Difference equations and pseudo-differential operators on Z n
Difference equations and pseudo-differential operators on Z n
复制标题
Z n 上的差分方程和伪微分算子
DOI：
10.1016/j.jfa.2020.108473
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发表时间：
2020
影响因子：
1.7
通讯作者：
Botchway L
中科院分区：
文献类型：
--
作者：
Botchway L
本文研究了格Zn上的伪微分算子的微积分，我们称之为伪差分算子.这个演算的一个有趣的特点是，全球的频率空间（T n）是紧凑的，所以符号类的定义的行为相对于格变量。我们建立公式的组成，伴随，转置，和参数的椭圆算子。我们还给出了条件的W22，加权W22，和W2p有界的运营商和他们的紧凑性W2p。我们描述了一个链接到环面T n的环形量子化，并应用它给条件的Schatten类W22（Z n）。此外，我们讨论了格上的一类Fourier积分算子，并给出了它们有界的条件。结果被应用到格Z n上的差分方程的解的估计。此外，我们建立了Gårding和sharp Gårding不等式，并应用于格Z n上抛物方程的唯一可解性。
In this paper we develop the calculus of pseudo-differential operators on the lattice Z n, which we can call pseudo-difference operators. An interesting feature of this calculus is that the global frequency space (T n) is compact so the symbol classes are defined in terms of the behaviour with respect to the lattice variable. We establish formulae for composition, adjoint, transpose, and for parametrix for the elliptic operators. We also give conditions for the ℓ 2, weighted ℓ 2, and ℓ p boundedness of operators and for their compactness on ℓ p. We describe a link to the toroidal quantization on the torus T n, and apply it to give conditions for the membership in Schatten classes on ℓ 2 (Z n). Furthermore, we discuss a version of Fourier integral operators on the lattice and give conditions for their ℓ 2-boundedness. The results are applied to give estimates for solutions to difference equations on the lattice Z n. Moreover, we establish Gårding and sharp Gårding inequalities, with an application to the unique solvability of parabolic equations on the lattice Z n.