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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.09013 |
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Table of Contents:
- 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 $1<r_{1},r_{2}<\infty$, the bilinear operators satisfy $$M^\sharp T_a(f_1,f_2)(x) \lesssim \mathcal{M}_{\vec{r}}(f_1,f_2)(x).$$ This extends the parameter range from the restrictive condition $0 \leq δ\leq \varrho < 1$ to the general setting $0 \leq \varrho \leq 1$, $0 \leq δ< 1$ with $δ> \varrho$ 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.