论文标题
关于派象和变量的变化
On paracomposition and change of variables in Paradifferential operators
论文作者
论文摘要
在本文中，我们重新审视定义“ paracostosition”操作员所需的假设，这是与S. alinhac在[3]中首先引入的低规律性设置中经典背背操作的类似物。更确切地说，我们在两个方向上这样做。首先，我们删除差异假说。其次，我们给出全球Sobolev和Zygmund空间的估计。因此，我们完全概括了Bony的经典副层次定理，从而对Sobolev和Zygmund空间中的组成进行了清晰的估计。为了证明由范例分量运算符组成的新型操作效果符号符号属性，我们讨论了伪差异和范例分量型操作员的背包，然后成为傅立叶积分运算符。在此讨论中，我们表明，当且仅当它们被变化的变化时，且仅当它们被带来的差异时，那些被撤回获得的傅立叶积分运算符是pseudododiventen或paradiffertent的操作员。我们给出了范例分量运算符中变量变化的证明。 最终，我们研究了定义范例分量运算符的截止，并通过连续组成的稳定性。众所周知，在每种组合后，截止量变得更糟，我们给出了霍曼德（Hörmander）在[14]中提出的截止的略微精制版本，为此给出了对组成后截止的支持的最佳估计。
In this paper we revisit the hypothesis needed to define the "paracomposition" operator, an analogue to the classic pull-back operation in the low regularity setting, first introduced by S. Alinhac in [3]. More precisely we do so in two directions. First we drop the diffeomorphism hypothesis. Secondly we give estimates in global Sobolev and Zygmund spaces. Thus we fully generalize Bony's classic paralinearasition theorem giving sharp estimates for composition in Sobolev and Zygmund spaces. In order to prove that the new class of operations benefits of symbolic calculus properties when composed by a paradifferential operator, we discuss the pull-back of pseudodifferential and paradifferential operators which then become Fourier Integral Operators. In this discussion we show that those Fourier Integral Operators obtained by pull-back are pseudodifferential or paradifferential operators if and only if they are pulled-back by a diffeomorphism that is a change of variable. We give a proof of the change of variables in paradifferential operators. Finally we study the cutoff defining paradifferential operators and it's stability by successive composition. It is known that the cutoff becomes worse after each composition, we give a slightly refined version of the cutoffs proposed by Hörmander in [14] for which give an optimal estimate on the support of the cutoff after composition.
