An Hardy estimate for commutators of pseudo-differential operators

TL;DR

This paper establishes boundedness of commutators of certain pseudo-differential operators on local Hardy spaces under a specific integrability condition on the function b, extending previous results in harmonic analysis.

Contribution

The paper proves boundedness of commutators of pseudo-differential operators with symbols in Hörmander classes on local Hardy spaces, under a new integrability condition on b.

Findings

Boundedness of commutators on local Hardy space h^1.

Extension of previous results for specific symbol classes.

Improvement over recent related theorems.

Abstract

Let $T$ be a pseudo-differential operator whose symbol belongs to the H\"ormander class $S^m_{\rho,\delta}$ 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 $$\sup_{{\rm balls}\; B\subset \mathbb R^n} \frac{\log(e+ 1/|B|)}{(1+ |B|)^\theta} \frac{1}{|B|}\int_{B} \Big|f(x)- \frac{1}{|B|}\int_{B} f(y) dy\Big|dx <\infty$$ 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}.