Second Derivative of the Dirac Delta function

Question

I want to simplify:

$\int dp dp' f(p,p') \delta''(p-p')$

where f(p,p') is an unknown function.

How do I deal with the second derivative of the delta function?

Answer 1

You should integrate by parts twice ti shift the derivatives off the delta and on to the function f(p, p'). Doing this, and assuming the boundary terms cannot contribute (they should not saturate the delta function) will leave $$(-1)^2\int dp dp' f''(p, p') \delta(p-p'). $$

Can you finish from here?