Estimates for Pseudo‐Differential Operators of Class S0,0
Estimates for Pseudo‐Differential Operators of Class S0,0
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S0,0 类伪微分算子的估计
DOI：
10.1002/mana.19871330109
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发表时间：
1987
影响因子：
1
通讯作者：
Akihiko Miyachi
中科院分区：
文献类型：
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作者：
Akihiko Miyachi
(1.1)其中d表示傅里叶变换。函数a（z，. 5)称为算子a（X，D）的符号。一个符号a（x，5）被称为Sg类，其中m是一个红数，如果它满足所有多指标a和，8的不等式，其中（E）=（1+（512）1 '2。本文所考虑的符号是满足（1.2）的修正形式的符号。注意，如果a（z，6）满足（1.2），则对于类8（W）的所有函数f（SCHWartz的快速下降光滑函数类），（1.1）的右侧是明确定义的。3= 0，且对于某个m，我们用k（k，k '）表示满足（1.2）的所有符号a（z，E）的集合，其中（a（sk）和I/？(sk'（此符号与后续章节中使用的符号不同）。关于符号在Sco（k，k '）中的伪微分算子的有界性，下面的定理是基本的。(In定理，我们将简单地说，a（S，D）是从Y到2的有界算子，其中Y和2是Rs上的函数空间，如果a（S，D）可以扩展为从Y到2的有界算子。
(1.1) where d denotes the FOURIER transform. The function a (z,. 5) is called the symbol of the operator a (X, D). A symbol a (x, 5) is said to be of class Sg,, where m is a red number, if it satisfies the inequalities for all multi-indices a and, 8, where (E)=(1+(512) 1’2. The symbols considered in this paper are the ones which satisfy (1.2) in a modified form. Note that the righthand side of (1.1) is well-defined for all functions f of class 8 (W)(SCHWARTZ’S class of rapidly decreasing smooth functions) if a (z, 6) satisfies (1.2) for a=,! 3= 0 and for some m.For the moment, we shall denote by &’$(k, k’) the set of all symbols a (z, E) which satisfy (1.2) for (a ( sk and I/?(sk ‘(this notation is different from the one used in the subsequent sections). As to the boundedness of the pseudo-differential operators with symbols in Sco (k, k ‘), the following theorem is fundamental.(In the theorem, we shall simply say that a (5, D) is bounded from Y into 2, where Y and 2 are function spaces on Rs, if a (S, D) can be extended to a bounded operator from Y into 2.)