Usually, a distribution $T$ applied to a test function $\varphi$ can be written as $ \langle T, \varphi \rangle$ instead of $T(\varphi)$. But, there is also this notation with quotes that I found in our notebook $$ \forall \varphi\in\mathcal{D}: \langle "T(ax)", \varphi \rangle = \frac{1}{|a|} \langle T, "\varphi(\frac{x}{a})" \rangle $$
I know that we can use $\varphi(\frac{\cdot}{a})$ instead, but we can't write $T(a \times \cdot)$, I assume.
So, is there any better way to write the definition above?
Edit: I am sorry if my question caused confusion, but it is not about the dilation operation, but rather about the "quote" notation itself.
For example, $$ \langle \delta, "\varphi(x+a)" \rangle $$ or, $$ \langle "T_{\exp(x)}", \varphi \rangle $$
Is there any general notation that can be used for these cases instead of double quotes?
I guess if you let $M_a$ be the operator $M_a\varphi(x)=\varphi(ax)$, then for any $f\in L^1_\mathrm{loc}$ we have $$ \langle M_af,\varphi\rangle=\frac{1}{\lvert a\rvert}\langle f,M_{a^{-1}}\varphi\rangle $$ by a change of variables, so it makes sense to define $\langle M_aT,\varphi\rangle:=\lvert a\rvert^{-1}\langle T,M_{a^{-1}}\varphi\rangle$, where of course $a\neq 0$.