Pseudodifferential p-adic vector fields and pseudodifferentiation of a composite p-adic function

TL;DR

This paper explores how p-adic pseudodifferential operators transform under automorphisms, deriving formulas for pseudodifferentiation of composite functions and introducing wavelet frames and vector field operators in p-adic spaces.

Contribution

It provides new transformation rules for p-adic pseudodifferential operators, including formulas for composite functions and the development of wavelet frames and vector field operators in multidimensional p-adic analysis.

Findings

Derived transformation formulas for pseudodifferential operators

Established pseudodifferentiation rule for composite p-adic functions

Introduced wavelet frames for automorphism groups in p-adic spaces

Abstract

We discuss transformation of p-adic pseudodifferential operators (in the one-dimensional and multidimensional cases) with respect to p-adic maps which correspond to automorphisms of the tree of balls in the corresponding p-adic spaces. In the dimension one we find a rule of transformation for pseudodifferential operators. In particular we find the formula of pseudodifferentiation of a composite function with respect to the Vladimirov p-adic fractional differentiation operator. We describe the frame of wavelets for the group of parabolic automorphisms of the tree of balls in the p-adic field. In many dimensions we introduce the group of mod p-affine transformations, the family of pseudodifferential operators corresponding to pseudodifferentiation along vector fields on the tree of balls in p-adic miltidimensional space and obtain a rule of transformation of the introduced pseudodifferential operators with respect to mod p-affine transformations.