Singular integrals in quantum Euclidean spaces
Singular integrals in quantum Euclidean spaces
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DOI：
10.1090/memo/1334
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发表时间：
2017-05
影响因子：
1.9
通讯作者：
A. Gonz'alez-P'erez;M. Junge;Javier Parcet
中科院分区：
文献类型：
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作者：
A. Gonz'alez-P'erez;M. Junge;Javier Parcet
我们将在非交换几何的原型代数上建立奇异积分理论和伪微分学的核心：欧几里得空间和环面的量子形式。我们的结果超越了cones的旋转代数的伪微分演算，这要归功于在这些空间上的一种新形式的Calderón-Zygmund理论，它关键地包含了非卷积核。我们推导了期望秩中正则符号、奇异符号和禁止符号的L_p -有界性和Sobolev p -估计。在l2 l2水平上，对于奇异符号和禁止符号的Calderón-Vaillancourt定理和Bourdaud定理也被推广到量子环境。作为本方法的一个基本应用，我们证明了椭圆型偏微分方程解的L_p -正则性。
We shall establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes’ pseudodifferential calculus for rotation algebras, thanks to a new form of Calderón-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduce L p L_p -boundedness and Sobolev p p -estimates for regular, exotic and forbidden symbols in the expected ranks. In the L 2 L_2 level both Calderón-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we prove L p L_p -regularity of solutions for elliptic PDEs.