An Hardy estimate for commutators of pseudo-differential operators

Abstract

Let $T$ be a pseudo-differential operator whose symbol belongs to the Hörmander class $S_{\rho ,\delta }^m$ with $0\le \delta <1,0<\rho \le 1,\delta \le \rho$ and $-(n+1)<m\le -(n+1)(1-\rho )$. In present paper, we prove that if $b$ is a locally integrable function satisfying $$\underset{\mathrm{balls}B\subset {\mathbb{R}}^n}{\sup }\frac{\log (e+1/|B|)}{(1+|B|{)}^{\theta }}\frac{1}{|B|}{\int }_B|f(x)-\frac{1}{|B|}{\int }_{B}f(y)dy|dx<\infty$$ 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 \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}.

Taiwanese J. Math. (to appear)