Linear change of variables on mixed-(quasi)norm Lebesgue spaces

Question

Assume $0<p\neq q\leq\infty$. I need a general counterexample that shows that, in general, if $f\in L^{p,q}(\mathbb{R}^{2n})$, where $L^{p,q}(\mathbb{R}^{2n})$ is the quasi-Banach space of measurable functions $g=g(x,y):\mathbb{R}^{2n}\to\mathbb{C}$ such that the quasi-norm $$ \Vert g\Vert_{p,q}:=\Vert y\mapsto \Vert g(\cdot,y)\Vert_p \Vert_q $$ is finite, then $f\circ L\notin L^{p,q}(\mathbb{R}^{2n})$. I know that the "converse" holds if $$L=\begin{pmatrix} A & B \\ C & D\end{pmatrix}\in GL(2n,\mathbb{R}),$$ where $C=0$ is the zero-matrix $n\times n$ and $A,B,C\in Mat(n,\mathbb{R})$, that is if $L$ has this form, then $f\circ L\in L^{p,q}$ for every $f\in L^{p,q}$. But what can we say if $C\neq0$?