Bilinear operators associated with generalized Schrödinger operators

  • Nan Hu
  • Yu LiuEmail author


Let \(\mathcal {L}=-\Delta + \mu \) be the generalized Schrödinger operator on \(\mathbb {R}^d\), \(d\ge 3 \), where \(\Delta \) is the Laplacian, \(\mu \) is a nonnegative Radon measure on \(\mathbb {R}^d\) and \({H}^1_{\mathcal {L}}(\mathbb {R}^d)\) is the Hardy type space associated to \(\mathcal {L}\). In this paper we establish an estimate for the bilinear operators \(T^{+}\) or \(T^{-}\) defined by
$$\begin{aligned} T^{\pm }(f,g)(x)=(T_{1}f)(x)(T_{2}g)(x)\pm (T_{2}f)(x)(T_{1}g)(x), \end{aligned}$$
where \(T_{1}\) and \(T_{2}\) are Calderón-Zygmund operators related to \(\mathcal {L}\). Specifically, we prove that either \(T^{+}\) or \(T^{-}\) is bounded from \({L}^p(\mathbb {R}^d)\times {L}^q(\mathbb {R}^d)\) to \( {H}^1_{\mathcal {L}}(\mathbb {R}^d)\) for \(1<p, q<\infty \) with \(1/p+1/q=1\).


Bilinear operators Riesz transform Schrödinger operators Hardy space 

Mathematics Subject Classification

42A20 42A30 35J10 



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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingPeople’s Republic of China

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