Dilation operators in Besov spaces over local fields

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 )= max(1/r - 1, 0) which depends on the constant k unlike the case of Euclidean spaces. In Rn, it is independent of constant k, the constant appears for limiting case s = 0 and s = sigma(r). In local fields, the limiting case is still open.