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On page 41 (on Green's functions) of "Quantum Field Theory and the Standard Model" by Matthew D. Schwartz there is an equation $$-\int d^4y[\Box_y\Pi(x,y)]h(y)=-\int d^4 y \Pi(x,y)\Box_y h(y) \tag{3.81}$$ that he says is derived using integration by parts. The same calculation also comes up again on page 81 (on Position-space Feynman rules).

I could not figure out that derivation, so to at least gain plausibility I tried using integration by parts on the two sides separately but in a grossly simplified form:

\begin{align*} \int f(y)g''(y)dy &= f(y)g'(y)| - \int g'(y)f'(y)dy\\ \int g(y)f''(y)dy &= g(y)f'(y)| - \int f'(y)g'(y)dy\quad\text{so}\\ \int f(y)g''(y)dy - \int g(y)f''(y)dy &= f(y)g'(y)| - g(y)f'(y)|\\ \end{align*}

The best would be to forget about plausibility and provide the exact details for his derivation, but it would be good to know whether the plausibility is on the right track, and just needs more detail on, say, the limits above and below that are omitted. Maybe that's where I am not going far enough.

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Your proof strategy is basically correct; the final ingredient is we usually assume functions vanish fast enough at infinity that boundary terms are zero (e.g. you use this in deriving the Euler-Lagrange equation), which simplifies each use of integration by parts.

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