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

Abstract

Let L ₁ = −Δ + V be a Schr:dinger operator and let L ² = (−Δ)² + V ² be a Schrödinger type operator on ℝⁿ (n ⩾ 5), where V≠ 0 is a nonnegative potential belonging to certain reverse Hölder class B _s for s ⩾ n/2. The Hardy type space $H_{{L_2}}^1$ is defined in terms of the maximal function with respect to the semigroup $\left\{ {{e^{ - t{L_2}}}} \right\}$ and it is identical to the Hardy space $H_{{L_1}}^1$ established by Dziubański and Zienkiewicz. In this article, we prove the L ^p-boundedness of the commutator R _b = bRf - R(bf) generated by the Riesz transform $R = {\nabla ^2}L_2^{ - 1/2}$, where $b \in BM{O_\theta }(\varrho )$, which is larger than the space BMO(ℝⁿ). Moreover, we prove that R _b is bounded from the Hardy space $H_{\mathcal{L}_1 }^1$ into weak $L_{weak}^1 (\mathbb{R}^n )$.