must specify free variable for consistent operator on L^2(R)?

Suppose I want to define a "dilation" operator D on L^2(R) such that
D x(t) = x(2t) ... (t is the free variable).

What about D x(u(t))?
Is it possible to define a *consistent* dilation operator D that
"dilates" with respect to the argument u(t) rather than the free
variable t?

That is, is it possible to define D in a consistent manner such that
D x(u(t)) = x(2u(t)) ... (rather than D x(u(t)) = x(u(2t)) )

In this case, D seems inconsistent because
x(t) = (3t+1)^2 ==> D x(t) = (6t+1)^2
y(t) = t^2 ==> D y(3t+1) = (6t+2)^2

which is inconsistent because
D (3t+1)^2 = Dx(t) = (6t+1)^2
not=
(6t+2)^2 = D y(3t+1) = D (3t+1)^2

So then my question is, when defining an operator on L^2(R), is it
always necessary to also specify the free variable (e.g. "t")? Because
without defining exactly what the free variable is, won't this lead to
inconsistent definitions of operators on L^(R)?

For example, when defining a "dilation" operator D on L^2(R), must I
also specify exactly what the free variable is?

Many thanks in advance,
Dan Greenhoe

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