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 $B{S}_{\varrho ,\delta }^m({\mathbb{R}}^n)$ in the previously unexplored regime $0\le \varrho <\delta <1$. 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 $$m\le m_{\varrho }(p,q)-\frac{n\max \{\delta -\varrho ,0\}}{\max \{r,2\}},$$ where $m_{\varrho }(p,q)$ is the critical order in the case $0\le \delta \le \varrho <1.$ Furthermore, we develop refined pointwise estimates via sharp maximal functions, establishing that for $m\le -n(1-\varrho )(\frac{1}{\min \{r_1,2\}}+\frac{1}{\min \{r_2,2\}})$ with $1<r_1,r_2<\infty$, the bilinear operators satisfy $$M^{♯}{T}_a(f_1,f_2)(x)\lesssim {\mathcal{M}}_{\vec{r}}(f_1,f_2)(x).$$ This extends the parameter range from the restrictive condition $0\le \delta \le \varrho <1$ to the general setting $0\le \varrho \le 1$, $0\le \delta <1$ with $\delta >\varrho$ 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.