΔ 我们通过对$（\ Mathsf {h} _t）_ {t \ geq 0} $的加热内核进行适当的解释来终止。我们在没有任何局部紧凑或加倍的情况下显示出它的存在，并得出了它的基本估计和特性。
We study the canonical heat flow $(\mathsf{H}_t)_{t\geq 0}$ on the cotangent module $L^2(T^*M)$ over an $\mathrm{RCD}(K,\infty)$ space $(M,\mathsf{d},\mathfrak{m})$, $K\in\boldsymbol{\mathrm{R}}$. We show Hess-Schrader-Uhlenbrock's inequality and, if $(M,\mathsf{d},\mathfrak{m})$ is also an $\mathrm{RCD}^*(K,N)$ space, $N\in(1,\infty)$, Bakry-Ledoux's inequality for $(\mathsf{H}_t)_{t\geq 0}$ w.r.t. the heat flow $(\mathsf{P}_t)_{t\geq 0}$ on $L^2(M)$. Variable versions of these estimates are discussed as well. In conjunction with a study of logarithmic Sobolev inequalities for $1$-forms, the previous inequalities yield various $L^p$-properties of $(\mathsf{H}_t)_{t\geq 0}$, $p\in [1,\infty]$. Then we establish explicit inclusions between the spectrum of its generator, the Hodge Laplacian $\smash{\vecΔ}$, of the negative functional Laplacian $-Δ$, and of the Schrödinger operator $-Δ+K$. In the $\mathrm{RCD}^*(K,N)$ case, we prove compactness of $\smash{\vecΔ^{-1}}$ if $M$ is compact, and the independence of the $L^p$-spectrum of $\smash{\vecΔ}$ on $p \in [1,\infty]$ under a volume growth condition. We terminate by giving an appropriate interpretation of a heat kernel for $(\mathsf{H}_t)_{t\geq 0}$. We show its existence in full generality without any local compactness or doubling, and derive fundamental estimates and properties of it.
