薛定谔群的尖锐端点$L^p$估计
Sharp endpoint $L^p$ estimates for Schrödinger groups
摘要：
设$L$是作用在$L^2(X)$上的非负自伴算子，其中$X$是具有维数$n$的齐型空间。设热算子$e^{-tL}$对某些$1\leq p_0 < 2$满足$m$阶的广义Gaussian $(p_0, p'_0)$ -估计.本文证明了薛定谔群$e^{itL}$的{PDMATHABMUsharp}端点$L^p$-Sobolev界，即对每个$p\in (p_0, p'_0)$，存在一个与$t$无关的常数$C=C(n,p)>0$，使得\begin{eqnarray*} \left\| (I+L)^{-{s}}e^{itL} f\right\|_{p} \leq C(1+|t|)^{s}\|f\|_{p}, \ \ \ t\in{\mathbb R}, \ \ \ s\geq n\big|{1\over 2}-{1\over p}\big|. \end{eqnarray*}.因此，当$L$的热核满足高斯上界时，上述估计对所有$1<p<\infty$成立.这扩展了经典的结果，由于费曼和斯坦因，和Miyachi的拉普拉斯算子的欧几里德空间${\mathbb R}^n$。我们也给出了一个应用程序，以获得PDMATHAMV-有界的Riesz平均的薛定谔方程的解决方案的端点估计。
英文摘要：
Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat operator $e^{-tL}$ satisfies the generalized Gaussian $(p_0, p'_0)$-estimates of order $m$ for some $1\leq p_0 < 2$. In this paper we prove {\it sharp} endpoint $L^p$-Sobolev bound for the Schrödinger group $e^{itL}$, that is for every $p\in (p_0, p'_0)$ there exists a constant $C=C(n,p)>0$ independent of $t$ such that \begin{eqnarray*} \left\| (I+L)^{-{s}}e^{itL} f\right\|_{p} \leq C(1+|t|)^{s}\|f\|_{p}, \ \ \ t\in{\mathbb R}, \ \ \ s\geq n\big|{1\over 2}-{1\over p}\big|. \end{eqnarray*} As a consequence, the above estimate holds for all $1<p<\infty$ when the heat kernel of $L$ satisfies a Gaussian upper bound. This extends classical results due to Feffermann and Stein, and Miyachi for the Laplacian on the Euclidean spaces ${\mathbb R}^n$. We also give an application to obtain an endpoint estimate for $L^p$-boundedness of the Riesz means of the solutions of the Schrödinger equations.
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