Question:
Let $f,g$ be piecewise continous square-integrable functions. Then, is the following limit integral well-defined (and finite)? $$ \lim_{\epsilon\to 0} \int dx dy ~ \frac{f(x) g(y)}{y-x + i\epsilon} < \infty. $$ If not, what would be a counter-example. If so, how can this be proven?
Thoughts and ideas:
Differentiable functions would work:
If $f$ and $g$ would be differentiable, existence of the above integral would be clear. I would write $1/(y-x + i\epsilon)$ as the derivative of a regular function and partially integrate. Then, the limit in $\epsilon$ can be taken without problems.
One integral is not enough:
I did not expect that such integrals exist because they do not exist in one variable. For example, let $s(x) = \begin{cases} 1 & x \in (0,1) \\ 0 & \text{otherwise}\end{cases}$. Then, using Sokhotsky's formula one can show that:
$$ \lim_{\epsilon\to 0} \int dx~ \frac{s(x)}{x + i\epsilon} = -i \pi s(0) + \lim_{\epsilon \to 0} \int_\epsilon^1 \frac{1}{x} = -\lim_{\epsilon \to 0} \ln(\epsilon) = \infty. $$
However, it seems that another integral further regularizes the simple pole as $\epsilon \to 0$. For example, when setting $f$ and $g$ equal to $s$, the integral in question can be computed (which I did using again Sokhotsky's formula).
Again, an integral in one variable would be unproblematic for differentiable functions with sufficient fall-off behavior. Possible problems are due to the restriction to piecewise continous functions.
Background:
In quantum field theories, so-called matrix elements of operators $A$ have distributional behavior. They have to be integrated with states from the Hilbert space $\mathcal H$ to be well-defined:
$$ \langle \psi| A | \phi \rangle = \int d^m p d^n q~ \underbrace{\psi(p_1, .., p_m) \phi(q_1, .., q_n)}_{\text{states } \in \mathcal H} ~ \underbrace{\langle p_1, .., p_m| A |q_1, .., q_n\rangle}_{\text{matrix element}} < \infty $$
The Hilbert space usually contains $L^2$ function of several variables (symmetrized or anti-symmetrized for bosons and fermions, respectively). It is possible to restrict to subspace (e.g., Schwartz functions) if necessary.
One particular type of poles are kinematic poles, which are of the form $\frac{1}{p_i - q_j + i0}$. I am asking whether it is necessary to restrict the Hilbert space to allow for such poles.