Pseudodifferential operators on Adele rings and wavelet bases
Let \(\mathbb Q_p\) be the field of \(p\)-adic numbers and \(\mathbb Z_p=\{x\in\mathbb Q_p: |x|_p\leq1\}\) be the subring of integers in \(\mathbb Q_p\). Suppose that there exists a prime \(P = P(x)\) such that \(x_p\in\mathbb Z_p\) for all \(p>P\). The authors define the adele ring \(\mathbb A\) as the set of infinite sequences \(x'= (x_2,\dots, x_p,\dots)\), where \(x_p\in \mathbb Q_p\) for \(p = 2,3,\dots\). They construct wavelet bases in \(L^2(\mathbb Q_p)\) for \(p\in V_\mathbb Q\) and in \(L^2(\mathbb A)\). Then they introduce Lizorkin spaces \(\Phi(\mathbb A)\) on adeles and wavelets, leading to Theorem~1. A~class of pseudodifferential operators is defined on \(\Phi(\mathbb A)\), leading to Theorem~2. A~fractional adelic operator is introduced and eigenfunctions of pseudodifferential operators are considered in Theorem~3.
scientific article; zbMATH DE number 6106461