This question is motivated by another one, asked by Cameron Williams: Adjoint of an integral operator
Let us say that a Borel function $k:\mathbb R\times \mathbb R\to\mathbb C$ defines an operator on $L^2$ if the following holds for every $f\in L^2(\mathbb R)$:
(i) for almost every $y\in \mathbb R$, the function $x\mapsto k(x,y)f(x)$ is integrable on $\mathbb R$.
(ii) The function $T_kf$ (almost everywhere) defined by $$T_kf(y)=\int_{\mathbb R} k(x,y)f(x)\, dx$$ belongs to $L^2(\mathbb R)$.
Now here is the question: Is it true that if $k$ defines an operator on $L^2$, then so does $\vert k\vert$?
This question is purely "academic", i.e. it is very likely to have no "practical" interest. Usually, when one knows that a kernel $k$ defines an operator on $L^2$ in the above sense, this is because $k$ satisfies some condition (e.g. Schur's test) which is also satisfied by $\vert k\vert$. Still, I would like to know the answer just by curiosity. I have absolutely no intuition about it.
