Bilinear operators associated with Schrödinger operators
Volume 205 / 2011
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.
