Fourier积分算子Hardy空间上的粗糙伪微分算子II
Rough pseudodifferential operators on Hardy spaces for Fourier integral operators II
摘要：
我们得到了傅里叶积分算子的Hardy空间上具有粗糙符号的伪微分算子的改进的界.符号$a(x,η)$是在$x$变量中具有有限规则性的$C^{r}_{*}S^{m}_{1,δ}$类的元素。我们证明了相关的伪微分算子$a(x,D)$在傅里叶积分算子的Hardy空间上的Sobolev空间$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$和$\mathcal{H}^{t,p}_{FIO}(\mathbb{R}^{n})$之间的映射$\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$。我们的主要结果是：对于所有的$r>0$、$m=0$和$δ=1/2$，在$2$周围存在一个$p$的区间，使得$a(x,D)$有界地作用于$\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ .
英文摘要：
We obtain improved bounds for pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $a(x,η)$ are elements of $C^{r}_{*}S^{m}_{1,δ}$ classes that have limited regularity in the $x$ variable. We show that the associated pseudodifferential operator $a(x,D)$ maps between Sobolev spaces $\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ and $\mathcal{H}^{t,p}_{FIO}(\mathbb{R}^{n})$ over the Hardy space for Fourier integral operators $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$. Our main result is that for all $r>0$, $m=0$ and $δ=1/2$, there exists an interval of $p$ around $2$ such that $a(x,D)$ acts boundedly on $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$.
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