An Hardy estimate for commutators of pseudo-differential operators

Let $T$ be a pseudo-differential operator whose symbol belongs to the Hörmander class $S^m_{ρ,δ}$ with $0\leq δ<1, 0< ρ\leq 1, δ\leq ρ$ and $-(n+1)< m \leq - (n+1)(1-ρ)$. In present paper, we prove that if $b$ is a locally integrable function satisfying $\sup_{{\rm balls}\; B\subset \mathbb R^n} \frac{\log(e+ 1/|B|)}{(1+ |B|)^θ} \frac{1}{|B|}\int_{B} \Big|f(x)- \frac{1}{|B|}\int_{B} f(y) dy\Big|dx <\infty$ for some $θ\in [0,\infty)$, 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 $ρ=1$ and $m=0$, we obtain an improvement of a recent result by Yang, Wang and Chen \cite{YWC}.

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