# 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…

## 11 Citations

### Sharp endpoint estimates for Schr\"odinger groups on Hardy spaces.

- Mathematics
- 2019

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 kernel of $L$ satisfies a Gaussian upper bound.…

### The Schrödinger equation in $L^{p}$ spaces for operators with heat kernel satisfying Poisson type bounds

- MathematicsJournal of the Mathematical Society of Japan
- 2021

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 $n$. In this paper, we study sharp endpoint $L^p$-Sobolev estimates for…

### Weak Type (1,1) Bounds for Schrödinger Groups

- MathematicsMichigan Mathematical Journal
- 2021

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 kernel of $L$ satisfies a Gaussian upper bound.…

### $L^{p}$ and $\mathcal{H}^{p}_{FIO}$ regularity for wave equations with rough coefficients, Part I.

- Mathematics
- 2020

We consider wave equations with time-independent coefficients that have $C^{1,1}$ regularity in space. We show that, in two space dimensions, the standard inhomogeneous initial value problem for the…

### On Boundedness of Oscillating Multipliers on Stratified Lie Groups

- Materials ScienceThe Journal of Geometric Analysis
- 2022

In this paper, we study the oscillating spectral multipliers associated with the sub-Laplacian L on an arbitrary stratified Lie group G. We prove the boundedness of the operators…

### Sharp Estimates for Schrödinger Groups on Hardy Spaces for 0<p≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<p\le

- Materials ScienceJournal of Fourier Analysis and Applications
- 2022

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.

### Sharp endpoint $L_p$ estimates of quantum Schr\"{o}dinger groups

- Mathematics
- 2023

A bstract . In this article, we establish sharp endpoint L p estimates of Schr¨odinger groups on general measure spaces which may not be equipped with good metrics but admit submarkovian semigroups…

### Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces

- MathematicsMathematics
- 2022

Assume that (X,d,μ) is a metric measure space that satisfies a Q-doubling condition with Q>1 and supports an L2-Poincaré inequality. Let 𝓛 be a nonnegative operator generalized by a Dirichlet form E…

### REGULARITY FOR WAVE EQUATIONS WITH ROUGH COEFFICIENTS

- Mathematics
- 2022

We consider wave equations with time-independent coefficients that have C regularity in space. We show that, for nontrivial ranges of p and s, the standard inhomogeneous initial value problem for the…

## References

SHOWING 1-10 OF 51 REFERENCES

### Sharp $L^p$ estimates for Schrödinger groups on spaces of homogeneous type

- MathematicsRevista 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…

### A Hörmander-type spectral multiplier theorem for operators without heat kernel

- Mathematics
- 2003

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…

### Spectral multiplier theorems of H\"ormander type on Hardy and Lebesgue spaces

- 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…

### Calderón-Zygmund theory for non-integral operators and the $H^{\infty}$ functional calculus

- 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.…

### LP ESTIMATES FOR SCHRODINGER EVOLUTION EQUATIONS

- Mathematics
- 2010

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…

### Schrödinger Semigroups

- Mathematics
- 2007

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…

### Heat Kernel and Analysis on Manifolds

- Mathematics
- 2012

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…

### Calderón-Zygmund theory for non-integral operators and the H∞ functional calculus

- 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.…

### Sharp L^p estimates for Schrödinger groups

- Mathematics
- 2014

Consider a non-negative self-adjoint operator H in L2(Rd). We suppose that its heat operator e−tHe satisfies an off-diagonal algebraic decay estimate, for some exponents p0∈[0,2]. Then we prove sharp…

### Singular integral operators with non-smooth kernels on irregular domains

- Mathematics
- 1999

Let ? be a space of homogeneous type. The aims of this paper are as follows.
i) Assuming that T is a bounded linear operator on L2(?), we give a sufficient condition on the kernel of T such that T is…