AN HARDY ESTIMATE FOR COMMUTATORS OF PSEUDO-DIFFERENTIAL OPERATORS

Abstract

Let $T$ be a pseudo-differential operator whose symbol belongs to the Hörmander class $S^m_{\rho,\delta}$ with $0\leq \delta\lt 1, 0\lt \rho\leq 1, \delta \leq \rho$ and $-(n+1)\lt m \leq - (n+1)(1-\rho)$. In present paper, we prove that if $b$ is a locally integrable function satisfying $$\sup_{{\rm balls}\; B\subset \mathbb R^n} \frac{\log(e+ 1/|B|)}{(1+ |B|)^\theta} \frac{1}{|B|}\int_{B} \Big|f(x)- \frac{1}{|B|}\int_{B} f(y) dy\Big|dx \lt \infty$$ for some $\theta\in [0,\infty)$, then the commutator $[b,T]$ is bounded on the local Hardy space $h^1(\mathbb R^n)$ introduced by Goldberg [9].

As a consequence, when $\rho=1$ and $m=0$, we obtain an improvement of a recent result by Yang, Wang and Chen [21].