Some estimates for commutators of Riesz transform associated with Schrödinger type operators

Liu, Yu ; Zhang, Jing ; Sheng, Jie-Lai ; Wang, Li-Juan

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 169-191.

Résumé

Let ${\mathcal{L}}_1=-\mathrm{\Delta }+V$ be a Schrödinger operator and let ${\mathcal{L}}_2=(-\mathrm{\Delta }{)}^2+V^2$ be a Schrödinger type operator on ${\mathbb{R}}^n$ $(n\ge 5)$, where $V\ne 0$ is a nonnegative potential belonging to certain reverse Hölder class $B_s$ for $s\ge n/2$. The Hardy type space $H_{{\mathcal{L}}_2}^1$ is defined in terms of the maximal function with respect to the semigroup $\{{\mathrm{e}}^{-t{\mathcal{L}}_2}\}$ and it is identical to the Hardy space $H_{{\mathcal{L}}_1}^1$ established by Dziubański and Zienkiewicz. In this article, we prove the $L^p$-boundedness of the commutator ${\mathcal{R}}_b=b\mathcal{R}f-\mathcal{R}(bf)$ generated by the Riesz transform $\mathcal{R}={\mathrm{\nabla }}^2{\mathcal{L}}_2^{-1/2}$, where $b\in {\mathrm{B}\mathrm{M}\mathrm{O}}_{\theta }(\rho )$, which is larger than the space $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$. Moreover, we prove that ${\mathcal{R}}_b$ is bounded from the Hardy space $H_{{\mathcal{L}}_2}^1({\mathbb{R}}^n)$ into weak $L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^1({\mathbb{R}}^n)$.

Détail

DOI : 10.1007/s10587-016-0248-z

Classification : 35J10, 42B20, 42B30, 42B35
Mots-clés : commutator; Hardy space; reverse Hölder inequality; Riesz transform; Schrödinger operator; Schrödinger type operator