# Sharp endpoint $L^p$ estimates for the Schrodinger groups.

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a homogeneous space 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 $1\leq p_0 < 2$ and $m\geq 2$. In this paper we prove sharp endpoint $L^p$-Sobolev bounds for the Schr\"odinger groups $e^{itL}$ that for every $p\in (p_0, p'_0)$, there exists a constant $C=C(n,p)>0$ independent of $t$ such that for $s= n\big|{1/2}-{1/p}\big|$

\begin{eqnarray*}

\left\| (I+L)^{-{s}}e^{itL} f\right\|_{p} \leq C(1+|t|)^{s}\|f\|_{p}, \ \ \ \ \ t\in{\mathbb R} \end{eqnarray*} and

\begin{eqnarray*}

\left\| I_{s}(t)(L) f\right\|_{p} \leq C \|f\|_{p}, \ \ \ \ t\in {\mathbb R}\backslash\{0\}, \end{eqnarray*} where $I_{s}(t)(L)$ is the Riesz means for the Schr\"odinger group defined by $I_{s}(t)(L)=t^{-s} \int_0^t (t-\lambda)^{s-1} e^{-i\lambda L} d\lambda$ for $t>0$, and $I_{s}(t)(L)={\overline I}_{s}(-t)(L)$ for $t<0$. As a consequence, the above estimates hold for all $1<p<\infty$ when the heat kernel of $L$ satisfies a Gaussian upper bound. This extends the classical results due to Feffermann and Stein, Miyachi, and Sj\"ostrand for the Laplacian on the Euclidean spaces ${\mathbb R}^n$.

Publisher URL: http://arxiv.org/abs/1811.03326

DOI: arXiv:1811.03326v2

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