论文标题
关于伪差异操作员端点估计值的一些注释
Some notes on endpoint estimates for pseudo-differential operators
论文作者
论文摘要
我们研究伪差异操作员 \ begin {equation*} t_a f \ left（x \ right）= \ int _ {\ mathbb {r}^n} \ end {equation*} 其中符号$ a $在hörmander类$ s^{m} _ {ρ，1} $或更一般的hörmander类$ l^{\ infty} s^{\ infty} s^{m} s^{m}_ρ$带有$ m \ in \ mathbb {r} $和$ρ\ in [0,0,1] $。众所周知，$ t_a $在$ l^1（\ mathbb {r}^n）$上限制为$ m <n（ρ-1）$。在本文中，当$ M $等于关键指数$ n（ρ-1）$时，我们主要研究其有限属性。 对于任何$ 0 \ leqρ\ leq 1 $，我们构造一个符号$ a \ in s^{n（ρ-1）} _ {ρ，1} $，这样$ t_a $在$ l^1 $上没有结合，此外，它不是弱的$（1,1）$ρ= 0 $ρ= 0 $。另一方面，我们证明$ t_a $从$ h^1 $到$ l^1 $，如果$ 0 \ leqρ<1 $，并构造一个符号$ a \ in s^0_ {1,1} $，这样$ t_a $从$ h^1 $ to $ l^1 $中取下。 最后，作为补充，对于任何$ 1 <p <\ infty $，我们给出了一个示例$ a \ in s^{ - 1/p} _ {0,1} $，这样$ t_a $在$ l^p（\ mathbb {r}）上是不可接受的。
We study the pseudo-differential operator \begin{equation*} T_a f\left(x\right)=\int_{\mathbb{R}^n}e^{ix\cdotξ}a\left(x,ξ\right)\widehat{f}\left(ξ\right)\,\textrm{d}ξ, \end{equation*} where the symbol $a$ is in the Hörmander class $S^{m}_{ρ,1}$ or more generally in the rough Hörmander class $L^{\infty}S^{m}_ρ$ with $m\in\mathbb{R}$ and $ρ\in [0,1]$. It is known that $T_a$ is bounded on $L^1(\mathbb{R}^n)$ for $m<n(ρ-1)$. In this paper we mainly investigate its boundedness properties when $m$ is equal to the critical index $n(ρ-1)$. For any $0\leq ρ\leq 1$ we construct a symbol $a\in S^{n(ρ-1)}_{ρ,1}$ such that $T_a$ is unbounded on $L^1$ and furthermore it is not of weak type $(1,1)$ if $ρ=0$. On the other hand we prove that $T_a$ is bounded from $H^1$ to $L^1$ if $0\leq ρ<1$ and construct a symbol $a\in S^0_{1,1}$ such that $T_a$ is unbounded from $H^1$ to $L^1$. Finally, as a complement, for any $1<p<\infty$ we give an example $a\in S^{-1/p}_{0,1}$ such that $T_a$ is unbounded on $L^p(\mathbb{R})$.
