关于仿复合与仿微分算子中的变量变换
On paracomposition and change of variables in Paradifferential operators
AI总结：
本文推广仿复合算子，去掉微分同胚假设，给出Sobolev和Zygmund空间的全局估计，并证明变量变换下仿微分算子的性质及截断的优化。
摘要：
本文重新审视了定义“仿复合”算子所需的假设，该算子是低正则性设定下经典拉回运算的类比，最初由S. Alinhac在文献[3]中引入。更确切地说，我们从两个方向进行探讨。首先，我们去掉微分同胚假设。其次，我们给出全局Sobolev和Zygmund空间中的估计。因此，我们完全推广了Bony经典的仿线性化定理，给出了Sobolev和Zygmund空间中复合的精确估计。为了证明当与仿微分算子复合时，新一类运算具有符号演算性质，我们讨论了伪微分算子和仿微分算子的拉回，这些算子随后成为傅里叶积分算子。在此讨论中，我们表明，通过拉回获得的那些傅里叶积分算子是伪微分或仿微分算子，当且仅当它们是由作为变量变换的微分同胚拉回的。我们给出了仿微分算子中变量变换的证明。最后，我们研究了定义仿微分算子的截断及其在连续复合下的稳定性。已知截断在每次复合后变差，我们给出了Hörmander在文献[14]中提出的截断的一个略微精炼版本，并对其给出了复合后截断支撑集的最优估计。
英文摘要：
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.
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