Dilation Operators in Besov Spaces over Local Fields

TL;DR

This paper studies dilation operators on Besov spaces over local fields, providing operator norm estimates that depend on a constant unlike Euclidean spaces, and explores localization properties of these spaces.

Contribution

It introduces new estimates for dilation operators on Besov spaces over local fields, highlighting differences from Euclidean cases and addressing localization properties.

Findings

Operator norm depends on a constant k over local fields.

In Euclidean spaces, the norm is independent of such constants.

Localization properties of Besov spaces over local fields are characterized.

Abstract

We consider a dilation operator on Besov spaces $(B^s_{r,t}(K))$ over local fields and estimate an operator norm on such a field for $s > \sigma_r = \text{max}\big(\frac{1}{r} -1,~0\big)$ which depends on the constant $k$ unlike the case of Euclidean spaces. In $\mathbb{R}^n$, it is independent of constant. A constant $k$ appears for liming case $s=0$ and $s=\sigma_r$. In case of local fields, the limig case is still open. Further we also estimate the localization property of Besov spaces over local fields.