Hardy spaces, commutators of singular integral operators related to Schrödinger operators and applications
Résumé
Let $L= -\Delta+ V$ be a Schrödinger operator on $\mathbb R^d$, $d\geq 3$, where $V$ is a nonnegative function, $V\ne 0$, and belongs to the reverse Hölder class $RH_{d/2}$. The purpose of this paper is three-fold. First, we prove a version of the classical theorem of Jones and Journé on weak$^*$-convergence in $H^1_L(\mathbb R^d)$. Secondly, we give a bilinear decomposition for the product space $H^1_L(\mathbb R^d)\times BMO_L(\mathbb R^d)$. Finally, we study the commutators $[b,T]$ for $T$ belongs to a class $\mathcal K_L$ of sublinear operators containing almost all fundamental operators in harmonic analysis related to $L$. More precisely, when $T\in \mathcal K_L$, we prove that there exists a bounded subbilinear operator $\mathfrak R= \mathfrak R_T: H^1_L(\mathbb R^d)\times BMO(\mathbb R^d)\to L^1(\mathbb R^d)$ such that \begin{equation}\label{abstract 1} |T(\mathfrak S(f,b))|- \mathfrak R(f,b)\leq |[b,T](f)|\leq \mathfrak R(f,b) + |T(\mathfrak S(f,b))|, \end{equation} where $\mathfrak S$ is a bounded bilinear operator from $H^1_L(\mathbb R^d)\times BMO(\mathbb R^d)$ into $L^1(\mathbb R^d)$ which does not depend on $T$. In the particular case of the Riesz transforms $R_j= \partial_{x_j}L^{-1/2}$, $j=1,...,d$, and $b\in BMO(\mathbb R^d)$, we prove that the commutators $[b, R_j]$ are bounded on $H^1_L(\mathbb R^d)$ iff $b\in BMO^{\rm log}_L(\mathbb R^d)$-- a new space of $BMO$ type, which coincides with the space $LMO(\mathbb R^d)$ when $L=-\Delta +1$. Furthermore, $$\|b\|_{BMO_L^{\rm log}}\approx \|b\|_{BMO}+ \sum_{j=1}^d \|[b, R_j]\|_{H^1_L\to H^1_L}.$$ The subbilinear decomposition (\ref{abstract 1}) explains why almost all commutators of the fundamental operators are of weak type $(H^1_L,L^1)$, and when a commutator $[b,T]$ is of strong type $(H^1_L,L^1)$.
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