Q1: Let $f \in \mathcal{L}^1_{loc}(\mathbb{R}^n)$ and $A$ be an invertible matrix. Is the function $x \mapsto f(Ax)$ also locally integrable?

A1: Yes!

A2: No, never!

A3: No, not necessarily. There are counterexamples.

Q2: Let $T \in \mathcal{D}^\prime(\mathbb{R}^n)$ and $A$ be an invertible matrix. What is the correct definition for the distribution $T \circ A$?

A1: $$ \langle T \circ A , \varphi \rangle = \langle T, \varphi \rangle$$

A2: $$ \langle T \circ A , \varphi \rangle = \langle T(x), \varphi(A x) \rangle$$

A3: $$ \langle T \circ A , \varphi \rangle = \left< T(x), \frac{1}{|\det(A)|}\varphi(A^{-1} x) \right>$$

A4: $$ \langle T \circ A , \varphi \rangle = \left< T(x), \frac{1}{|\det(A^{-1})|}\varphi(A x) \right>$$

Q3: Let $A$ be a matrix where $A^T A$ is the identity matrix. Which claim is correct for the delta distribution?

A1: $ \delta \circ A = \delta $

A2: $ \delta \circ A = 0 $

A3: $ \delta \circ A = \frac{1}{2} \delta $

A4: $ \delta \circ A = -\delta$