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