Weighted and unweighted regularity of bilinear pseudo-differential operators with symbols in general Hörmander classes

Abstract

This paper investigates the boundedness of bilinear pseudo-differential operators with symbols in the Hörmander class $BS_{\varrho,δ}^m(\mathbb{R}^n)$ in the previously unexplored regime $0 \leq \varrho < δ< 1$. We establish boundedness from $H^p(\mathbb{R}^n) \times H^q(\mathbb{R}^n)$ to $L^r(\mathbb{R}^n)$ (with $L^r$ replaced by $\mathrm{BMO}$ when $p=q=r=\infty$) under the probably optimal condition on the order $$m \leq m_\varrho(p,q) - \frac{n\max\{δ-\varrho,0\}}{\max\{r,2\}},$$ where $m_\varrho(p,q)$ is the critical order in the case $0\leqδ\leq\varrho<1.$ Furthermore, we develop refined pointwise estimates via sharp maximal functions, establishing that for $m \leq -n(1-\varrho)(\frac{1}{\min\{r_1,2\}}+ \frac{1}{\min\{r_2,2\}})$ with $1 \varrho$ permitted, and generalizes previous results of Park and Tomita to distinct exponent pairs. Consequently, we obtain weighted norm inequalities for bilinear pseudo-differential operators under multilinear $A_{\vec{p},(\vec{r},\infty)}$ weights.