Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, where $\alpha\delta-\beta\gamma\ne 0$. It is well known that an integral operator belongs to the Hilbert-Schmidt class $\mathfrak S_2$ if and only if its kernel is square-summable; therefore, $K\in\mathfrak S_2$ if and only if $L\in\mathfrak S_2$. Is the same true if we replace the class $\mathfrak S_2$ by $\mathfrak S_1$, or by $\mathfrak S_p$ with $p\ge 1$? Can such an equivalence be proved for other properties? (References to known results would be appreciated.)

This question was not answered at SE. I was not able to use hints suggested in the discussion there.

In particular, this contains another question as a special case: does the integral operator on $L^2(\mathbb R)$, whose kernel is the indicator of the rhombus $\{|x|+|y|<1\}$, belong to the trace class?

but this was based on an error in my thinkingand on 2nd thought I don't see how the comments at MSE can be used to answer the question $\endgroup$