Endpoint estimates for commutators of singular integrals related to Schrödinger operators

Abstract

Let $L=-\Delta +V$ be a Schrödinger operator on ${\mathbb{R}}^d$, $d\ge 3$, where $V$ is a nonnegative potential, $V\ne 0$, and belongs to the reverse Hölder class $R{H}_{d/2}$. In this paper, we study the commutators $[b,T]$ for $T$ in a class ${\mathcal{K}}_L$ of sublinear operators containing the 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 $\mathfrak{R}={\mathfrak{R}}_T:H_L^1({\mathbb{R}}^d)\times \mathrm{BMO}({\mathbb{R}}^d)\to L^1({\mathbb{R}}^d)$ such that

where $\mathfrak{S}$ is a bounded bilinear operator from $H_L^1({\mathbb{R}}^d)\times \mathrm{BMO}({\mathbb{R}}^d)$ into $L^1({\mathbb{R}}^d)$ which does not depend on $T$. The subbilinear decomposition $(\star )$ allows us to explain why commutators with the fundamental operators are of weak type $(H_L^1,L^1)$, and when a commutator $[b,T]$ is of strong type $(H_L^1,L^1)$.

Also, we discuss the $H_L^1$-estimates for commutators of the Riesz transforms associated with the Schrödinger operator $L$.