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 norm such field for $$s > \sigma _r = \text {max}\big (\frac{1}{r} -1,\,0\big )$$ which depends the constant k unlike case of Euclidean spaces. In $${\mathbb {R}}^n,$$ it is independent k, appears limiting $$s=0$$ $$s=\sigma _r.$$ fields, still open.