Dilation Operators in Besov Spaces over Local Fields

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.