Uniqueness of dilation invariant norms

Let delta(a) be a nontrivial dilation. We show that every complete norm parallel to.parallel to on L-1(R-N) that makes delta(a) from (L-1(R-N); parallel to.parallel to) into itself continuous is equivalent to parallel to.parallel to(1.)delta(a) also determines the norm of both C-0(R-N) and L-p(R-N) with 1 < p < infinity in a weaker sense. Furthermore, we show that even all the dilations do not determine the norm on L-infinity(R-N).