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Guangqing Wang

Publications and source records attributed to Guangqing Wang.

5 recordsLinked to original sources
arXiv2026

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

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.

math.AP↗
arXiv2025

Notes on Regularity of Fourier integral operators with symbol in $S^{m}_{0,δ}$

Let $T_{a,φ}$ be a Fourier integral operator defined with $a\in S^{m}_{0,δ}(0\leqδ<1)$ and $φ\in Φ^{2}$ satisfying the strong non-degenerate condition. We demonstrate that when the order satisfies $$m\leq-\frac{n}{2}-\frac{n}{p}δ+\frac{n}{p},$$ the operator $T_{a,φ}$ becomes bounded on $L^{p}(\mathbb{R}^n)$ for $2< p<\infty$ and maps $L^{\infty}(\mathbb{R}^n)$ to $BMO(\mathbb{R}^n)$ when $p=\infty$. Furthermore, the derived bound on $m$ is sharp for $L^{p}$ estimates in the case $δ=0$, and for $(L^{\infty},BMO)$ when $0\leqδ<1$.

math.CA↗
arXiv2025

Weighted weak-type (1, 1) inequalities for pseudo-differential operators with symbol in $S^{m}_{0,δ}$

Let $T_a$ be a pseudo-differential operator defined by exotic symbol $a$ in Hörmander class $S^m_{0,δ}$ with $m \in \mathbb{R} $ and $0 \leq δ\leq 1 $. It is well-known that the weak type (1,1) behavior of $T_a $ is not fully understood when the index $m $ is equal to the possibly optimal value $-\frac{n}{2} - \frac{n}{2} δ$ for $0 \leq δ< 1 $, and that $T_a $ is not of weak type (1,1) when $m = -n$ and $δ= 1 $. In this note, we prove that $T_a $ is of weighted weak type (1,1) if $a \in S^{-n}_{0, δ}$ with $0 \leq δ< 1 $. Additionally, we show that the dual operator $T_a^* $ is of weighted weak type (1,1) if $a \in L^\infty S^{-n}_0 $. We also identify $m = -n$ as a critical index for these weak type estimates. As applications, we derive weighted weak type (1,1) estimates for certain classes of Fourier integral operators.

math.AP↗
arXiv2022

Sharp function and weighted $L^{p}$ estimates for pseudo-differential operators with symbols in general Hörmander classes

The purpose of this paper is to prove pointwise inequalities and to establish the boundedness on weighted $L^{p}$ spaces for pseudo-differential operators $T_{a}$ defined by the symbol $a\in S^{m}_{\varrho,δ}$ with $0\leq\varrho\leq1,$ $0\leqδ<1$. Firstly, we prove that if $m\leq-n(1-\varrho)/2$, then $$(T_{a}u)^{\sharp}(x)\lesssim M(|u|^{2})^{1/2}(x)$$ for all $x\in\mathbb{R}^{n}$ and all Schwartz function $u$. Secondly, it is shown that if $1\leq r\leq2$ and $m\leq-\frac{n}{r}(1-\varrho)$, then for any $ω$ belongs to the class of Muckenhoupt weights $A_{p/r}$ with $r<p<\infty$, these operators are bounded on $L^{p}_ω$. Moreover, these results are sharp on the bound of $m$.

math.AP↗