Bilinear operators associated with Schrödinger operators

Volume 205 / 2011

Chin-Cheng Lin, Ying-Chieh Lin, Heping Liu, Yu Liu Studia Mathematica 205 (2011), 281-295 MSC: Primary 42A20, 42A30, 35J10. DOI: 10.4064/sm205-3-4

Abstract

Let $L=-\Delta+V$ be a Schrödinger operator in $\Bbb R^d$ and $H^1_L(\Bbb R^d)$ be the Hardy type space associated to $L$. We investigate the bilinear operators $T^+$ and $T^-$ defined by $$T^{\pm}(f,g)(x)=(T_1f)(x)(T_2g)(x)\pm(T_2f)(x)(T_1g)(x),$$ where $T_1$ and $T_2$ are Calderón–Zygmund operators related to $L$. Under some general conditions, we prove that either $T^+$ or $T^-$ is bounded from $L^p(\Bbb R^d)\times L^q(\Bbb R^d)$ to $H^1_L(\Bbb R^d)$ for $1< p,q< \infty$ with ${1}/{p}+{1}/{q}=1$. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails.

Authors

  • Chin-Cheng LinDepartment of Mathematics
    National Central University
    Chung-Li 320, Taiwan
    e-mail
  • Ying-Chieh LinDepartment of Mathematics
    National Central University
    Chung-Li 320, Taiwan
    e-mail
  • Heping LiuLMAM, School of Mathematical Sciences
    Peking University
    Beijing 100871, China
    e-mail
  • Yu LiuSchool of Mathematics and Physics
    University of Science and Technology Beijing
    Beijing 100083, China
    e-mail

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