Pseudodifferential \(p\)-adic vector fields and pseudodifferentiation of a composite \(p\)-adic function. (English) Zbl 1288.47048
The authors study the transformation of \(p\)-adic pseudodifferential operators under certain \(p\)-adic mappings corresponding to automorphisms of the tree of balls in \(p\)-adic spaces. An important tool is provided by \(p\)-adic complex-valued wavelets introduced by S. V. Kozyrev [Izv. Math. 66, No. 2, 367–376 (2002; Zbl 1016.42025)]; for further developments, see [S. Albeverio, A. Yu. Khrennikov and V. M.Shelkovich, Theory of \(p\)-adic distributions. Linear and nonlinear models. London Mathematical Society Lecture Note Series 370. Cambridge: Cambridge University Press (2010; Zbl 1198.46001)].
Since \(p\)-adic wavelets are eigenfunctions of \(p\)-adic pseudodifferential operators, the latter can be interpreted as local operators on the wavelet fiber bundle on the tree of balls \(\mathcal T (\mathbb Q_p)\). In this sense, such an operator can be called a \(p\)-adic vector field. For a natural group of mappings \(\mathbb Q_p\to \mathbb Q_p\), the orbit of its action on wavelets is shown to be a tight frame. A formula of commutation with Vladimirov’s fractional differentiation operator is found.
For the multi-dimensional case, the authors introduce a natural subgroup of the group of automorphisms of the tree of balls, and also find a transformation formula for appropriate pseudodifferential operators. In this case, too, a fiber bundle interpretation clarifies the geometric meaning of the constructions.
Since \(p\)-adic wavelets are eigenfunctions of \(p\)-adic pseudodifferential operators, the latter can be interpreted as local operators on the wavelet fiber bundle on the tree of balls \(\mathcal T (\mathbb Q_p)\). In this sense, such an operator can be called a \(p\)-adic vector field. For a natural group of mappings \(\mathbb Q_p\to \mathbb Q_p\), the orbit of its action on wavelets is shown to be a tight frame. A formula of commutation with Vladimirov’s fractional differentiation operator is found.
For the multi-dimensional case, the authors introduce a natural subgroup of the group of automorphisms of the tree of balls, and also find a transformation formula for appropriate pseudodifferential operators. In this case, too, a fiber bundle interpretation clarifies the geometric meaning of the constructions.
Reviewer: Anatoly N. Kochubei (Kyïv)
MSC:
| 47G30 | Pseudodifferential operators |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 26E30 | Non-Archimedean analysis |
| 05E18 | Group actions on combinatorial structures |
| 47S10 | Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory |
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