Sharp endpoint $$L^p$$ estimates for Schrödinger groups

@article{Chen2018SharpE,
  title={Sharp endpoint \$\$L^p\$\$ estimates for Schr{\"o}dinger groups},
  author={Peng Chen and Xuan Thinh Duong and Ji Li and Lixin Yan},
  journal={Mathematische Annalen},
  year={2018}
}
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 $1\leq p_0 0$ 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= n\big|{1\over 2}-{1\over p}\big|. \end{eqnarray*} As a consequence, the above… 

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It is shown that the kernel of the jats:inline-formula satisfies paper shows that the nonnegative self-adjoint operator of L, the non-negative doubling order of X, satisfies the paper's paper requirements.

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