An Hardy estimate for commutators of pseudo-differential operators

Let $T$ be a pseudo-differential operator whose symbol belongs to the H"ormander class $S_{\rho ,\delta }^m$ with $0\leq \delta<1, 0< \rho\leq 1, \delta \leq \rho$and$-(n+1)< m \leq - (n+1)(1-\rho)$. In present paper, we prove that if $b$ is a locally integrable function satisfying $$\underset{\mathrm{b}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}B\subset {\mathbb{R}}^n}{\sup }\frac{\log (e+1\mathrm{/}\mathrm{\mid }B\mathrm{\mid })}{(1+\mathrm{\mid }B\mathrm{\mid }{)}^{\theta }}\frac{1}{\mathrm{\mid }B\mathrm{\mid }}{\int }_B\mid f(x)-\frac{1}{\mathrm{\mid }B\mathrm{\mid }}{\int }_{B}f(y)dy\mid dx<\mathrm{\infty }$$ for some $\theta \in [0,\mathrm{\infty })$, then the commutator $[b,T]$ is bounded on the local Hardy space $h^1({\mathbb{R}}^n)$ introduced by Goldberg \cite{Go}. As a consequence, when $\rho =1$ and $m=0$, we obtain an improvement of a recent result by Yang, Wang and Chen \cite{YWC}.