# 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}
}
• Published 8 November 2018
• Mathematics
• Mathematische Annalen
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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## References

SHOWING 1-10 OF 51 REFERENCES

• Mathematics
Revista Matemática Iberoamericana
• 2019
We prove an $L^{p}$ estimate $$\|e^{-itL} \varphi(L)f\|_{p}\lesssim (1+|t|)^s\|f\|_p, \qquad t\in \mathbb{R}, \qquad s=n\left|\frac{1}{2}-\frac{1}{p}\right|$$ for the Schr\"odinger group
• Mathematics
• 1995
$L^{p}(R^{d})$ , where $H=-\Delta+V(x)$ is a Schr\"odinger operator defined primarily as a self-adjoint operator in $L^{2}(R^{d})$ . For $H_{0}=-\Delta$ , mapping properties of $f(H_{0})$ between
Hormander’s famous Fourier multiplier theorem ensures the L_p-boundedness of F(-\Delta _{\mathbb{R}} D) whenever F\in \mathcal{H}(s) for some s>\frac{D}{2}, where we denote by \mathcal{H} (s) the set
• Mathematics
• 2012
Let $X$ be a space of homogeneous type and let $L$ be an injective, non-negative, self-adjoint operator on $L^2(X)$ such that the semigroup generated by $-L$ fulfills Davies-Gaffney estimates of
• Mathematics
• 2003
We modify Hormander's well-known weak type (1,1) condition for integral operators (in a weakened version due to Duong and McIn- tosh) and present a weak type (p,p) condition for arbitrary operators.
We prove that for Cauchy data in L1(Rn), the solution of a Schrodinger evolution equation with constant coefficients of order 2m is uniformly bounded for t £ 0, with bound (1 4|t|-c), where c is an
Let H = \L + V be a general Schrödinger operator on R" (v~> 1), where A is the Laplace differential operator and V is a potential function on which we assume minimal hypotheses of growth and
Laplace operator and the heat equation in $\mathbb{R}^n$ Function spaces in $\mathbb{R}^n$ Laplace operator on a Riemannian manifold Laplace operator and heat equation in $L^{2}(M)$ Weak maximum
• Mathematics
• 2004
We modify Hörmander’s well-known weak type (1,1) condition for integral operators (in a weakened version due to Duong and McIntosh) and present a weak type (p, p) condition for arbitrary operators.