L p estimates for Schrödinger operators with certain potentials

Zhongwei Shen

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 2, page 513-546
  • ISSN: 0373-0956

Abstract

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We consider the Schrödinger operators - Δ + V ( x ) in ℝ n where the nonnegative potential V ( x ) belongs to the reverse Hölder class B q for some q ≥ n / 2 . We obtain the optimal L p estimates for the operators ( - Δ + V ) i γ , ∇ 2 ( - Δ + V ) - 1 , ∇ ( - Δ + V ) - 1 / 2 and ∇ ( - Δ + V ) - 1 where γ ∈ ℝ . In particular we show that ( - Δ + V ) i γ is a Calderón-Zygmund operator if V ∈ B n / 2 and ∇ ( - Δ + V ) - 1 / 2 , ∇ ( - Δ + V ) - 1 ∇ are Calderón-Zygmund operators if V ∈ B n .

How to cite

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Shen, Zhongwei. "$L^p$ estimates for Schrödinger operators with certain potentials." Annales de l'institut Fourier 45.2 (1995): 513-546. <http://eudml.org/doc/75127>.

@article{Shen1995,
abstract = {We consider the Schrödinger operators $-\Delta +V(x)$ in $\{\Bbb R\}^n$ where the nonnegative potential $V(x)$ belongs to the reverse Hölder class $B_q$ for some $q\ge n/2$. We obtain the optimal $L^p$ estimates for the operators $(- \Delta +V)^\{i\gamma \},\nabla ^2 (- \Delta +V)^\{-1\}, \nabla (- \Delta +V)^\{-1/2\}$ and $\nabla (- \Delta +V)^\{-1\}$ where $\gamma \in \{\Bbb R\}$. In particular we show that $(- \Delta +V)^\{i\gamma \}$ is a Calderón-Zygmund operator if $V\in B_\{n/2\}$ and $\nabla (- \Delta +V)^\{-1/2\}, \nabla (- \Delta +V)^\{-1\}\nabla $ are Calderón-Zygmund operators if $V\in B_n$.},
author = {Shen, Zhongwei},
journal = {Annales de l'institut Fourier},
keywords = { estimates; reverse Hölder class},
language = {eng},
number = {2},
pages = {513-546},
publisher = {Association des Annales de l'Institut Fourier},
title = {$L^p$ estimates for Schrödinger operators with certain potentials},
url = {http://eudml.org/doc/75127},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Shen, Zhongwei
TI - $L^p$ estimates for Schrödinger operators with certain potentials
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 2
SP - 513
EP - 546
AB - We consider the Schrödinger operators $-\Delta +V(x)$ in ${\Bbb R}^n$ where the nonnegative potential $V(x)$ belongs to the reverse Hölder class $B_q$ for some $q\ge n/2$. We obtain the optimal $L^p$ estimates for the operators $(- \Delta +V)^{i\gamma },\nabla ^2 (- \Delta +V)^{-1}, \nabla (- \Delta +V)^{-1/2}$ and $\nabla (- \Delta +V)^{-1}$ where $\gamma \in {\Bbb R}$. In particular we show that $(- \Delta +V)^{i\gamma }$ is a Calderón-Zygmund operator if $V\in B_{n/2}$ and $\nabla (- \Delta +V)^{-1/2}, \nabla (- \Delta +V)^{-1}\nabla $ are Calderón-Zygmund operators if $V\in B_n$.
LA - eng
KW - estimates; reverse Hölder class
UR - http://eudml.org/doc/75127
ER -

References

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Citations in EuDML Documents

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  1. Yu Liu, Jing Zhang, Jie-Lai Sheng, Li-Juan Wang, Some estimates for commutators of Riesz transform associated with Schrödinger type operators
  2. Zhongwei Shen, The magnetic Schrödinger operator and reverse Hölder class
  3. Ali Akbulut, Vagif Guliyev, Rza Mustafayev, On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces
  4. Kazuhiro Kurata, Satoko Sugano, Fundamental solution, eigenvalue asymptotics and eigenfunctions of degenerate elliptic operators with positive potentials
  5. Nguyen Ngoc Trong, Le Xuan Truong, Generalized Morrey spaces associated to Schrödinger operators and applications
  6. Jacek Dziubański, A note on Schrödinger operators with polynomial potentials
  7. Pascal Auscher, Besma Ben Ali, Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials

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