3 years ago

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

Peng Chen, Xuan Thinh Duong, Ji Li, Lixin Yan

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

You might also like
Discover & Discuss Important Research

Keeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free.

  • Download from Google Play
  • Download from App Store
  • Download from AppInChina

Researcher displays publicly available abstracts and doesn’t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article.