Dilation Operators in Besov Spaces over Local Fields

Abstract

We consider a dilation operator on Besov spaces $(B_{r,t}^s(K))$ over local fields and estimate an operator norm on such a field for $s>{\sigma }_r=\text{max}(\frac{1}{r}-1,0)$ 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.